Noncommutative spaces and schemes.

Alexander L. Rosenberg

Introduction.

1. Spaces and algebras. The correspondence between spaces and commutative algebras is a fundamental fact of functional analysis discovered by I. M. Gelfand in the late thirties. A. Connes extended this correspondence to the noncommutative setting identifying ’noncommutative spaces’ with noncommutative C^{∗}-algebras. This lead to the creation of noncommutative differential geometry [C1], [C2].

The category of affine spaces in noncommutative algebraic geometry was defined by V. Drinfeld [Dr] (who calls them ’quantum spaces’) in a similar way, as the category dual to the category of algebras, forcing to the noncommutative case the duality [algebras ↔ affine spaces] of commutative algebraic geometry.

2. Spaces and categories. Noncommutative projective spaces were introduced by Yu.I. Manin [M1] via a formal extension of the Serre’s description of the category of quasi-coherent sheaves on a projective space [S]: the category of quasi-coherent sheaves on the projective spectrum of an associative graded algebra R is the quotient category of the category of graded R-modules by the subcategory of locally finite ones (this approach was futher developed in [V1], [V2], [A2], [AZ], [OW], and in a number of other works). Thus a noncommutative projective space X over k is represented by a category, Qcoh_{X}, regarded as the category of quasi-coherent sheaves on X, together with a (global section) functor Qcoh_{X} ⟶ k − mod.

3. Affine schemes over a category. This viewpoint is well adapted to the affine case: for any associative ring R, the category Qcoh_{Spec(R)} is identified with the category R−mod of left R-modules, and to any morphism Spec(R) ⟶ Spec(S), there corresponds a (direct image) functor Qcoh_{Spec(S)} ⟶ Qcoh_{Spec(R)}. Here we identify the category Aff of affine noncommutative schemes with the category Rings^{op} dual to the category of rings, and denote by Spec(R) the object of Aff corresponding to the ring R.

One can characterise direct image functors of affine morphisms in abstract nonsense terms obtaining this way a notion of an affine scheme over an arbitrary category C (cf. [R3], or Section 2 of this work). If C is the category of left modules over a ring R, we recover affine noncommutative schemes over Spec(R).

4. Noncommutative algebras and non-symmetric monoidal categories. Local objects of commutative algebraic geometry over Spec(k) are commutative algebras in the symmetric monoidal category k − mod^{∼} = (k − mod, ⊗_{k}, k) of left k-modules. Local objects of the super-geometry are commutative algebras in the symmetric monoidal category of ℤ/2ℤ-graded k-modules. Affine noncommutative schemes over Spec(R) are defined by ring morphisms R ⟶ S. These are naturally identified with algebras in the monoidal category R − bimod^{∼} = (R − bimod, ⊗_{R}, R) of R-bimodules. Note that if the ring R is noncommutative, the category R − bimod^{∼} is not symmetric (or braided) any more, so that the notion of a commutative algebra in R − bimod^{∼} is not defined.

5. Affine schemes in a monoidal category. Quasi-coherent modules. For an arbitrary monoidal category A^{∼}, we define the category Aff_{A^{∼}} of affine schemes in A^{∼} as the category dual to the category of algebras in A^{∼}. A link between the formal notion of an affine scheme in a monoidal category and a more geometric notion of an affine scheme over a category C is given by fixing an action, ℭ = (A^{∼}, C, Φ : A ⟶ EndC), of the monoidal category on the category C. This action allows to define for any algebra S in the monoidal category A^{∼}, the category S − mod_{C} of S-modules in C which is called the category of quasi-coherent modules on Spec(S)/ℭ, and the forgetful functor S −mod_{C} ⟶ C regarded as a direct image functor of the morphism of Spec(S) to the base.

A standard noncommutative example is the monoidal category R − bimod^{∼} = (R − bimod, ⊗_{R}, R) of bimodules over an associative ring R acting (by tensoring over R) on the category R − mod of left R-modules. For any algebra S in the monoidal category R − bimod^{∼} (that is a ring morphism R ⟶ S), the category of quasi-coherent modules on Spec(S)/ℭ is naturally equivalent to the category S − mod of left S-modules.

A non-trivial example is given by the monoidal category A^{∼} = S − V ec_{k}^{∼} of S-spaces objects of which are families of representations of all symmetric groups, S_{n}, n ≥ 1, in vector spaces over a field k, and the tensor product is the so called plethysm product. Algebras in the monoidal category S − V ec_{k}^{∼} are called k-linear operads. The monoidal category of S-spaces acts canonically on the category C = V ec_{k} of k-vector spaces. For each k-linear operad P, the corresponding category of quasi-coherent sheaves on Spec(P) is called the category of P-algebras.

6. Noncommutative spaces and schemes. Commutative schemes, or more general ”varieties” (like algebraic spaces of Artin and Moishezon) are ”spaces” obtained by glueing affine schemes. The data required to give a sense to such definition consists of (a) a category Spaces, (b) a functor from the category Aff = Rings^{op} to the category Spaces, (c) a topology on Aff. Two categories of spaces are used: the category of locally ringed spaces and the category of sheaves of sets on the site (Aff, 𝔗_{fl}), where 𝔗_{fl} is a flat topology. Schemes are spaces (in any of the two senses) obtained by glueing affine schemes for the Zariski topology. The algebraic spaces of Artin and Moishenson are sheaves of sets obtained by glueing affine schemes for the étale topology.

The known examples of noncommutative spaces appear either as categories (of quasicoherent sheaves) over a base category, like the Proj of a graded noncommutative ring mentioned in 2 above, or the flag variety of a quantized enveloping algebra (see [R3], [LR2]), or as sheaves of sets on the category of noncommutative affine schemes, like projective spaces and Grassmannians introduced in [KR]. Accordingly, we introduce the notions of a scheme and that of a locally affine space (noncommutative projective spaces of [KR] are locally affine, but not schemes or algebraic spaces) for appropriate choices of the category Spaces and a topology on the category of noncommutative affine schemes.

7. Virtually noncommutative geometries. The main purpose of this paper is to introduce noncommutative affine schemes in an appropriate way, define general ’locally affine spaces’, in particular schemes, as ’spaces’ glued from affine schemes, and give some important examples of noncommutative schemes. This occupies the most of the first six sections of the paper. The geometry developed in these sections is really noncommutative: commutative schemes enter into the picture, but the basic categorical operations, like taking direct products, send commutative schemes into noncommutative ones. For instance, the product of two copies of a line, Specℂ[x], is the noncommutative affine scheme corresponding to the free algebra ℂ⟨x, y⟩. We recover back the commutative geometry, actually a whole family of commutative geometries depending on the choice of a symmetry, β, of the base monoidal category (see above), using the ’abelianization functor’. This functor assigns to a noncommutative scheme X a β-commutative scheme, X_{β}, and a closed immersion X_{β} ⟶ X. In the case of the base (k − mod, k − mod^{∼}) and the standard symmetry β : x ⊗ y ⟼ y ⊗ x, the abelianization functor maps an affine noncommutative scheme Spec(R) to the commutative scheme Spec(R_{β}), where R_{β} is the quotient of R by its commutant.

Taking infinitesimal neighborhoods of X_{β} in X defines ’nilpotezation functors’ mapping noncommutative schemes to schemes which are ’virtually noncommutative’, i.e. they are closed subschemes of formal noncommutative neighborhoods of commutative schemes. As a special case of this construction, we recover the ’non-commutative geometry based on commutator expansions’ introduced recently by Kapranov [K].

The paper is organized as follows.

In Section 1 we recall ’geometric language’ in which spaces are categories and morphisms are represented by (equivalence classes of) right exact functors called inverse image functors. This language appeared in [R], Ch.VII, and was used in [R3]. We introduce also the notion of a quasi-coherent presheaf.

In Section 2 we define affine morphisms and affine schemes over a category and reproduce analogs of main basic facts on conventional affine schemes in the new, noncommutative, context. In particular, we show that the category of affine schemes over a category C is dual to the category of so called continuous monads on the category C. If C is the category of R-modules, then the category of continuous monads on C is naturally equivalent to the category of rings over the ring R; in particular, if R = ℤ, it is equivalent to the category of rings.

In Section 3 we define schemes in monoidal categories and the categories of quasicoherent sheaves related to actions of monoidal categories.

In Sections 4 we develop an abstract version of a formalism of glueing new ’locally affine spaces’ starting from what we call a quasi-topology. In Section 5 we apply the formalism of Section 4 to introduce schemes and locally affine spaces over a given base.

Section 6 is dedicated to noncommutative versions of well known and important schemes: vector fibers, the group schemes GL_{V}, Grassmannians, flag varieties. Similarly to their commutative prototypes, they are defined as contravariant functors on the category of affine schemes. We show that these functors are representable.

It is worth to mention that most of interesting and natural examples of noncommutative spaces are not schemes. For instance, the Grassmannian presented in this work is a subspace of the natural Grassmannian introduced in [KR]. The latter is a locally affine space, where ’locally’ is understood in the sense of a flat topology.

In Section 7 we introduce abelianization and nilpotezation functors corresponding to a choice of a symmetry in the base monoidal category.

One of the purposes of this paper is to give a background to a series of works on noncommutative algebraic geometry, some of them have already appeared (like [LR1,2,3], [KR]), others are in progress. Some notions and constructions in this paper are given in a bigger generality than it is strictly necessary for examples presented in the text. This generality pays back in the continuations, where we study other examples and general properties of noncommutative spaces (which are not schemes), formal noncommutative geometry, differential calculus (including differential operators and D-modules), smooth non-commutative manifolds, and some applications of noncommutative geometry to mathematical physics and representation theory.

The paper was strongly influenced by conversations with I.M. Gelfand, A. Connes, and particularly with M. Kontsevich. Thus, the starting point of the work was an attempt to find a geometric meaning of quasi-determinants and (noncommutative) quasi-plückerian coordinates introduced by Gelfand and Retakh [GR]. This lead to the defining noncommutative Grassmannians and flag varieties. The project we are working on with M. Kontsevich dictated the degree of generality of the approach. Talking with A. Connes and attending his lectures have also influenced the setting of this work. I would like to thank D. Orlov for pointing out an error and for many useful discussions.

A considerable part of this work (actually its first version) was made during my staying at IHES. The present version was written at MPIM in Bonn. I would like to thank IHES and MPIM for the hospitality and excellent working conditions.

1. Categories as noncommutative ’spaces’.

Categories here are thought as categories of quasi-coherent or coherent sheaves on ’spaces’ and are identified with the ’spaces’. The equivalence classes of some functors represent morphisms.

1.0. Preliminaries on fibered categories. Fix a category B. Recall that a pseudo-functor from B to Cat is the following data:

(a) A map X ⟼ F_{X} from ObB to ObCat.

(b) A map which assigns to any morphism f : X ⟶ Y of B, a functor f^{∗} : F_{X} ⟶ F_{Y}.

(c) A map which assigns to any pair of composable morphisms X ⟶^{f} Y ⟶^{g} Z, a functor morphism c_{g,f} : f^{∗}g^{∗} ⟶ (gf)^{∗}.

This data satisfies the following conditions:

(i) id_{X}^{∗} = id_{F_{X}}.

(ii) c_{f,g} is identical if f or g is identical.

(iii) Given three composable morphisms X ⟶^{f} Y ⟶^{g} Z ⟶^{h} W, the diagram

f^{∗}(hg)^{∗} ⟶^{c_{hg,f}} (hgf)^{∗}
f^{∗}c_{h,g} ↑  ↑ c_{h,gf}
f^{∗}g^{∗}h^{∗} ⟶^{c_{f,g}h^{∗}} (gf)^{∗}h^{∗}

is commutative.

To each pseudo-functor B ⟶ Cat, there corresponds a prefibered category F over B defined as follows. Objects of F are pairs (X, M), where X ∈ ObB, M ∈ ObF_{X}. Morphisms from (X, M) to (Y, L) are pairs (f, ξ), where f is a morphism from X to Y and ξ is a morhism f^{∗}(L) ⟶ M. The composition of (f, ξ) : (X, M) ⟶ (Y, L) and (g, ν) : (Y, L) ⟶ (Z, N) is defined by (g, ν) ∘ (f, ξ) := (gf, ν ∘ g^{∗}ξ ∘ c_{f,g}(N) : (gf)^{∗}(N) ⟶ M).

We have a canonical fiber functor π : F ⟶ B, (X, M) ⟼ X, (f, ξ) ⟼ f. This together with morphisms {c_{g,f} : f^{∗}g^{∗} ⟶ (gf)^{∗}} defines a catégorie clivée which is, therefore, a prefibered category. The prefibered category π : F ⟶ B is fibered iff all morphisms c_{f,g} are isomorphisms.

1.0.1. Fibered category of quasi-coherent morphisms. Fix a fibered category π : F ⟶ B. Denote by 𝔔𝔠𝔬𝔥F the subcategory of F formed by all morphisms (f, ξ) such that ξ is an isomorphism. The composition of the embedding 𝔔𝔠𝔬𝔥F ⟶ F and the projection π is a fibered category which will be called the fibered category of quasi-coherent morphisms of F ⟶ B.

1.0.2. Presheaves and quasi-coherent presheaves. A presheaf in a fibered category F ⟶^{π} B (or an F-presheaf) is any section of F ⟶^{π} B, i.e. any functor G : B ⟶ F such that π_{F} ∘ G = Id_{B}. The category P reF of F-presheaves is the full subcategory of the category of functors B ⟶ F generated by F-presheaves.

We call an F-presheaf quasi-coherent if it takes values in the subcategory 𝔔𝔠𝔬𝔥F. The full subcategory of P reF generated by quasi-coherent presheaves will be denoted by P QcohF.

1.0.3. Lemma. Suppose the base B has a final object, X. Then the category P QcohF is naturally equivalent to the fiber F_{X} of F over X.

Proof. For any U ∈ ObB, denote by i_{U} the unique morphism U ⟶ X. An equivalence F_{X} ⟶ QcohF assigns to any object M of F_{T (X)} the presheaf M^{∼} : B ⟶ ModF which sends any object U of B to the pair (U, i_{U}^{∗}(M)) and any morphism g : U ⟶ V into (g, c_{i_{V} ,g}). A quasi-inverse functor assigns to any quasi-coherent presheaf M^{∼} its value on X.

1.0.4. Base change. Fix a fibered category F with a base B. Any functor T : T ⟶ B defines the induced fibered category F^{T} with the base T, usually denoted by F ×_{B} T . The fiber of F^{T} over U ∈ ObT is F_{T (U)}, the inverse image of an arrow f is (T f)^{∗} and the cocycle (gf)^{∗} ⟶ f^{∗}g^{∗} is c_{T f,T g}.

Let G : B ⟶ S be a functor. For any X ∈ ObS, we have the category G/X and the natural functor T : G/X ⟶ B. We denote the fibered category F ×_{B} G/X by F^{G/X}, or by F^{X}. The category of (resp. quasi-coherent) presheaves in F^{X} will be denoted by P re_{X}F (resp. P Qoh_{X}F). Its objects will be called (quasi-coherent) presheaves on X. Since the construction is functorial in X, it gives a rise to a fibered category over S having P Qcoh_{X}F as a fiber at an object X.

1.0.5. Continuous morphisms. Fix a fibered category F with a base B. We call a morphism f : X ⟶ Y of the base continuous if it has a direct image functor. (Dually, we call a morphism of a base of a cofibered category continuous if it has an inverse image functor.) The composition of continuous morphisms is continuous, and all isomorphisms of the category B are continuous. Thus continuous morphisms define a subcategory B_{c} of the category B, hence a bifibered subcategory π_{c} : F_{c} ⟶ B_{c}.

1.0.6. Monads and modules over monads. Recall that a monad in a category C is a pair (F, μ), where F is a functor C ⟶ C and μ is a functor morphism F^{2} ⟶ F having the following properties: μ∘F μ = μ∘μF and there exists a morphism η : Id_{C} ⟶ F such that μ ∘ F η = id_{F} = μ ∘ ηF. The morphism η is uniquely defined and is called the unit of the monad (F, μ). Let 𝔽 = (F, μ) and 𝔾 = (G, ν) be monads in C with units resp. η : Id_{C} ⟶ F and δ : Id_{C} ⟶ G. A morphism from 𝔽 to 𝔾 is a functor moprhism ϕ : F ⟶ G such that ϕ ∘ η = δ and ν ∘ ϕ ⊙ ϕ = ϕ ∘ μ. Here ϕ ⊙ ϕ := Gϕ ∘ ϕF : F^{2} ⟶ G^{2}. The composition of morphisms is defined in an obvious way. We denote the category of monads in C by 𝔐𝔬𝔫C. The category 𝔐𝔬𝔫C has a canonical initial object – the monad 𝕀_{C} = (Id_{C}, id).

Fix a monad 𝔽 = (F, μ) in C with the unit η. An 𝔽-module is a pair (M, m), where M ∈ ObC and m is a morphism F (M) ⟶ M such that m ∘ η(M) = id_{M} and m ∘ F m = m ∘ μ(M). A module morphism from (M, m) to (M′, m′) is a morphism f : M ⟶ M′ such that m′ ∘ F f = f ∘ m. The composition is induced by the composition in C. The category of 𝔽-modules will be defined by 𝔽 − mod.

Thus modules over monads on a given category C provide a natural example of a cofibered category 𝔐𝔬𝔡C: its base is the category of monads 𝔐𝔬𝔫C on C and the fiber at a monad 𝔾 is the category 𝔾 − mod of 𝔾-modules. Note that 𝔐𝔬𝔡C is not a fibered category in general, i.e. the pull-back functor corresponding to a monad morphism does not necessarily have a left adjoint.

Recall that a pair of arrows g_{1}, g_{2} : X ⇉ Y is called reflexive if there exists a morphism h : Y ⟶ X such that g_{1} ∘ h = id_{Y} = g_{2} ∘ h.

1.0.6.1. Lemma. Suppose the category C has cokernels of reflexive pairs. Let 𝔽 = (F, μ) be a monad in C such that the functor F preserves cokernels of all reflexive pairs of arrows. Then for any monad morphism ϕ : 𝔾 ⟶ 𝔽 the corresponding ’pull-back’ functor ϕ_{♯} : 𝔽 − mod ⟶ 𝔾 − mod is a direct image functor of a morphism (𝔾 − mod ⟶ C) ⟶ (𝔽 − mod ⟶ C).

Proof. For any 𝔾-module M = (M, m), we set 𝔽 ⊗_{𝔾} (M) := cok(F m, μ ∘ F ϕ(M)). Since F preserves cokernels of reflexive pairs of arrows, there is a unique 𝔽-module structure m′ on M′ = 𝔽 ⊗_{𝔾} (M) such that (F (M), μ) ⟶ (M′, m′) is an 𝔽-module moprhism.

1.0.7. Continuous morphisms and monads. Fix a fibered category F with a base B. For any object S of the base B, denote by (B/S)_{c} the full subcategory of the category B/S objects of which are pairs (X, f), where f is a continuous morphism X ⟶ S. Denote by F_{c}^{S} the fibered category with the base (B/S)_{c} induced by the natural functor (B/S)_{c} ⟶ B. To each object (X, f) of the category (B/S)_{c} there corresponds a monad (𝔾_{f}, μ_{f}) = (f_{∗}f^{∗}, f_{∗}ϵ_{f}f^{∗}) on the category S.

1.0.7.1. Proposition. The map f ⟼ (𝔾_{f}, μ_{f}) := (f_{∗}f^{∗}, f_{∗}ϵ_{f}f^{∗}) extends naturally to a morphism 𝔏^{∗} : F_{c}^{S} ⟶ 𝔐𝔬𝔡F_{S} of fibered categories.

Proof. Let h : A ⟶ B be a morphism from f : A ⟶ S to g : B ⟶ S; i.e. f = gh. And let c_{g,h} : h^{∗}g^{∗} ⟶ f^{∗} be the corresponding isomorphism. The composition of the canonical morphisms

η_{f}g_{∗} : g_{∗} ⟶ f_{∗}f^{∗}g_{∗}, f_{∗}c_{g,h}^{−1}g_{∗} : f_{∗}f^{∗}g_{∗} ⟶ f_{∗}h^{∗}g^{∗}g_{∗}, and f_{∗}h^{∗}ϵ_{g} : f_{∗}h^{∗}g^{∗}g_{∗} ⟶ f_{∗}h^{∗}

provides a morphism

ϕ_{h} = f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗})g^{∗} : g_{∗}g^{∗} ⟶ f_{∗}f^{∗}  (1)

We claim that ϕ_{h} is a monad morphism 𝔾_{g} ⟶ 𝔾_{f}, i.e. ϕ_{h}∘η_{g} = η_{f} and μ_{f} ∘ϕ_{h}⊙ϕ_{h} = ϕ_{h} ∘ μ_{g}. In fact,

ϕ_{h}∘η_{g} = f_{∗}c_{g,h}∘(f_{∗}h^{∗}ϵ_{g}∘f_{∗}c_{g,h}^{−1}g_{∗}∘η_{f}g_{∗})g^{∗}∘η_{g} = f_{∗}c_{g,h}∘f_{∗}h^{∗}ϵ_{g}g^{∗}∘f_{∗}c_{g,h}^{−1}g_{∗}g^{∗}∘f_{∗}f^{∗}η_{g}∘η_{f} =

f_{∗}c_{g,h} ∘ f_{∗}h^{∗}ϵ_{g}g^{∗} ∘ f_{∗}h^{∗}g^{∗}η_{g} ∘ f_{∗}c_{g,h}^{−1} ∘ η_{f} = f_{∗}c_{g,h} ∘ f_{∗}h^{∗}(ϵ_{g}g^{∗} ∘ g^{∗}η_{g}) ∘ f_{∗}c_{g,h}^{−1} ∘ η_{f} =

f_{∗}c_{g,h} ∘ f_{∗}c_{g,h}^{−1} ∘ η_{f} = η_{f}

since ϵ_{g}g^{∗} ∘ g^{∗}η_{g} = id_{g^{∗}}.

We have

ϕ_{h} ∘ μ_{g} = f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗})g^{∗} ∘ g_{∗}ϵ_{g}g^{∗} =
f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗} ∘ g_{∗}ϵ_{g})g^{∗}

and

μ_{f} ∘ ϕ_{h} ⊙ ϕ_{h} =

f_{∗}ϵ_{f}f^{∗}∘f_{∗}f^{∗}(f_{∗}c_{g,h}∘(f_{∗}h^{∗}ϵ_{g}∘f_{∗}c_{g,h}^{−1}g_{∗}∘η_{f}g_{∗})g^{∗})∘(f_{∗}c_{g,h}∘(f_{∗}h^{∗}ϵ_{g}∘f_{∗}c_{g,h}^{−1}g_{∗}∘η_{f}g_{∗})g^{∗})g_{∗}g^{∗} =

f_{∗}c_{g,h}∘f_{∗}ϵ_{f}h^{∗}g^{∗}∘f_{∗}f^{∗}(f_{∗}h^{∗}ϵ_{g}∘f_{∗}c_{g,h}^{−1}g_{∗}∘η_{f}g_{∗})g^{∗})∘(f_{∗}c_{g,h}∘(f_{∗}h^{∗}ϵ_{g}∘f_{∗}c_{g,h}^{−1}g_{∗}∘η_{f}g_{∗})g^{∗})g_{∗}g^{∗} =

f_{∗}(c_{g,h} ∘h^{∗}ϵ_{g} ∘c_{g,h}^{−1}g_{∗})∘f_{∗}ϵ_{f}f^{∗}g_{∗} ∘f_{∗}f^{∗}η_{f}g_{∗})g^{∗})∘(f_{∗}c_{g,h} ∘(f_{∗}h^{∗}ϵ_{g} ∘f_{∗}c_{g,h}^{−1}g_{∗} ∘η_{f}g_{∗})g^{∗})g_{∗}g^{∗} =

f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗})g^{∗} ∘ (f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗})g^{∗})g_{∗}g^{∗} =

f_{∗}c_{g,h} ∘ f_{∗}h^{∗}ϵ_{g}g^{∗} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗})g^{∗}g_{∗}g^{∗} =
f_{∗}c_{g,h} ∘ (f_{∗}h^{∗}ϵ_{g} ∘ f_{∗}c_{g,h}^{−1}g_{∗} ∘ η_{f}g_{∗} ∘ g_{∗}ϵ_{g})g^{∗}

In the last transformation we used the equality ϵ_{g} ∘ ϵ_{g}g^{∗}g_{∗} = ϵ_{g} ∘ g^{∗}g_{∗}ϵ_{g}.

The claimed morphism of fibered categories maps any object (X ⟶^{f} S, M) of the category ModF_{c}^{S} to the object (𝔾_{f}, (f_{∗}(M), μ_{f}(M))) of the category 𝔐𝔬𝔡F_{S} and any

morphism (h, ξ) : (X ⟶^{f} S, M) ⟶ (Y ⟶^{g} S, N) of the category ModF_{c}^{S} to the morphism (ϕ_{h}, ξ^{∼}), where ξ^{∼} is a morphism canonically defined by ξ and c_{g,h}^{−1}. Details are left to the reader.

Let Hom_{2}B denote the class of all continuous morphisms f : X ⟶ Y of the category B having the following properties:

(i) The category F_{X} has cokernels of all reflexive pairs of morphisms (recall that a pair of morphisms g_{1}, g_{2} : M ⟶ L is called reflexive if there exists a morphism h : L ⟶ M such that g_{1} ∘ h = id_{L} = g_{2} ∘ h).

(ii) A direct image functor of f preserves cokernels of all reflexive pairs of morphisms and reflects isomorphisms.

One can see that Hom_{2}B is closed under composition and contains identical morphisms of all objects X with the property (i) above.

Denote by (B/S)_{2} the full subcategory of the category B/S objects of which are pairs (X, f), where f : X ⟶ S belongs Hom_{2}B. Let F_{2}^{S} be the induced fibered subcategory of the fibered category F_{c}^{S}.

1.0.7.2. Lemma. The restriction 𝔏_{2} of the morphism 𝔏^{∗} to the fibered subcategory F_{2}^{S} is fully faithful (i.e. it induces a fully faithful functor from the category F_{2}^{S} to the category of modules 𝔐𝔬𝔡F_{S}).

Proof. It follows from the Barr-Beck theorem (cf. [ML], VI.7) that the morphism 𝔏_{2} induces an equivalence of categories on each fiber. The assertion follows now from SGA4, VI.4.2 (LNM 224, pp. 159–160).

1.0.8. Descent. Fix a fibered category F with a base B.

1.0.8.1. Comodules. Fix a set 𝔘 = {U ⟶^{u} X} of continuous morphisms of B. An 𝔘-comodule is a pair (F, ϕ), where F is a function which assigns to any (U ⟶^{u} X) ∈ 𝔘 an object F_{u} of F_{U} and ϕ assigns to any pair of arrows U ⟶^{u} X ⟵^{v} V of 𝔘, a morphism ϕ_{u,v} : F_{v} ⟶ v^{∗}u_{∗}(F_{u}) such that the following conditions hold.

(a) ϵ_{u} ∘ ϕ_{u,u} = id_{F_{u}}.

(b) For any three arrows of 𝔘, u : U ⟶ X, v : V ⟶ X, and w : W ⟶ X, the diagram

\begin{matrix}F_{w} & ⟶^{ϕ_{v,w}} & w^{∗}v_{∗}(F_{v}) \\ ϕ_{u,w} ↓ & & ↓ w^{∗}v_{∗}ϕ_{u,v} \\ w^{∗}u_{∗}(F_{u}) & ⟶^{w^{∗}η_{v}u_{∗}(F_{u})} & w^{∗}v_{∗}v^{∗}u_{∗}(F_{u}) \end{matrix}  (1)

is commutative.

For any two 𝔘-comodules (F, ϕ) and (G, ψ), a morphism from (F, ϕ) to (G, ψ) is a set of morphisms {g_{u} : F_{u} ⟶ G_{u}|u ∈ 𝔘} such that the diagram

\begin{matrix}F_{u} & ⟶^{g_{u}} & G_{u} \\ ϕ_{v,u} ↓ & & ↓ ψ_{v,u} \\ u^{∗}v_{∗}(F_{v}) & ⟶^{u^{∗}v_{∗}(g_{u})} & u^{∗}v_{∗}(G_{v}) \end{matrix}  (2)

is commutative for all u, v ∈ 𝔘. The compositon is defined in an obvious way. We denote the category of 𝔘-comodules by 𝔘 − Comod.

There is a natural functor F_{X} ⟶ 𝔘−Comod which assigns to any M ∈ ObF_{X} the 𝔘comodule (M^{∼}, ϕ^{M}), where M_{u}^{∼} = u^{∗}(M) and ϕ_{u,v}^{M} := v^{∗}η_{u}(M) : v^{∗}(M) ⟶ v^{∗}u_{∗}u^{∗}(M).

1.0.8.2. Proposition. Let 𝔘 = {U ⟶^{u} X} be a finite set of continuous morphisms of the base B.

1) The following properties are equivalent:

(a) A pair of arrows f, g : L ⟶ M of F_{X} has a kernel iff the pair (u^{∗}(f), u^{∗}(g)) has a kernel for all u ∈ 𝔘, and any functor u^{∗}, u ∈ 𝔘, maps a kernel of (f, g) (if any) into a kernel of (u^{∗}(f), u^{∗}(g)).

(b) The functor F_{X} ⟶ 𝔘 − Comod is an equivalence of categories.

2) If the equivalent conditions of 1) hold, then the cone of the functor morphisms

\begin{matrix}Id_{F_{X}} & ⟶^{η_{u}} & G_{u} \\ η_{v} ↓ & & ↓ η_{v}G_{u} \\ G_{v} & ⟶^{G_{v}η_{u}} & G_{v}G_{u} & u, v ∈ 𝔘 \end{matrix}  (3)

is terminal. Here G_{u} := u_{∗}u^{∗} and η_{u} is an adjunction morphism Id_{F_{X}} ⟶ G_{u}.

Proof. 1) If 𝔘 consists of one morphism, the assertion 1) follows from the Barr-Beck theorem (cf. [ML], VI.7). To the family of functors {u^{∗}|u ∈ 𝔘}, there corresponds a functor u^{∗} : X ⟶ ∏_{(U⟶^{u}X)∈𝔘} F_{U}. If the category F_{X} has products, then the functor u^{∗} has a right adjoint, and u^{∗} satisfies the condition (a).

2) Suppose that 𝔘 consists of one morphism u. Then the square

u^{∗}  ⟶^{u^{∗}η_{u}} u^{∗}G_{u}
u^{∗}η_{u} ↓  ↓ u^{∗}η_{u}G_{u}
u^{∗}G_{u}  ⟶^{u^{∗}G_{u}η_{u}} u^{∗}G_{u}G_{u}

is cartesian. Since u^{∗} reflects kernels of pairs of morphisms, it follows that the square

Id_{F_{X}}  ⟶^{η_{u}}  G_{u}
η_{u} ↓  ↓ η_{u}G_{u}
G_{u}  ⟶^{G_{u}η_{u}} G_{u}G_{u}

is cartesian. The general case can be reduced to the case of one morphism, as in 1) above. Details are left to the reader.

1.1. The category of ’spaces’. ’Spaces’ are categories which are equivalent to small categories with respect to some universum, 𝔘. We define a morphism f from a ’space’ A to a ’space’ B as an isomorphy class of right exact functors from B to A. Any functor B ⟶ A from f will be called an inverse image functor of f. And once we made a choice of an inverse image functor, we shall denote it by f^{∗}. The composition of morphisms is natural: f ∘ g = [g^{∗} ∘ f^{∗}]. Here [u] means ”all functors isomorphic to u”. Thus defined category of ’spaces’ shall be denoted by RCat_{𝔘} or simply by RCat.

The category RCat is a base of a naturally defined fibered category which we denote by RCat^{∼} and will call the fibered category of ’spaces’.

1.1.1. Proposition. (a) The category of ’spaces’ RCat has small colimits.

(b) For any category C, the category RCat/C has small colimits.

Proof. (a1) The coproduct of a family {A_{i}|i ∈ J} is the product of categories ∏_{i∈J} A_{i}. (a2) The category RCat has fibered coproducts, in particular it has cokernels of pairs

of morphisms.

In fact, let X ⟵^{p} Z ⟶^{q} Y be morphisms in RCat. Denote by W the category objects of which are triples (L, ϕ, M), where L ∈ ObX, M ∈ ObY, and ϕ an isomorphism p^{∗}(L) ⟶ q^{∗}(M). Morphisms from (L, ϕ, M) to (L′, ϕ′, M′) are pairs of morphisms s : L ⟶ L′, t : M ⟶ M′ such that the diagram

p^{∗}(L) ⟶^{ϕ} q^{∗}(M)
p^{∗}(s) ↓  ↓ q^{∗}(t)
p^{∗}(L′) ⟶^{ϕ′} q^{∗}(M′)

is commutative. The functors W ⟶ X, (L, ϕ, M) ⟼ L, and W ⟶ Y, (L, ϕ, M) ⟼ M, are right exact, hence can be regarded as inverse images of resp. morphisms q_{1} : X ⟶ W and p_{1} : Y ⟶ W. It follows that q_{1} ∘ p = p_{1} ∘ q. Any pair of morphisms X ⟶^{u} V ⟵^{v} Y such that u ∘ p = v ∘ q, defines uniquely up to isomorphism a functor g^{∗} : V ⟶ W which assigns to any N ∈ ObV the triple (u^{∗}(N), ϕ_{u,v}(N), v^{∗}(N)). Here ϕ_{u,v} is a functor isomorphism p^{∗}u^{∗} ⟶ q^{∗}v^{∗}, and ϕ_{u,v}(N) denotes its value on the object N. The functor g^{∗} is an inverse image functor of a morphism g : W ⟶ V uniquely defined by the equality u ∘ p = v ∘ q. Therefore W is a fibered coproduct of X ⟵^{p} Z ⟶^{q} Y.

It follows from (a1) and (a2) that RCat has colimits of arbitrary small diagrams.

(b) Let f : ∏_{i∈J} X_{i} ⟶ C be a morphism with the inverse image functor f^{∗} := ∏_{i∈J} f_{i}^{∗} : C ⟶ ∏_{i∈J} X_{i}. For any object (B, g : B ⟶ C) of the category RCat/C, the canonical map RCat/C((∏_{i∈J} X_{i}, f), (B, g)) ⟶ ∏_{i∈J} RCat/C((X_{i}, f_{i}), (Y, g)) is a bijection, hence (∏_{i∈J} X_{i}, f) is a coproduct of the family F in the category RCat/C.

To finish the argument, it suffices to show that RCat/C has fibered coproducts. Let (X, f) ⟵^{p} (Z, h) ⟶^{q} (Y, g) be morphisms in RCat/C. By definition, f ∘ p = h =

g ∘ q, or, equivalently, p^{∗}f^{∗} ≃ h^{∗} ≃ q^{∗}g^{∗}. In particular, we have a functorial isomorphism ψ : p^{∗}f^{∗} ⟶ q^{∗}g^{∗}, hence a functor t^{∗} : C ⟶ W = X ∐_{p,q} Y, E ⟼ (f^{∗}(E), ψ(E), g^{∗}(E)). It follows that t^{∗} is an inverse image functor of a morphism t : W ⟶ C and the canonical morphisms X ⟶^{q_{1}} W ⟵^{p_{1}} Y are morphisms over C, i.e. f = t ∘ q_{1} and g = t ∘ p_{1}. This proves the assertion.

A morphism f : B ⟶ A such that f^{∗} is a localization (i.e. a universal functor making invertible all arrows of Σ_{f} := {s ∈ HomA|f^{∗}(s) is invertible}) will be called by abuse of language a localization.

1.1.2. Proposition. Let A ⟶^{f} C ⟵^{g} B be morphisms of ’spaces’ such that A and B are categories with finite colimits and f (i.e. f^{∗}) is a localization. Then the fiber product A ×_{(f,g)} B exists and the canonical projection A ×_{(f,g)} B ⟶ B is a localization.

Proof. Denote by Σ_{f^{∗}} the class of morphisms of the category C the functor f^{∗} sends to isomorphisms. Since f^{∗} is right exact, Σ_{f} admits left fractions. Let Σ_{f,g} be the minimal class of morphisms of the category B containing g^{∗}(Σ_{f}) and admitting left fractions. Let q^{∗} : B ⟶ B′ := Σ_{f,g}^{−1}B be a localization at Σ_{f,g}. Since q^{∗} is right exact, it can be regarded as an inverse image functor of a morphism, q. The composition q^{∗} ∘ g^{∗} maps Σ_{f} to isomorphisms, and f^{∗} is a localization at Σ_{f}. Hence there exists a unique functor p^{∗} : A ⟶ B′ such that p^{∗}f^{∗} = q^{∗}g^{∗}. Since the functor q^{∗}g^{∗} is right exact, p^{∗} is right exact too. Thus we have a commutative square

\begin{matrix}A & ⟶^{f} & C \\ p ↑ & & ↑ g \\ B′ & ⟶^{q} & B \end{matrix}  (1)

The claim is that the square (1) is cartesian. In fact, let A ⟵^{u} B′′ ⟶^{v} B be morphisms such that fu = gv. Since v^{∗} is a right exact functor, the class Σ_{v} of morphisms v^{∗} maps to isomorphisms admits left fractions. Since v^{∗}g^{∗} being isomorphic to u^{∗}f^{∗} maps Σ_{f} to isomorphisms, the class Σ_{v} contains Σ_{f,g}. Therefore, v^{∗} = h^{∗}q^{∗} for a uniquely defined functor h^{∗} : B′ ⟶ B′′. Since the functor v^{∗} and the localization q^{∗} are right exact, h^{∗} is right exact too, hence it might be regarded as an inverse image functor of a morphism h : B′′ ⟶ B′. Thus h^{∗}p^{∗}f^{∗} = h^{∗}q^{∗}g^{∗} ≃ u^{∗}f^{∗}. Since f^{∗} is a localization, it follows from h^{∗}p^{∗}f^{∗} ≃ u^{∗}f^{∗} that h^{∗}p^{∗} ≃ u^{∗}, i.e. u = ph.

1.2. Monads and continuous morphisms. Fix a category C. Denote by RCat_{C} the full subcategory of the category RCat/C objects of which are continuous morphisms f : X ⟶ C. To any continuous morphism f : X ⟶ C, there corresponds a monad 𝔾_{f} = (G_{f}, μ_{f}) uniquely defined by the choice of the inverse and direct image functors, f^{∗} and f_{∗}, and an adjunction arrow ϵ_{f} : f^{∗}f_{∗} ⟶ Id_{X}. Namely G_{f} = f_{∗}f^{∗} and μ_{f} = f_{∗}ϵ_{f}f^{∗} : G_{f}^{2} ⟶ G_{f}. The adjunction arrow η_{f} : Id_{C} ⟶ f_{∗}f^{∗} = G_{f} is the unit of 𝔾_{f}. One can check that to different choices of (f^{∗}, f_{∗}, ϵ_{f}) there correspond isomorphic monads.

1.2.1. Proposition. For any category C with cokernels of reflexive pairs of morphisms, the canonical morphism of fibered categories 𝔐𝔬𝔡C ⟶ RCat_{2}^{∼C} is an equivalence.

Proof. The fact follows from 1.0.7.2.

1.3. Coproducts in RCat_{C}.

1.3.1. Proposition. Let F = {f_{i} : X_{i} ⟶ C|i ∈ J} be a family of morphisms. Suppose that C has products of sets of |J| objects. If all morphisms f_{i} are continuous, then their coproduct in the category RCat/C is continuous, hence it is a coproduct in the category RCat_{C}.

Proof. (a) For any (B, g : B ⟶ C), the canonical map

RCat/C((∏_{i∈J} X_{i}, f), (B, g)) ⟶ ∏_{i∈J} RCat/C((X_{i}, f_{i}), (Y, g))

is a bijection.

(b) Let f_{i∗} be a direct image functor of the morphism f_{i}. Then M = (M_{i}|i ∈ J) ⟼ ∏_{i∈J} f_{i∗}(M_{i}) is a direct image functor of f.

1.3.2. The monad associated with a coproduct. Let {f_{i} : X_{i} ⟶ C|i ∈ J} be a family of continuous morphisms. Let f be their coproduct. Assume that the condition of 1.3.1 holds, hence f is continuous. The monad (f_{∗}f^{∗}, μ_{f}) corresponding to f is isomorphic to the product ∏_{i∈J} 𝔾_{f_{i}} of monads 𝔾_{f_{i}} = (f_{i∗}f_{i}^{∗}, μ_{f_{i}}) corresponding to morphisms f_{i}.

1.4. Glueing two ’spaces’.

1.4.1. Proposition. Let X ⟵^{p} Z ⟶^{q} Y be morphisms of ’spaces’ such that q (i.e. q^{∗}) is a localization (resp. a continuous localization). Then the canonical morphism q_{1} : X ⟶ X ∐_{(p,q)} Y is a localization (resp. a continuous localization).

Proof. (a) Denote by Σ_{q_{1}^{∗}} the class of all morphisms s of the category X ∐_{(p,q)} Y such that q_{1}^{∗}(s) is invertible. Note that Σ_{q_{1}^{∗}} consists of all morphisms (s, t) : (L, ϕ, M) ⟶ (L′, ϕ′, M′) such that s is an isomorphism, hence t ∈ Σ_{q^{∗}}. Since q^{∗} is a localization, for any L ∈ ObX, there exists an M ∈ ObY such that p^{∗}(L) ≃ q^{∗}(M). This defines a functor X ⟶ Σ_{q_{1}^{∗}}^{−1}(X ∐_{(p,q)} Y) quasi-inverse to the canonical functor Σ_{q_{1}^{∗}}^{−1}(X ∐_{(p,q)} Y) ⟶ X, (L, ϕ, M) ⟼ L.

(b) Suppose the morphism q is continuous, i.e. there exists a direct image functor q_{∗}. Fix adjunction morphisms η_{q} : Id_{Y} ⟶ q_{∗}q^{∗} and ϵ_{q} : q^{∗}q_{∗} ⟶ Id_{Z}. Since q^{∗} is a localization, the morphism ϵ_{q} is an isomorphism. In particular, we have a functor q_{1∗} : X ⟶ X ∐_{(p,q)} Y, L ⟼ (L, ϵ_{q}^{−1}(p^{∗}(L)), q_{∗}p^{∗}(L)). This functor is a right adjoint to the functor q_{1}^{∗} : (L, ϕ, M) ⟼ L. The adjunction morphism ϵ_{q_{1}} : q_{1}^{∗} ∘ q_{1∗} ⟶ Id_{X} is identical; the adjunction morphism η_{q_{1}} : Id_{X ∐_{(p,q)} Y} ⟶ q_{1∗}q_{1}^{∗} assigns to any object

(L, ϕ, M) of the category X ∐_{(p,q)} Y the morphism

(id_{L}, q_{∗}(ϕ^{−1}) ∘ η_{q}(M)) : (L, ϕ, M) ⟶ q_{1∗}q_{1}^{∗}(L) := (L, ϵ_{q}^{−1}(p^{∗}(L)), q_{∗}p^{∗}(L))

hence the assertion.

1.4.2. Proposition. Let X ⟵^{p} Z ⟶^{q} Y be morphisms of ’spaces’ such that q and p (i.e. q^{∗} and p^{∗}) are localizations. Then the square

Z ⟶^{q}  Y
p ↓  ↓ p_{1}
X ⟶^{q_{1}} X ∐_{(p,q)} Y

is cartesian.

Proof. Let X ⟵^{u} W ⟶^{v} Y be morphisms such that q_{1} ∘ u = p_{1} ∘ v. In other words, there exists an isomorphism ϕ : u^{∗} ∘ q_{1}^{∗} ⟶^{∼} v^{∗} ∘ p_{1}^{∗}. Let s : M ⟶ M′ be any morphism of Σ_{q^{∗}}. Since p is a localization, there exists L ∈ ObX and an isomorphism ϕ : p^{∗}(L) ⟶ q^{∗}(M). Consider the morphism s′ = (id_{L}, s) : (L, ϕ, M) ⟶ (L, q^{∗}(s) ∘ ϕ, M′) of the category X ∐_{(p,q)} Y. By definition of q_{1}^{∗} and p_{1}^{∗} (cf. the argument of 1.1.1), q_{1}^{∗}(s′) = id_{X} and p_{1}^{∗}(s′) = s, hence u^{∗} ∘ q_{1}^{∗}(s′) = id_{W} and v^{∗} ∘ p_{1}^{∗}(s′) = v^{∗}(s). Since u^{∗} ∘ q_{1}^{∗} ≃ v^{∗} ∘ p_{1}^{∗}, it follows that v^{∗}(s) = id_{W}. Therefore the functor v^{∗} maps Σ_{q} into invertible morphisms. Since q^{∗} is a localization, there exists a unique functor w^{∗} : Y ⟶ W such that v^{∗} = w^{∗}∘q^{∗}. Since the functors q^{∗} and v^{∗} are right exact, the functor w^{∗} is right exact, hence can be regarded as an inverse image functor of a morphism w : Z ⟶ W. Thus, the morphism v is uniquely represented as a composition w ∘ q. Similarly, the morphism u = u′ ∘ p for a uniquely defined morphism u′ : Z ⟶ W, hence the assertion.

1.4.3. Almost affine localizations. We call a continuous morphism f : X ⟶ Y almost affine, if its direct image functor is faithful and exact.

Note that f_{∗} is exact iff it is right exact. The composition of almost affine morphisms is almost affine.

1.4.3.1. Proposition. Let X ⟵^{p} Z ⟶^{q} Y be morphisms of ’spaces’ such that q (i.e. q^{∗}) is a localization. If q is almost affine, then the canonical morphism q_{1} : X ⟶ X ∐_{(p,q)} Y is an almost affine localization.

Proof. By the argument of 1.3.4.1(b), q_{1∗} maps any object L of the category X into (L, ϵ_{q}^{−1}(p^{∗}(L)), q_{∗}p^{∗}(L)). By 1.3.4.1, q_{1∗} is a right adjoint to a localization, hence (fully) faithful. By assumption, the functor q_{∗} is right exact, hence q_{∗}p^{∗} is right exact which implies that q_{1∗} is right exact.

1.5. The category of cospaces. We denote by LCat, or by LCat_{𝔘}, the category objects of which are categories equivalent to small categories with respect to some universum, 𝔘. A morphism f from A to B as an isomorphy class of left exact functors from B to A. Any functor B ⟶ A from f will be called a direct image functor of f. Once we made a choice of a direct image functor, we denote it by f_{∗}. The composition of morphisms is defined by f ∘ g = [f_{∗} ∘ g_{∗}]. Here [u] means ’all functors isomorphic to u’.

The category LCat is a base of a naturally defined cofibered category which we denote by LCat^{∼} and will call the cofibered category of ’cospaces’.

We denote by R_{c}Cat the subcategory of the category RCat formed by continuous morphisms and by R_{c}Cat^{∼} the corresponding bifibered category. We have natural embedding R_{c}Cat ⟶ RCat and the faithful functor R_{c}Cat ⟶ LCat which maps any morphism to f : X ⟶ Y to the isomorphy class of [f_{∗}] : X ⟶ Y of its direct image functor.

The functor Cat ⟶ Cat which assigns to any small subcategory its dual and to any functor the corresponding dual functor induces an equivalence between the categories RCat^{op} and LCat. Thus the dual versions of assertions about RCat hold in LCat. For instance, LCat is a category with limits, in particular with fiber products.

The existence of a limit or a colimit of a diagram 𝔇 : D ⟶ R_{c}Cat is usually a ruther subtle problem. The following trivial observation may be regarded as a starting point. If X ⟶ 𝔇 is a cone such that the image of this cone in LCat is an initial cone, then the cone itself is initial, i.e. X is a limit of 𝔇 in R_{c}Cat.

1.6. Closed ’subspaces’ and cosubspaces. For a morphism f : X ⟶ Y of a category A, denote by Λ_{f} the class of all pairs of morphisms u_{1}, u_{2} : X ⇉ V equalizing f. A morphism f : Y ⟶ X is called a strict monomorphism if any morphism g : Z ⟶ X such that Λ_{f} ⊆ Λ_{g} has a unique decomposition g = f ∘ g′. It follows that any strict monomorphism is a monomorphism. Suppose a morphism f : Y ⟶ X is such that there exists a fibered coproduct X ∐_{Y} X. Then f is a strict monomorphism iff it is a kernel of the coprojections Y ⇉ Y ∐_{X} Y. We denote the class of strict monomorphisms of the category C by 𝔐_{s}(C), or by 𝔐_{s}. The class 𝔈_{s} = 𝔈_{s}(C) of strict epimorphisms is defined dually. We say that an object Y is a strict subquotient of an object X if there exists a diagram X ⟵^{m} K ⟶^{e} Y, where m is a strict monomorphism and e is a strict epimorphism.

We call a full subcategory B of a category A topologizing if B contains all strict subquotients of any of its objects and limits and colimits of all finite diagrams D ⟶ B taken in A (if any).

We call a full topologizing subcategory B of a category A a closed ’subspace’ of A (resp. a closed cosubspace of A), if the inclusion functor J : B ⟶ A has a left (resp. right) adjoint. Thus if B is a closed ’subspace’ of A, the inclusion functor J : B ⟶ A can be regarded as a direct image functor of an embedding B ⟶ A. Closed cosubspaces of A are exactly closed ’subspaces’ of A^{op}.

We call a full subcategory B of A a bisubspace of A if it is both a closed ’subspace’ and a closed cosubspace. Sometimes bisubspaces will be called Zariski closed ’subspaces’.

1.6.1. Example: closed ’subspaces’ of the category of modules. Let A be the category R−mod of left R-modules over a ring R. Then bisubspaces of A are in one-to-one correspondence with two-sided ideals of the ring R: to any two-sided ideal α of R, there corresponds the bisubspace R/α − mod – the full subcategory of R − mod generated by all R-modules annihilated by the ideal α (see [R], III.6.4.1).

1.6.2. Example: closed cosubspaces of the category of modules. Fix a left ideal m of a ring R. Let [R/m] denote the minimal cosubspace of the category R − mod of left R-modules containing the module R/m. This cosubspace admits the following description.

For two left ideals, m and n, we write m ≤ n if there exists a finite subset x of elements of R such that the left ideal (m : x) := {r ∈ R| rx ⊆ n} is contained in n. The cosubspace [R/m] is a full subcategory of R − mod objects of which are all R-modules M such that for any element z of M, m ≤ Ann(z). Here Ann(z) denotes the annihilator of z.

Note that if the ideal m is two-sided, then m ≤ n iff m ⊆ n. In particular, if m is a two-sided ideal, the category [R/m] coincides with the bisubspace R/m − mod of Example 1.6.1.

2. Affine schemes.

2.1. Continuous and 𝔖-continuous functors. Fix a class of small categories 𝔖. For any two categories A and B (objects of Cat), denote by F un_{𝔖}(A, B) the full subcategory of the category F un(A, B) of functors from A to B objects of which are all functors preserving colimits of functors from 𝔖.

2.1.1. 𝔖_{2}-continuous functors. A special case of this construction is the category F un_{2}(A, B) objects of which are functors preserving cokernels of reflexive pairs of arrows. The corresponding class consists of one category, 𝔖_{2}, with two objects, a, b, and six arrows: id_{a}, id_{b}, e_{i} : a ⟶ b, h : b ⟶ a, and f_{i} : a ⟶ b, i = 1, 2, such that e_{1}^{2} = e_{2} ∘ e_{1} = e_{1}, e_{2}^{2} = e_{1} ∘ e_{2} = e_{2}, h ∘ f_{i} = e_{i}, i = 1, 2.

2.1.2. Right exact functors. Another special case is the category F un_{fin}(A, B) objects of which are right exact functors (i.e. functors which preserve colimits of all finite diagrams).

2.1.3. ω-continuous functors. The category F un_{ω}(A, B) objects of which are functors preserving colimits of countable diagrams will also play a particular role in this work beginning from 2.6.

2.1.4. The categories Cat_{𝔖}. For any three categories, A, B, C, the composition map

F un_{𝔖}(A, B) × F un_{𝔖}(B, C) ⟶ F un(A, C), (F, G) ⟼ G ∘ F,

takes values in F un_{𝔖}(A, C). This defines a subcategory Cat_{𝔖} of the category Cat. In particular, we have the subcategories Cat_{2}, Cat_{fin}, and Cat_{ω}.

2.1.5. Continuous functors. We call a functor F : A ⟶ B continuous if it has a right adjoint. The full subcategory of F un(A, B) objects of which are continuous functors will be denoted by F un_{c}(A, B). Clearly the composition of continuous functors is a continuous functor. Hence we have a subcategory Cat_{c} of the category Cat.

2.2. Proposition. Let f : X ⟶ Y be a continuous morphism. Suppose the category X has cokernels of reflexive pairs of arrows and a direct image functor f_{∗} of f preserves cokernels of reflexive pairs and reflects isomorphisms. Then

(a) The functor f_{∗} preserves colimits of functors from 𝔖 iff the functor f_{∗}f^{∗} has this property.

(b) The functor f_{∗} has a right adjoint iff f_{∗}f^{∗} has a right adjoint.

Proof. (a) If f_{∗} preserves colimits of functors from 𝔖, then f_{∗}f^{∗} enjoys this property (without any additional conditions), since f^{∗} preserves all colimits.

Conversely, suppose f_{∗}f^{∗} preserves colimits of functors from 𝔖. Then the forgetful functor 𝔾_{f} −mod ⟶ Y, where 𝔾_{f} is the monad (f_{∗}f^{∗}, μ_{f}), has the same property. By the Barr-Beck theorem, the canonical functor K_{f} : X ⟶ 𝔾_{f} − mod, M ⟼ (f_{∗}(M), μ_{f}(M)) is an equivalence, in particular it preserves all colimits. The composition of the functor K_{f} with the forgetful functor coincides with f_{∗}, hence the assertion.

(b) If f_{∗} is a right adjoint, f^{!}, then f_{∗}f^{!} is a right adjoint to the functor f_{∗}f^{∗}.

Suppose 𝔽 = (F, μ) is a monad on Y such that the functor F has a right adjoint, F^{!}. Then the forgetful functor 𝔽 − mod ⟶ Y has a right adjoint ([R3], 4.4.1). Since f_{∗} is the composition of the forgetful functor 𝔾_{f} − mod ⟶ Y and the equivalence K_{f} : X ⟶ 𝔾_{f} − mod, M ⟼ (f_{∗}(M), μ_{f}(M)), it has a right adjoint too.

2.3. Relative affine and 𝔖-affine schemes. For a class of small categories 𝔖, a morphism f : X ⟶ Y is called 𝔖-affine if the object (X, f) of the category RCat/Y is isomorphic to the object ((F, μ) − mod_{Y} , f_{Y}) for some monad (F, μ) on Y such that F is a 𝔖-continuous functor. Here f_{Y} is the canonical morphism (F, μ) − mod_{Y} ⟶ Y with a direct image functor equal to the forgetful functor.

The object (X, f), where f : X ⟶ Y is a 𝔖-affine morphism, will be called an 𝔖affine Y -scheme. We denote by Aff_{𝔖}/Y the full subcategory of the category RCat/Y objects of which are 𝔖-affine Y -schemes and by Aff_{𝔖}^{∼}/Y the corresponding fibered category.

2.3.1. Almost affine and ω-affine relative schemes. We single out two important special cases of 𝔖-affine relative schemes:

A morphism f : X ⟶ Y is called almost affine (resp. ω-affine), if (X, f) is isomorphic to ((F, μ) − mod_{Y} , f_{Y}) for some monad (F, μ) such that F is a right exact functor (resp. F preserves colimits of countable diagrams). The object (X, f) of RCat/Y will be called resp. almost affine, or ω-affine Y -scheme. Thus we have the category Aff_{fin}/Y of almost affine Y -schemes and the category Aff_{ω}/Y of ω-affine Y -schemes and the corresponding fibered categories, resp. Aff_{fin}^{∼}/Y and Aff_{ω}^{∼}/Y.

2.3.2. Affine relative schemes. A morphism f : X ⟶ Y is called affine, if (X, f) is isomorphic to ((F, μ) − mod_{Y} , f_{Y}) for some monad (F, μ) such that F is continuous, i.e. has a right adjoint. An object (X, f) of the category RCat/Y will be called an affine Y -scheme if the morphism f : X ⟶ Y is affine. The full subcategory of RCat/Y objects of which are affine Y -schemes will be denoted by Aff/Y.

It follows from 2.1.4 and 2.1.5 that the composition of affine (resp. 𝔖-affine) morphisms is affine (resp. 𝔖-affine).

2.3.3. Proposition. Let (X, f) ∈ ObRCat/Y. Suppose a category X has cokernels of reflexive pairs of morhisms. Then (X, f) is a 𝔖-affine (resp. almost affine, resp. affine) Y -scheme iff the morphism f is continuous and its direct image functor preserves cokernels of reflexive pairs of morphisms and colimits of functors from 𝔖 (resp. is exact, resp. has a right adjoint) and reflects isomorphisms.

Proof. The assertion follows from 2.2 and the Barr-Beck theorem (cf. [ML], VI.7).

2.4. Relative 𝔖-affine schemes and 𝔖-monads. Fix a class of small categories 𝔖. For a category Y, denote by 𝔐𝔬𝔫_{𝔖}Y the full subcategory of the category of 𝔐𝔬𝔫Y of monads on Y objects of which are monads (F, μ) such that F preserves cokernels of reflexive pairs of arrows and colimits of functors from 𝔖. As particular cases, we have the subcategory 𝔐𝔬𝔫_{fin}Y of right exact monads on Y objects of which are monads (F, μ) such that the functor F is right exact, and the subcategory 𝔐𝔬𝔫_{ω}Y of ω-monads on Y which are monads (F, μ) such that F preserves colimits of countable diagrams. Finally, we denote by 𝔐𝔬𝔫_{c}Y the full subcategory of 𝔐𝔬𝔫Y object of which are continuous monads on Y, that is monads (F, μ) with F having a right adjoint.

Assume that Y has cokernels of reflexive pairs of arrows. Then, by 1.0.6.1, we have natural functors

Spec_{𝔖}/Y : (𝔐𝔬𝔫_{𝔖}Y)^{op} ⟶ Aff_{𝔖}/Y.  (1)
Spec_{c}/Y : (𝔐𝔬𝔫_{c}Y)^{op} ⟶ Aff/Y.  (2)

which are compositions of natural pseudo-functors respectively

Spec_{𝔖}^{∼}/Y : (𝔐𝔬𝔫_{𝔖}Y)^{op} ⟶ Aff_{𝔖}^{∼}/Y.  (1′)
Spec_{c}^{∼}/Y : (𝔐𝔬𝔫_{c}Y)^{op} ⟶ Aff^{∼}/Y.  (2′)

with the corresponding projections.

We single out two special cases of (1):

Spec_{fin}/Y : (𝔐𝔬𝔫_{fin}Y)^{op} ⟶ Aff_{fin}/Y.  (3)

and

Spec_{ω}/Y : (𝔐𝔬𝔫_{ω}Y)^{op} ⟶ Aff_{ω}/Y.  (4)

It follows from 1.0.7.1 that the functors Spec_{𝔖}/Y and Spec_{c}/Y are full. They are not faithful in general.

2.5. Affine and almost affine k-schemes and k-’spaces’. If C is the category k − mod of left modules over an associative ring k, we shall say affine (resp. almost affine, resp. 𝔖-affine) k-schemes instead of affine (resp. almost affine, resp. 𝔖-affine) C-schemes.

Fix a pseudo-functor

(RCat)^{op} ⟶ Cat, ((X, f) ⟶^{h} (Y, g)) ⟼ (Y ⟶^{h^{∗}} X); c_{h,t} : t^{∗}h^{∗} ⟶^{∼} (ht)^{∗} (1)

We associate with (1) the fibered category (RCat/C)^{∼} ⟶^{π} RCat/C (see 1.0) and a natural quasi-coherent presheaf (i.e. a section of the fibered category (RCat/C)^{∼} ⟶ RCat/C taking values in the category of quasi-coherent morphisms, cf. 1.0.2)

O : RCat/C ⟶ (RCat/C)^{∼}

which assigns to any object (X, f) of RCat/C the object (X, f^{∗}(k)) of the category (RCat/C)^{∼} and to any morphism h : (X, f) ⟶ (Y, g) the morphism

(h, c_{g,h}(k) : h^{∗}(g^{∗}(k)) ⟶ f^{∗}(k)).

Fix an object (A, f) of RCat/C such that f : A ⟶ C is a continuous morphism. Then it is defined uniquely up to isomorphism by the object O = f^{∗}(k).

In fact, we have functorial isomorphisms A(f^{∗}(k), X) ≃ C(k, f_{∗}(X)) ≃ f_{∗}(X) which shows that the direct image functor f_{∗} of f is naturally isomorphic to the functor X ⟼ A(f^{∗}(k), X). Therefore the inverse image functor f^{∗} (representing f) is defined uniqely up to isomorphism (being a left adjoint to the functor f_{∗}) by the object f^{∗}(k). Note that since f^{∗} preserves colimits, there exists a coproduct of any set of copies of O = f^{∗}(k).

2.5.0. Note. Consider the category Cat_{∗} objects of which are pairs (A, O), where A is a category and O is an object of A. Morphisms from (A, O) to (A′, O′) are pairs (f^{∗}, ϕ), where f^{∗} is a right exact functor from A to A′ and ϕ is an isomorphism from f^{∗}(O′) to O. Suppose that (A, O) is an object of the category Cat_{∗} such that the category A is abelian and there exists a coproduct of any set of copies of O. Then the functor M ⟼ A(O, M) from A to the category K − mod, where K = A(O, O)^{o}, is a direct image of a continuous morphism from A to K − mod ([BD], Proposition 6.6.23).

Now fix an additive category A and a continuous morphism f : A ⟶ C = k − mod. And set O = f^{∗}(k). The functor f_{∗} is faithful iff O is a generator of the category A. In this case, A has a structure of a k-linear category.

Since f_{∗} ≃ A(O, −), the functor f_{∗} is exact iff O is a projective object.

Thus f is almost affine iff O is a projective generator. Finally, f is affine iff O is a projective generator of finite type.

2.5.1. Proposition. (a) For any continuous morphism f : A ⟶ C = k − mod, there is a canonical functor morphism

ψ_{f} : A(O, O)^{o}⊗_{k} ⟶ A(O, f^{∗}−)  (1)

such that ψ_{f}(V) is an isomorphism for any free k-module V of finite type.

(b) If f is almost affine, then ψ_{f}(V) is an isomorphism for any finitely presented k-module V. In particular, if k is left noetherian, then ψ_{f}(V) is an isomorphism for any finitely generated k-module V.

(c) The morphism ψ_{f} is an isomorphism if and only if f is affine.

(d) The morphism f is affine if and only if the functor

A(O, −) : A ⟶ A(O, O)^{o} − mod

is an equivalence of categories.

Proof. (a) For any additive functor F : k − mod ⟶ k − mod, the module F (k) has a natural k-bimodule there is a canonical functor morphism ψ_{F} : F (k)⊗_{k} ⟶ F (see for instance, [Bass], Ch.I). Recall that, for any k-module V, the morphism ψ_{F} (V) is the image of id_{V} with respect to the composition

Hom_{k}(V, V)  ⟶  Hom_{k}(V, Hom_{k}(k, V))
↓
Hom_{k}(F (k) ⊗_{k} V, F (V)) ⟵ Hom_{k}(V, Hom_{k}(F (k), F (V)))

Since ψ_{F} (k) is an isomorphism and the functor F is additive, ψ_{F} (V) is an isomorphism for any free k-module V of finite rank.

(b) If the functor F is right exact, i.e. if it preserves cokernels, ψ_{F} (V) is an isomorphism for any finitely presented object V, since finitely presented objects are exactly cokernels of morphisms between free objects of finite rank.

(c) The morphism ψ_{F} is an isomorphism iff the functor F preserves arbitrary colimits (or, equivalently, has a right adjoint).

The assertions (a)–(c) of the lemma are just specializations of these facts for the functor f_{∗} ∘ f^{∗} ≃ A(O, f^{∗}−).

(d) If f : A ⟶ k − mod is almost affine, the canonical functor A ⟶ 𝔾_{f} − mod is an equivalence of categories. The assertion (c) implies that if (and only if) f is affine, the monad 𝔾_{f} is naturally isomorphic to the monad (A(O, O)⊗_{k}, m), where m is induced by the multiplication in A(O, O)^{o}. The category (A(O, O)⊗_{k}, m) − mod is isomorphic to the category of left modules over A(O, O)^{o}.

2.5.2. Remark. The analysis above shows that when the ring k is commutative, affine schemes over C = k −mod are affine schemes in the sense of M. Artin and J.J. Zhang [AZ].

2.6. Product of ω-affine relative schemes.

2.6.1. The free product of continuous morphisms. Let f : A ⟶ C and g : B ⟶ C be continuous morphisms with direct image functors resp. f_{∗} and g_{∗}. Denote by A ⋆_{f,g} B the category objects of which are triples (X, ϕ, Y), where X ∈ ObA, Y ∈ ObB, and ϕ is an isomorphism f_{∗}(X) ⟶ g_{∗}(Y). Morphisms between triples are defined in an obvious way. There are canonical functors p_{∗} : A ⋆_{f,g} B ⟶ A, (X, ϕ, Y) ⟼ X and q_{∗} : A ⋆_{f,g} B ⟶ B, (X, ϕ, Y) ⟼ Y. It follows that f_{∗}p_{∗} ≃ g_{∗}q_{∗}.

2.6.1.1. Remark. Let f : A ⟶ C and g : B ⟶ C be continuous morphisms.

In terms of 1.5, the category A ⋆_{f,g} B is a fiber product of the image A ⟶^{[f_{∗}]} C ⟵^{[g_{∗}]} B

of the diagram A ⟶^{f} C ⟵^{g} B in the category LCat. If the canonical functor π_{∗} : A ⋆_{f,g} B ⟶ C, (X, ϕ, Y) ⟼ f_{∗}(X) has a left adjoint, that is π_{∗} is a direct image functor

of a continuous morphism π : A ⋆_{f,g} B ⟶ C, then this morphism is a product of A ⟶^{f} C and B ⟶^{g} C, or, in other words, a fiber product of A and B over C (cf. the end of 1.5).

Note however, that in spite of suggestive notations, the functors p_{∗} and q_{∗} do not necessarily have left adjoints.

2.6.1.2. Lemma. Let F, G be two monads in a category C. Set A = F − mod, B = G − mod, and let f : F − mod ⟶ C, g : G − mod ⟶ C be the corresponding canonical morphisms. Then the category by A ⋆_{f,g} B is equivalent to the category 𝔐_{f,g} objects of which are triples (m_{F} , M, m_{G}), where M ∈ ObC and m_{F} : F (M) ⟶ M and m_{G} : G(M) ⟶ M are structures of resp. an F -module and a G-module on M. Morphisms from (m_{F} , M, m_{G}) to (m′_{F} , M′, m′_{G}) are morphisms M ⟶ M′ which are morphisms of F -modules and G-modules. In other words, A ⋆_{f,g} B is equivalent to its full subcategory objects of which are triples (X, id, Y).

Proof. The functor quasi-inverse to the canonical embedding 𝔐_{f,g} ⟶ A ⋆_{f,g} B assigns to any object (X, ϕ, Y) of the category A ⋆_{f,g} B the module (m_{X}, f_{∗}(X), ϕ^{−1} ∘ m_{Y} G ∘ ϕ). Here m_{X} is the F -module structure of X = (m_{X}, f_{∗}(X)), and m_{Y} is the G-module structrure of Y = (m_{Y} , g_{∗}(Y)).

2.6.2. The free product of monads.

2.6.2.1. Lemma. Let (Φ, ϕ) be a monoidal functor from D^{∼} = (D, ⊗, ...) to End^{∼}C = (EndC, ∘, Id_{C}). Suppose there exists a colimit, F_{Φ}, of the functor Φ and the functors Φ(X), X ∈ ObD, preserve colimits of functors from D (say continuous). Then F_{Φ} has a canonical structure of a monad, m_{ϕ} : F_{Φ}^{2} ⟶ F_{Φ}.

If Φ takes values in the subcategory of continuous functors and C is a category with limits, then the monad (F_{Φ}, m_{ϕ}) (i.e. the functor F_{Φ}) is continuous.

Proof. For any two objects, X and Y of the category D, denote by m_{X,Y} the composition of ϕ_{X,Y} : Φ(X) ∘ Φ(Y) ⟶ Φ(X ⊗ Y) and the canonical morphism Φ(X ⊗ Y) ⟶ F_{Φ}. If Φ(X) preserves colimits, the morphisms {m_{X,Y} |Y ∈ ObD} define a morphism m_{X} : Φ(X) ∘ F_{Φ} ⟶ F_{Φ}. The family of morphisms {m_{X}|X ∈ ObD} defines a morphism m_{ϕ} : F_{Φ}^{2} ⟶ F_{Φ} which is a monad structure on F_{Φ}.

If each functor Φ(X) has a right adjoint Φ(X)^{∧}, and there exists a limit of the family of functors {Φ(X)^{∧}|X ∈ ObD}, then this limit is a right adjoint to the functor F_{Φ}.

2.6.2.2. A monoidal functor associated with two monads. Let F, G be two monads with the units resp. η_{F} and η_{G} and the multiplications μ_{F} and μ_{G}. Denote by D_{F,G} the category objects of which are ’monomials’ F^{i_{1}}G^{j_{1}}...F^{i_{k}}G^{j_{k}} in G and F, i_{k}, j_{k} ≥ 0. We set F^{0} = G^{0} = 1 and 1 · 1 = 1. Morphisms of D_{F,G} are identical morphisms and those obtained by composing morphisms of the form Xη_{G}Y, Xη_{F} Y, Xμ_{G}Y, Xμ_{F} Y. Here X, Y are monomials in F and G, and μ_{F} : F^{2} ⟶ F, η_{F} : 1 ⟶ F, μ_{G} : G^{2} ⟶ G, η_{G} : 1 ⟶ G, are morphisms such that μ_{F} ∘ F η_{F} = id_{F} = μ_{F} ∘ η_{F} F and μ_{G} ∘ Gη_{G} = id_{G} = μ_{G} ∘ η_{G}G.

We turn D_{F,G} into a strict monoidal category with the unit object 1 (’strict’ means that the associativity constraint and the left and right multiplications by the unit object are identical morphisms) by setting

(F^{i_{1}}G^{j_{1}}...F^{i_{k}}G^{j_{k}}) ⊗ (F^{i′_{1}}G^{j′_{1}}...F^{i′_{k}}G^{j′_{k}}) := F^{i_{1}}G^{j_{1}}...F^{i_{k}}G^{j_{k}}F^{i′_{1}}G^{j′_{1}}...F^{i′_{k}}G^{j′_{k}}

and defining ⊗ on the morphisms in an obvious way.

There is a natural strict monoidal functor Φ = (Φ, id) from D_{F,G} to the category EndC^{∼} = (EndC, ∘) assigning to any object of the category D_{F,G} the corresponding functor C ⟶ C.

2.6.2.3. Proposition. (a) Let C be a category with colimits of countable diagrams. And let (F, μ) and (G, ν) be monads in the category C such that the functors F and G preserve colimits of countable diagrams. Then there exists a monad, F ⋆G, associated with the monoidal functor Φ, μ : D_{F,G} ⟶ End^{∼}C.

(b) The monad F ⋆ G preserves countable colimits, in particular it is right exact.

(c) If the category C has countable limits and the monads F and G are continuous, then the monad F ⋆ G is continuous.

Proof. (a) If the category C has countable (co)limits, then the category EndC has countable (co)limits. Therefore the assertion follows from 2.6.2.1.

(b) If the monads F and G preserve countable colimits, then all compositions of F and G have this property, hence F ⋆ G, being a colimit of such compositions preserves countable colimits.

(c) The assertion follows from (the argument of) 2.6.2.1.

The monad F ⋆ G of Proposition 2.6.2.3 will be called the free product, or the starproduct of the monads F and G.

2.6.2.4. Proposition. Let f : A ⟶ C and g : B ⟶ C be ω-affine schemes over C and 𝔾_{f}, 𝔾_{g} associated monads. Suppose C is a category with countable colimits. Then 𝔾_{f} ⋆ 𝔾_{g} exists and is an ω-continuous monad.

The canonical functor A ⋆_{f,g} B ⟶ C is a direct image of an ω-affine morphism isomorphic to 𝔾_{f} ⋆ 𝔾_{g} − mod ⟶ C. The ω-affine scheme A ⋆_{f,g} B ⟶ C is a product of ω-affine schemes (A, f) and (B, g) in the category R_{c}Cat/C of bispaces over C (cf. 1.5).

Proof. By Proposition 2.6.2.3, the monad 𝔾_{f} ⋆ 𝔾_{g}(M) is continuous, hence the morphism 𝔾_{f} ⋆ 𝔾_{g} − mod ⟶ C (and isomorphic to it A ⋆_{f,g} B ⟶ C) is affine. The assertion follows now from 2.6.2.4 and 2.2.5 (see the argument of 2.5.2.4).

2.6.2.4.1. Corollary. Let f : A ⟶ C and g : B ⟶ C be affine schemes over C and 𝔾_{f}, 𝔾_{g} associated monads. Suppose C is a category with countable colimits and limits. Then 𝔾_{f} ⋆ 𝔾_{g} exists and is a continuous monad.

The canonical functor A ⋆_{f,g} B ⟶ C is a direct image of an affine morphism isomorphic to 𝔾_{f} ⋆ 𝔾_{g} − mod ⟶ C. The affine scheme A ⋆_{f,g} B ⟶ C is a product of affine schemes (A, f) and (B, g) in the category R_{c}Cat/C of bispaces over C.

2.6.3. The free product with a localization.

2.6.3.1. Proposition. Suppose C is a category with countable colimits. Let A ⟶^{f} C be an ω-affine morphism, and let B ⟶^{g} C be an ω-affine localization. Then the canonical morphism A ⋆_{f,g} B ⟶^{p} A is an ω-affine localization.

Proof. Let 𝔾 = (G, μ_{G}) be a monad in C. Suppose the morphism g : 𝔾 − mod ⟶ C (i.e. its inverse image functor g^{∗}) is a localization. This is equivalent to the following condition: the direct image functor g_{∗} : (M, m) ⟼ M is fully faithful.

The category 𝔾 ⋆ 𝔽-modules is equivalent to the category A ⋆ B of triples (ν, M, m), where (M, m) is a 𝔾-module and (M, ν) is a 𝔽-module. The direct image functor p′_{∗} of the canonical morphism p′ : 𝔾 ⋆ 𝔽 − mod ⟶ 𝔽 − mod assigns to any triple (ν, M, m) the 𝔽-module (M, ν). Clearly the functor p′_{∗} is faithful. Since the functor g_{∗} : (M, m) ⟼ M is full, the functor p′_{∗} is full. The assertion follows now from 2.6.2.4.

2.6.3.2. Corollary. Suppose C is a category with countable colimits and limits. Let f : A ⟶ C be an affine morphism, and let g : B ⟶ C be an affine localization. Then the canonical morphism p : A ⋆_{f,g} B ⟶ A is an affine localization.

2.6.3.3. Example. Suppose C is the category R − mod of left modules over an associative ring R. Let f : X ⟶ C and g : Y ⟶ C be affine morphisms. This means that X ≃ A − mod, Y ≃ B − mod, and there are ring morphisms R ⟶ A and R ⟶ B corresponding to the morphisms resp. f and g. The morphism g is an affine localization iff the following conditions hold:

(a) B is flat as a right R-module, i.e. the functor B⊗_{R} is left exact;

(b) the canonical morphism B ⊗_{R} B ⟶ B induced by the multiplication on B is an isomorphism.

It follows from 2.6.3.2 that (A ⋆_{R} B) is a flat right A-module and the morphism (A ⋆_{R} B) ⊗_{A} (A ⋆_{R} B) ⟶ A ⋆_{R} B induced by the multiplication is an isomorphism.

2.6.4. Faithfully flat morphisms. A morphism ϕ : (X, f) ⟶ (S, h) in RCat_{C} is called flat (resp. faithful, resp. faithfully flat) if a corresponding inverse image functor is exact (resp. faithful, resp. faithful and exact).

2.6.4.1. Lemma. Suppose C is a category with countable colimits.

(a) Let (X, f) ⟶^{ϕ} (S, h) ⟵^{ψ} (Y, g) be morphisms in Aff/S such that ϕ is faithful. Then the projection π : (X, f) ×_{(ϕ,ψ)} (Y, g) ⟶ (Y, g) is faithful.

(b) If both morphisms ϕ and ψ are flat, then the projection π is flat.

Proof. (i) Suppose first that (S, h) = (C, id). Then (X, f) ×_{(ϕ,ψ)} (Y, g) ≃ 𝔾_{f} ⋆ 𝔾_{g} and the π is the natural morphism 𝔾_{f} ⋆ 𝔾_{g} − mod ⟶ 𝔾_{g} − mod having 𝔾_{f} ⋆ 𝔾_{g}⊗_{𝔾_{g}} as an inverse image functor.

(a) Since the functor f_{∗} is faithful, the functor f^{∗} is faithful iff G_{f} := f_{∗}f^{∗} is faithful. It follows from 2.6.2.3 that the composition of this functor with the forgetful functor p_{∗} : 𝔾_{f} ⋆ 𝔾_{g} − mod ⟶ C maps a 𝔾_{g}-module (M, m) into G_{f}(M) ⊕ G′(M) for a certain functor G′. Since the functor G_{f} is faithful, the functor G_{f} ⊕ G′ is faithful.

(b) If both functors g^{∗} and f^{∗} are exact, then by 2.6.2.3 the functor G_{p} is exact. (ii) Replacing the category C by the category S ≃ 𝔾_{h} − mod and A by the corre-

sponding category of 𝔾_{h}-bimodules, we reduce the assertion to the case (S, h) = (C, id). Details are left to the reader.

3. Affine schemes in a monoidal category. Quasi-coherent modules.

Affine schemes in a monoidal category A^{∼} are defined as the category opposite to the category of algebras (in the terminology of [ML] monoids) in A^{∼}. A relation with ’spaces over S’ appeares when we have an action of the monoidal category A^{∼} on the category S. Taking as A^{∼} the monoidal subcategory End_{𝔖}^{∼}S of endofunctors preserving colimits of the type 𝔖, or the subcategory End_{c}^{∼}S of continuous endofunctors (cf. 2.3), we recover back 𝔖-affine and affine schemes over S.

3.0. Preliminaries: algebras and (bi)modules in monoidal categories. Fix a monoidal category A^{∼} = (A, ⊗, 1, a, l, r). Here A is a category, ⊗ is a functor from A×A to A, a is a functor isomorphism ⊗∘(Id_{A}×⊗) ⟶ ⊗∘(⊗×Id_{A}) (associativity constraint), and l : Id_{A} ⟶ 1⊗Id_{A}, r : Id_{A} ⟶ Id_{A} ⊗1 are functor isomorphisms compatible with the associativity constraint a. An algebra (or monoid) in A^{∼} is a pair (R, μ) where R ∈ ObA and μ is a morphism R⊗R ⟶ R such that μ∘(μ⊗id_{R})∘a_{R,R,R} = μ∘(id_{R}⊗μ). The unit of an algebra (R, μ) is a morphism η : 1 ⟶ R such that μ∘η⊗id_{R}∘l_{R} = id_{R} = μ∘id_{R}⊗η∘r_{R}. The unit (if it exists) is unique. We assume that all algebras considered here are unital. Algebras in A^{∼} form a category which we denote by AlgA^{∼}.

A left module over an algebra (R, μ) is a pair (M, m), where M ∈ ObA, m is a morphism R⊗M ⟶ M such that m∘id_{R}⊗m = m∘μ⊗id_{M} ∘a_{R,R,M} and m∘η⊗id_{M} = id_{M}. Left modules over R^{∼} = (R, μ) form a category R^{∼} − mod.

The category mod − R^{∼} of right R^{∼}-modules is defined in an obvious way. Note that right modules are just left modules in the opposite monoidal category. A triple (m, M, m′), where (m, M) and (M, m′) are resp. left and right R^{∼}-modules is called an R^{∼}-bimodule if m ∘ id_{R} ⊗ m′ = m′ ∘ m ⊗ id_{R} ∘ a_{R,M,R}.

Suppose that for any X ∈ ObA, the functor X ⟼ X ⊗ − preserves cokernels of reflexive pairs of arrows. Then there is a well defined functor ⊗_{R} : mod−R^{∼}×R^{∼}−mod ⟶ A which assigns to any pair of resp. right and left R^{∼}-modules (M, m), (ν, N) the cokernel of the pair of morphisms id_{M} ⊗ν, m⊗ν∘a_{M,R,N} : M ⊗(R⊗N) ⟶ M ⊗N. The functor ⊗_{R} induces a structure of a monoidal category on the category R^{∼} − bimod of R^{∼}-bimodules.

Let β be a symmetry of the monoidal category A^{∼}. An algebra R^{∼} = (R, μ) is called β-commutative (or commutative if β is fixed) if μ ∘ β_{R,R} = μ. The full subcategory of AlgA^{∼} formed by β-commutative algebras will be denoted by Alg_{β}A^{∼}.

For any β-commutative algebra R^{∼}, the map (m, M) ⟼ (m, M, m ∘ β_{M,R}) defines a functor, Δ_{β}, identifying the category R^{∼} −mod of left R^{∼}-modules with a full subcategory of the category R^{∼} − bimod of R^{∼}-bimodules.

Suppose the functor X ⟼ X ⊗ − is right exact for any X ∈ ObA. Then the functor Δ_{β} identifies R^{∼} − mod with a monoidal subcategory of R^{∼} − bimod. And the symmetry β induces a symmetry on R^{∼} − mod.

3.1. Noncommutative affine schemes in a monoidal category. Fix a monoidal category A^{∼} = (A, ⊗, 1, a). We define the category Aff_{A^{∼}} of affine schemes in A^{∼} as the opposite category to the category of algebras: Aff_{A^{∼}} := (AlgA^{∼})^{op}. Thus if A^{∼} = (k − mod, ⊗_{k}, k) for some commutative ring k, then Aff_{A^{∼}} is the category opposite to the category Alg_{k} of k-algebras.

We denote by Spec(R^{∼}), or Spec_{A^{∼}}(R^{∼}) the object of Aff_{A^{∼}} corresponding to the algebra R^{∼}.

3.1.1. Monoidal functors and affine schemes. Recall that a monoidal functor from a monoidal category A^{∼} = (A, ⊗, 1, a, l, r) to a monoidal category A^{′∼} = (A′, ⊗′, 1′, a′, l′, r′) is a triple Φ^{∼} = (Φ, ϕ, ϕ_{0}), where Φ is a functor A ⟶ A′, ϕ is a functor morphism {ϕ_{X,Y} : Φ(X) ⊗′ Φ(Y) ⟶ Φ(X ⊗ Y), ϕ_{0} a morphism 1′ ⟶ Φ(1 satisfying natural compatibility conditions.

Any monoidal functor Φ^{∼} : A^{∼} ⟶ A′ induces a functor

Φ_{Alg}^{∼} : AlgA^{∼} ⟶ AlgA^{′∼}, (R, m) ⟼ Φ^{∼}(R, m) := (Φ(R), Φ(m) ∘ ϕ_{R,R}), (1)

hence a functor

Φ_{Aff}^{∼} : Aff_{A^{∼}} ⟶ Aff_{A^{′∼}}

3.2. Coproducts and fiber products of affine schemes. Fix a monoidal category A^{∼} = (A, ⊗, 1, a).

3.2.1. Lemma. Suppose that the category A has products of |J| objects. Then the category AlgA^{∼} has products of |J| objects.

Proof. Let R_{i}^{∼} = (R_{i}, μ_{i}), i ∈ J, be a set of algebras in A^{∼}. By assumption, there exists a product ∏_{i∈J} R_{i} of the set of objects {R_{i}| i ∈ J}, and the canonical projections p_{j} : ∏_{i∈J} R_{i} ⟶ R_{j} together with multiplications μ_{j} provide morphisms

∏_{i∈J} R_{i} ⊗ ∏_{i∈J} R_{i} ⟶^{p_{j}⊗p_{j}} R_{j} ⊗ R_{j} ⟶^{μ_{j}} R_{j}, j ∈ J.  (1)

By the universal property of products, there exists a unique morphism μ : ∏_{i∈J} R_{i} ⊗ ∏_{i∈J} R_{i} ⟶ ∏_{i∈J} R_{i} such that for any i ∈ J, p_{i} ∘ μ = μ_{i} ∘ p_{i} ⊗ p_{i}. We leave to the reader to check that thus defined morphism μ is a structure of an associative unital algebra on ∏_{i∈J} R_{i} and that the algebra (∏_{i∈J} R_{i}, μ) is a product of the set of algebras {R_{i}^{∼} = (R_{i}, μ_{i}), i ∈ J}.

3.2.2. Proposition. Suppose A^{∼} = (A, ⊗, 1, a) is a monoidal category such that A has countable coproducts. Then

For any algebras R^{∼} = (R, μ) and S^{∼} = (S, ν) in A^{∼} such that the functors R ⊗ − and S ⊗ − are compatible with countable coproducts, there exists a free product R^{∼} ⋆ S^{∼} which is a coproduct in the category AlgA^{∼}. In particular, Spec_{A^{∼}}(R^{∼} ⋆ S^{∼}) is a product of Spec_{A^{∼}}(R^{∼}) and Spec_{A^{∼}}(S^{∼}) in the category Aff_{A^{∼}} of affine schemes in A^{∼}.

Proof. Let 𝔏_{A^{∼}}^{∼} = (𝔏, a) be the canonical monoidal functor ”of left multiplication”

A^{∼} ⟶ End^{∼}A, M ⟼ M ⊗ −

The monoidal functor 𝔏_{A^{∼}}^{∼} induces a functor from AlgA^{∼} to 𝔐𝔬𝔫A which assigns to algebras R^{∼} = (R, μ) and S^{∼} = (S, ν) the monads (𝔏(R), μ_{R}) and (𝔏(S), ν_{S}) such that the functors 𝔏(R) and 𝔏(S) preserve countable coproducts. The assertion follows now from Proposition 2.6.2.3.

3.2.3. Proposition. Let A be a category with countable colimits. And suppose that for any M ∈ ObA, the functor M ⊗ − : A ⟶ A is compatible with countable colimits. Then the category Aff_{A^{∼}} of affine schemes in A^{∼} has fiber products.

Proof. Let R^{∼} ⟵ S^{∼} ⟶ T^{∼} be morphisms of AlgA^{∼}. Since the functor M ⟼ M ⊗ − preserves cokernels of coreflexive pairs of arrows, we have well defined monads 𝔽 = R^{∼} ⊗_{S^{∼}} − and 𝔾 = T^{∼} ⊗_{S^{∼}} − on the category S^{∼} − mod of left S^{∼}-modules.

The fact that the functors R ⊗ − and T ⊗ − preserve countable coproducts implies that the functors R ⊗_{S^{∼}} − and T ⊗_{S^{∼}} − from S^{∼} − mod to S^{∼} − mod have the same property.

By 2.6.2.3, there exists the coproduct 𝔽⋆𝔾 which we denote by (F′, μ′). To the monad (F′, μ′) on S^{∼} − mod, there corresponds a monad (f_{∗}F′f^{∗}, μ′_{f}) on A. Here f_{∗} denotes the canonical forgetful functor S^{∼} − mod ⟶ A and f^{∗} is its left adjoint: L ⟼ (S ⊗ L, m_{L}), where m_{L} is the canonical action induced by the multiplication on S. The multiplication μ′_{f} is the composition of the canonical morphism f_{∗}F′f^{∗}f_{∗}F′f^{∗} ⟶ f_{∗}F^{′2}f^{∗} induced by the adjunction morphism f^{∗}f_{∗} ⟶ Id and the morphism f_{∗}μ′f^{∗} : f_{∗}F^{′2}f^{∗} ⟶ f_{∗}F′f^{∗}. With the monad (f_{∗}F′f^{∗}, μ′_{f}) one can associate the algebra (f_{∗}F′f^{∗}(1), μ′′) = (f_{∗}F′(S), μ′′), where μ′′ is a naturally defined multiplication. This algebra is a fiber coproduct of R^{∼} and T^{∼} over S^{∼}. A less formal (more constructive) argument uses the explicit description of f_{∗}F′(S) in terms of tensor product of copies of R^{∼} and S^{∼} over S^{∼} (following 2.6.2) in which the multiplication appears immediately and the universal property is evident. We leave this argument to the reader.

3.2.4. Example. Let A^{∼} = (k − mod^{∼}) = (k − mod, ⊗_{k}, k) for a commutative associative ring k. Algebras in k − mod^{∼} are just k-algebras, and the coproduct (or ⋆-product) of algebras A and B is their free product. For instance, the coproduct of two copies of polynomial algebra in one variable is isomorphic to the free algebra in two variables: k[x] ⋆ k[y] ≃ k⟨x, y⟩.

3.3. A ’base space’ and quasi-coherent modules. A link between the formal notion of an affine scheme in a monoidal category and a more geometric notion of an affine scheme over a category C is given by fixing an action of the monoidal category on the category C, i.e. a triple (A^{∼}, C, Φ^{∼}), where C is a category, A^{∼} = (A, ⊗, 1, a) is a monoidal category, Φ^{∼} = (Φ, ϕ) is a monoidal functor A^{∼} ⟶ End_{2}^{∼}C. Recall that End_{2}C denotes the full subcategory of the category EndC objects of which are functors compatible with cokernels of reflexive pairs of morphisms (cf. 1.2.2). In other words, Φ^{∼} is a unital action of the monoidal category A^{∼} on a category C which preserves cokernels of reflexive pairs of arrows.

We shall call the triple (A^{∼}, C, Φ^{∼}) a ’base space’, or simply a base.

3.3.1. Quasi-coherent modules on an affine scheme. The action Φ^{∼} induces a functor, R^{∼} ⟼ Φ^{∼}(R^{∼}), from the category AlgA^{∼} to the category of monads on the category C (cf. 3.1.1(1)). In particular, we have the category Φ^{∼}(R^{∼})−mod_{C} of Φ^{∼}(R^{∼})- modules which we call simply R^{∼}-modules in C, or quasi-coherent O_{Spec(R^{∼})}-modules over ℭ, and denote by Qcoh_{Spec(R^{∼})/ℭ}.

3.3.2. A fibered category associated with a base. We associate with ℭ a cofibered category F^{ℭ} ⟶^{π} (AlgA^{∼})^{op} = Aff_{A^{∼}} in a natural way: objects of the category F^{ℭ} are pairs (R^{∼}, M), where R^{∼} = (R, μ) is an algebra in A^{∼}, M is a Φ^{∼}(R^{∼})-module. A morphism from (R^{∼}, M) to (S^{∼}, L) is a pair (ψ, ξ), where ψ is an algebra morphism R^{∼} ⟶ S^{∼}, ξ is a morphism M ⟶ ψ_{∗}(L), where ψ_{∗} is the pull-back functor Φ^{∼}(R^{∼}) − mod_{C} ⟶ Φ^{∼}(S^{∼}) − mod_{C} induced by the algebra morphism ψ. The composition is defined in a standard way. The projection π assigns to any object (R^{∼}, M) the algebra R^{∼} and to any morphism (ψ, ξ) the algebra morphism ψ. It follows from the assumptions and 1.0.6.1 that F^{ℭ} ⟶^{π} Aff_{A^{∼}} is a bifibered category: the functors ψ_{∗} have left adjoints.

3.4. Examples.

3.4.1. A typical noncommutative example is the monoidal category R − bimod^{∼} = (R − bimod, ⊗_{R}, R) of bimodules over an associative ring R acting (by tensoring over R) on the category of left R-modules. For any algebra S in the monoidal category R−bimod^{∼} (that is a ring morphism R ⟶ S), the category of quasi-coherent modules on Spec(S)/ℭ is naturally equivalent to the category S − mod of left S-modules.

3.4.2. The left and right ’base spaces’ of a monoidal category. Let A^{∼} = (A, ⊗, 1) be a monoidal category such that for any X ∈ ObA, the functor X ⊗ − : V ⟼ X ⊗ V preserves cokernels of reflexive pairs of morphisms. The ’left base space’ of the monoidal category A^{∼} is the triple (A, A^{∼}, (𝔏, a)), where 𝔏 = 𝔏_{A^{∼}} is the functor A ⟶ EndA, X ⟼ X ⊗−, a is the associativity constraint. The functor Φ^{∼} is faithful, because X ⟼ X ⊗ 1 is a faithful functor.

Replacing the action X ⟼ X ⊗ − by X ⟼ − ⊗ X, we obtain the notion of the right base space of the monoidal category A^{∼}.

3.4.3. Operads and algebras over operads. Fix a symmetric additive monoidal category C^{∼} = (C, ⊗, 1, a, l, r; β) (here β is a symmetry, β_{X,Y} : X ⊗ Y ⟶ Y ⊗ X). Let S denote the category objects of which are sets [n] := {1, ..., n}, n ≥ 1, and [0] := ∅ and morphisms are bijections. Denote by C^{S} the category of functors S^{op} ⟶ C. In other words, objects of C^{S} are collections M = (M(n)| n ≥ 0), where M_{n} is an object of C with an action of the symmetric group S_{n}.

The category C^{S} acts on the category C by polynomial functors:

M : V ⟼ M(V) = ⊕_{n≥0}M(n) ⊗_{S_{n}} V^{⊗n}  (1)

The composition of polynomial functors is again a polynomial functor. This defines a tensor product, ⊙, on C^{S} called the plethism product. We denote the corresponding monoidal category (C^{S}, ⊙, 1_{S}) by C^{∼S}. Here 1_{S} is the unit object 1_{S}. One can see that 1_{S}(n) = 0 if n ≠ 1 and 1_{S}(1) is the unit object of the category C^{∼}. Thus we have an action ℭ of the monoidal category C^{S} on the category C.

Algebras in the monoidal category C^{S} are called operads or C^{∼}-operads. For each operad R, the corresponding category of quasi-coherent sheaves on Spec(R) is usually called the category of R-algebras.

3.5. Affine schemes and relative affine schemes. Fix a base ℭ = (A^{∼}, C, Φ^{∼}). Let Spec_{ℭ} denote the functor (AlgA^{∼})^{op} = Aff_{A^{∼}} ⟶ RCat/C which assigns to any affine scheme X = Spec(R^{∼}) in A^{∼} the object (Φ^{∼}(R^{∼}) − mod_{C}, f_{C}) := (Qcoh_{X/ℭ}, f_{C}), where f_{C} is the canonical (𝔖_{2}-affine) morphism Φ^{∼}(R^{∼}) − mod_{C} ⟶ C, and to any affine morphism Spec(R^{∼}) ⟶ Spec(S^{∼}) the (𝔖_{2}-affine) morphism direct image of which is the corresponding pull-back functor. There is a canonical morphism from the fibered category F^{ℭ} ⟶^{π} (AlgA^{∼})^{op} to the fibered category RCat_{2}^{∼C} which assigns to any object (R^{∼}, M) of F^{ℭ} the object (Spec_{ℭ}(R^{∼}), M).

3.5.1. ℭ-affine morphisms. Morphisms of the form Spec_{ℭ}(ψ) will be called ℭaffine, or, loosely, affine. They are not usually affine in the sense of 2.3, as one can see taking as a base the monoidal category End_{2}^{∼}C and as Φ^{∼} the identical monoidal functor.

Let ℭ be the ’left base space’ (A^{∼}, 𝔏^{∼}) of a monoidal category A^{∼} satisfying the conditions of 3.4.2.

3.5.2. Lemma. Let R = (R, μ) and S = (S, ν) be algebras in A^{∼} and ψ an algebra morphism S ⟶ R. The morphism Spec_{ℭ}(ψ) : R − mod ⟶ S − mod is affine iff the inner hom Hom_{S}(R, M) exists for all left S-modules M.

Proof. Suppose the inner hom Hom_{S}(R, M) exists for all left S-modules M. It has a canonical left action of R. For any left R-module L, we have a canonical functorial isomorphism

Hom_{R}(L, Hom_{S}(ψ_{∗}(R), M) ⟶^{∼} Hom_{S}(ψ_{∗}(R ⊗_{R} L), M) = Hom_{S}(ψ_{∗}(L), M) (1)

which shows that the functor M ⟼ Hom_{S}(ψ_{∗}(R, M) is a right adjoint to the direct image functor ψ_{∗}. The isomorphism (1) can be regarded as a definition of Hom_{S}(ψ_{∗}(R, M). In particular, the existence of a right adjoint to ψ_{∗} implies that of Hom_{S}(ψ_{∗}(R), M).

3.5.2.1. Note. In the case A^{∼} = k − mod^{∼} := (k − mod, ⊗_{k}, k) for a commutative ring k, algebras in A^{∼} are k-algebras and, given an algebra morphism S ⟶ R = (R, μ) and an S-module M, the inner hom, Hom_{S}(R, M), coincides with Hom_{S}(R, M).

3.6. The base change. A morphism from a base ℭ′ = (A^{′∼}, C′, Φ^{′∼}) to a base ℭ = (A^{∼}, C, Φ^{∼}) is a triple (Ψ^{∼}, g^{∗}; λ), where Ψ^{∼} is a monoidal functor A^{′∼} ⟶ A^{∼} and g^{∗} an inverse image functor of a morphism C′ ⟶ C, and λ a functorial isomorphism C(M, ΦΨ(E′)(L)) ⟶^{∼} C′(g^{∗}(M), Φ′(E′)(g^{∗}(L))). We leave to the reader defining the composition. The main example is as follows.

3.6.1. Fix a ’base space’ ℭ = (A^{∼}, C, Φ^{∼}) such that the category C has cokernels of reflexive pairs of arrows. Let R^{∼} = (R, μ) be an algebra in the monoidal category A^{∼}. We assume that for any M ∈ ObA, the functors M ⊗ − and − ⊗ M preserve cokernels of reflexive pairs of arrows. In this case we have a well defined monoidal category R^{∼} − bimod^{∼} = (R^{∼} − bimod, ⊗_{R^{∼}}−, R^{∼}). Moreover, there is a naturally defined action of the monoidal category R^{∼} − bimod^{∼} on the category Φ^{∼}(R^{∼}) − mod_{C}.

In fact, for any R^{∼}-bimodule M = (m′, M, m′′) and for any Φ^{∼}(R^{∼})-module (L, ν), the action of M on (L, ν) assigns to (L, ν) the Φ^{∼}(R^{∼})-module Φ^{∼}(M) ⊗_{Φ^{∼}(R^{∼})} L. The latter is the cokernel of the pair of arrows Φ^{∼}(M)Φ^{∼}(R^{∼})(L) ⇉ Φ^{∼}(M)(L), where one arrow is the composition of the constraint

ϕ_{M,R}(L) : Φ^{∼}(M)Φ^{∼}(R^{∼})(L) ⟶ Φ^{∼}(M ⊗ R^{∼})(L)

and the morphism Φ^{∼}(m′′)(L) induced by the right action of R^{∼} on M. The other arrow is the image Φ^{∼}(M)(ν) of the left action on L.

We have a morphism g : R^{∼} − mod ⟶ C = A with the inverse image functor g^{∗} : L ⟼ R^{∼} ⊗ L and the functor Ψ : A ⟶ R^{∼} − bimod, M ⟼ R^{∼} ⊗ M ⊗ R^{∼}. The latter extends canonically to a monoidal functor Ψ^{∼} : A^{∼} ⟶ R^{∼} − bimod^{∼}.

3.7. The fibered category of bimodules. Fix a monoidal category A^{∼} such that for all M ∈ ObA, the functors M ⊗ − preserve colimits of reflexive pairs of arrows. We associate with A^{∼} a cofibered category Bi^{A^{∼}} ⟶^{π} (AlgA^{∼})^{op} = Aff_{A^{∼}} defined as follows. Objects of the category Bi^{A^{∼}} are pairs (R^{∼}, M), where R^{∼} = (R, μ) is an algebra in A^{∼}, M is an R^{∼}-bimodule. A morphism from (R^{∼}, M) to (S^{∼}, L) is a pair (ψ, ξ), where ψ is an algebra morphism R^{∼} ⟶ S^{∼}, ξ is an S^{∼}-bimodule morphism M ⟶ ψ_{∗}(L), where ψ_{∗} is the pull-back functor S^{∼}−bimod ⟶ R^{∼}−bimod induced by the algebra morphism ψ. The composition is defined in a standard way. The projection π assigns to any object (R^{∼}, M) the algebra R^{∼} and to any morphism (ψ, ξ) the algebra morphism ψ. It follows from the assumptions and 1.0.6.1 that Bi^{A^{∼}} ⟶^{π} Aff_{A^{∼}} is a bifibered category: the functors ψ_{∗} have left adjoints. For any R^{∼} ∈ ObAlgA^{∼}, the fiber R^{∼} − bimod over Spec(R^{∼}) will be regarded as a monoidal category with respect to ⊗_{R^{∼}}. For any algebra morphism ψ : R^{∼} ⟶ S^{∼}, the corresponding direct image functor ψ_{∗} is a monoidal functor (in a week sense). This monoidal structure is inherited by the fibered category 𝔔𝔠𝔬𝔥(Bi^{A^{∼}}) of quasi-coherent morphisms of Bi^{A^{∼}} (cf. 1.0.1) and, therefore, by the category Qcoh(Bi^{A^{∼}}) of quasi-coherent presheaves of bimodules (see 1.0.2).

Fix a ’base space’ ℭ = (A^{∼}, C, Φ^{∼}) such that the category C has cokernels of reflexive pairs of arrows. It follows from 3.6 that the action Φ^{∼} of the monoidal category A^{∼} on the category C induces an action of the fibered category Bi^{A^{∼}} on the fibered category F^{ℭ} defined in 3.4. This action induces an action of 𝔔𝔠𝔬𝔥(Bi^{A^{∼}}) on the corresponding fibered category 𝔔𝔠𝔬𝔥(F^{ℭ}) of quasi-coherent morphisms.

3.8. Fibered categories and quasi-coherent presheaves associated with a functor from Aff_{A^{∼}}. Let G be a functor Aff_{A^{∼}} := (AlgA^{∼})^{op} ⟶ S. For any X ∈ ObS, we have a natural functor G/X ⟶ Aff_{A^{∼}} and the corresponding pull-back, F^{ℭ,G/X} (see 1.0.4), of the fibered category F^{ℭ} (cf. 3.4). In particular, we have the category P Qcoh(F^{ℭ,G/X}) of quasi-coherent presheaves on X (cf. 1.0.4). Since the construction is functorial in X, it gives a rise to a fibered category over S having P Qcoh(F^{ℭ,G/X}) as a fiber at an object X.

Similarly, one can define the monoidal category of quasi-coherent bimodules on each object X of the category S as the category of quasi-coherent presheaves of the fibered category π_{G/X} : Bi^{A^{∼},G/X} ⟶ Aff_{A^{∼}} obtained as the pull-back by the canonical functor G/X ⟶ Aff_{A^{∼}}.

3.8.1. Quasi-coherent presheaves on a presheaf of sets. The case of a particular interest is when the functor G is the canonical embedding

h : Aff_{A^{∼}} ⟶ F un(Aff_{A^{∼}}^{op}, Sets), X ⟼ Aff_{A^{∼}}(−, X)

of the category Aff_{A^{∼}} to the category of presheaves of sets on Aff_{A^{∼}}. We have then the notion of the category P Qcoh_{X} of quasi-coherent presheaves on a presheaf of sets X and the monoidal category P Qcoh(Bi_{X}) of quasi-coherent bimodules on X together with an action of P Qcoh(Bi_{X}) on P Qcoh_{X}.

4. Locally affine 𝔗-spaces.

The contents of this section is a version of a formalism of glueing schemes and algebraic spaces from affine schemes [GrD], [A1], [Kn]. It is convenient for us to develop this formalism in a general setting, starting from a version of a Grothendieck topology (like in the first chapter of [Kn]).

The known examples of noncommutative locally affine spaces appear either as categories (of quasi-coherent sheaves) over a base category having affine covers, like, for instance, the flag variety of a quantized enveloping algebra (see [R3]), or as sheaves of sets on the category of noncommutative affine schemes, like projective spaces and Grassmannians introduced in [KR]. Respectively, we introduce the notions of a scheme and that of a locally affine space (noncommutative projective spaces of [KR] are locally affine spaces, but not schemes) in two different settings. Their common feature is as follows. We start from a category A with a subcanonical (quasi-)topology 𝔗 on A and a functor F : A ⟶ B. In terms of this data, we define the category of ’locally affine’ objects. In our applications the category A is the category of Aff_{A^{∼}} of affine schemes in a monoidal category A^{∼} and B is a bigger category of ’spaces’. The two basic settings are as follows:

(a) B is the category of sheaves on (A, 𝔗) and F is the canonical embedding.

(b) B = RCat/C for some category C and the functor F is determined by an action of the monoidal category A^{∼} on C.

The model commutative example of this setting is the site of affine schemes with étale or Zariski topology, the functor F being the embedding of the category of affine schemes into the category of locally ringed spaces. In this case the definition of (𝔗|F)-schemes coincides with the standard definition (i.e. the one via affine covers) of schemes. If 𝔗 is the étale topology, locally affine (𝔗|F)-spaces are algebraic spaces in the sense of Artin.

The main noncommutative setting is the site of affine A^{∼}-schemes with the fpqc topology corresponding to a fixed base ℭ = (A^{∼}, C, Φ^{∼}) and the embedding of the category of noncommutative affine schemes into the category RCat/C of ’spaces’ over C.

Section 4.1 contains preliminaries on sites.

In Sections 4.2, 4.3 we introduce a version of the fpqc topology and that of Zariski topology on the category of noncommutative affine schemes depending on a fixed base (i.e. on an action of a monoidal category on a category) and prove certain descent property of these topologies.

In Section 4.5 we define locally affine spaces and schemes on a subcanonical quasisite (A, 𝔗) (i.e. the topology 𝔗 is subcanonical) called respectively locally affine 𝔗-spaces and 𝔗-schemes. Informally speaking, these are sheaves of sets on the site (A, 𝔗) glued of representable sheaves.

In Sections 4.5 and 4.6, given a site (A, 𝔗) and a functor F : A ⟶ B, we introduce the category of (𝔗|F)-spaces and its full subcategories formed resp. by locally affine (𝔗|F)- spaces and by (𝔗|F)-schemes.

In section 4.6 we establish a connection between (locally affine) (𝔗|F)-spaces and (𝔗|F)-schemes and (locally affine) 𝔗-spaces and 𝔗-schemes. This connection is given by a natural functor from the category of (𝔗|F)-spaces to the category of 𝔗-spaces which are by definition sheaves of sets on the site (A, 𝔗).

4.1. Preliminaries on sites. Let A be a category. A sieve of A is a subcategory R of A such that for any X ∈ ObR all morphisms Y ⟶ X belong to R (in particular, R is a full subcategory of A). If F : A′ ⟶ A is a functor and R a sieve of A, then R^{F} := F^{−1}(R) is a sieve of A′ called inverse image of R by F.

An arrow f : X ⟶ Y in A induces a functor A/f : A/X ⟶ A/Y. If R is a sieve in A/Y, let R^{f} denote its inverse image by A/f. For a family of arrows U = (U_{i} ⟶ U| i ∈ J), denote by R_{U} the full subcategory of R/U objects of which are those X ⟶ U that there exists a morphism (X ⟶ U) ⟶ (U_{i} ⟶ U) for some i ∈ J. This is a sieve called the sieve of A/U generated by U.

If f : V ⟶ U and there exist U_{i} ×_{U} V, the inverse image R_{U}^{f} of R_{U} by f is the sieve generated by projections (U_{i} ×_{U} V ⟶ V | i ∈ J).

4.1.1. Definition of a quasi-topology. A quasi-topology τ on a category A is a function which assigns to any object X of the category A a set Cov_{τ} (X) of families of arrows U = {U_{i} ⟶ U| i ∈ J} such that

(a) For any X ∈ ObA, (id_{X}) ∈ Cov_{τ} (X).

(b) If {u_{i} : U_{i} ⟶ X|i ∈ J} ∈ Cov_{τ} (X) and {u_{ij} : U_{ij} ⟶ U_{i}|j ∈ J_{i}} ∈ Cov_{τ} (U_{i}) for any i ∈ J, then {u_{i} ∘ u_{ij} : U_{ij} ⟶ X|i ∈ J, j ∈ J_{i}} ∈ Cov_{τ} (X).

The cover {u_{i} ∘ u_{ij} : U_{ij} ⟶ X|i ∈ J, j ∈ J_{i}} is called a refinement of the cover {u_{i} : U_{i} ⟶ X|i ∈ J}.

(c) Any two covers of an object have a common refinement.

4.1.2. Continuous morphisms and continuous quasi-topologies. Given a quasi-topology τ on the category A, we call a morphism f : X ⟶ Y continuous, or τcontinuous, if for any U ∈ Cov_{τ} (Y), the sieve R_{U}^{f} of A/X (cf. 4.1.0) contains an element of Cov_{τ} (X).

We call a quasi-topology τ continuous if any morphism of A is continuous.

4.1.2.1. The subcategory A^{τ}. Fix a quasi-topology τ on A.

Denote by A^{τ} the class of all morphisms of A which belong to some τ-cover. It follows from 4.1.1(a) and 4.1.1(b) that A^{τ} is a subcategory of A. Clearly τ induces a quasi-topology on A^{τ}. The property 4.1.1(c) implies that this quasi-topology is continuous.

4.1.2.2. The subcategory A_{τ}. Denote by A_{τ} the class of all morphisms f : X ⟶ Y of A such that for any τ-cover U of Y, the sieve R_{U}^{f} (cf. 4.1) contains a τ-cover of X. Clearly the composition of morphisms of A_{τ} belongs to A_{τ} as well as all identical morphisms of the category A. This shows that A_{τ} is a subcategory of A and ObA_{τ} = ObA. The subcategory A_{τ} contains the subcategory A^{τ} (see 4.1.2.1), hence the quasi-topology τ induces a continuous quasi-topology on A_{τ}. Clearly, A_{τ} is a maximal subcategory of A having this property. In particular, if A_{τ} = A iff τ is continuous.

4.1.3. Continuous quasi-topologies and topologies.

4.1.3.1. Recall that a topology 𝔗 on a category A is a map which assigns to each X ∈ ObA a non-empty set 𝔗(X) of sieves of the category A/X such that

(a) For any morphism f : Y ⟶ X of A and any R ∈ 𝔗(X), R^{f} ∈ 𝔗(Y).

(b) For any S ∈ ObA and any R ∈ 𝔗(S), any sieve R_{1} of A/S such that R_{1}^{f} ∈ 𝔗(Y)

for all (Y ⟶^{f} S) ∈ R belongs to 𝔗(S).

For any X ∈ ObA, the elements of 𝔗(X) are called refinements of X. The pair (A, 𝔗) is called site.

One checks easily that for any X ∈ ObA, the intersection of two refinements of X is a refinement of X and that any sieve of A/X which contains a refinement is a refinement. In particular A/X is a refinement.

4.1.3.2. Covers of a topology. A set of arrows U = {U_{i} ⟶ U| i ∈ J} is called a cover in the topology 𝔗 if the sieve R_{U} generated by U belongs to 𝔗(U).

4.1.3.3. A topology associated with covers. Let τ be a function which assigns to any object X of the category A a set Cov_{τ} (X) of families of arrows U = {U_{i} ⟶ U| i ∈ J} such that for any X ∈ ObA, (id_{X}) ∈ Cov_{τ} (X). The topology generated by τ is the coarsest topology for which all U ∈ Cov_{τ} (X), X ∈ ObA, are covers. We denote this topology by 𝔗_{τ} and call τ a base of the topology 𝔗_{τ}.

4.1.3.4. Lemma. Suppose that τ is a continuous quasi-topology. Then the topology 𝔗_{τ} generated by τ is described as follows. For any X ∈ ObA, 𝔗_{τ} (X) consists of all sieves R of A/X containing some element of Cov_{τ} (X).

Proof. Denote by 𝔗 the function which assigns to any X ∈ ObA the set of sieves R of A/X such that R contains some U ∈ Cov_{τ} (X). Obviously 𝔗(X) ⊂ 𝔗_{τ} (X) for all X ∈ ObA. To prove the inverse inclusion it suffices to show that the function 𝔗 is a topology, i.e. it satisfies the conditions (a) and (b) of the definition of topology (cf. 4.1.3.1).

The condition (a) in follows from the fact that all morphisms of A are τ-continuous. It remains to check the condition (b) of 4.1.3.1.

Let R ∈ 𝔗(X), i.e. R is a sieve of A/X containing some U ∈ Cov_{τ} (X). And let R_{1} be a sieve of A/X such that R_{1}^{f} ∈ 𝔗 for any f ∈ R. In particular for any (U_{f} ⟶^{f} X) ∈ U, the sieve R_{1}^{f} contains some U^{f} ∈ Cov_{τ} (U_{f}). This implies (by definition of R_{1}^{f}) that R_{1} contains the composition of U and {U^{f}| f ∈ U}. Since this composition belongs to Cov_{τ} (X) by the definition of a base of a topology, R_{1} ∈ 𝔗(X). Thus 𝔗 is a topology, hence the assertion.

4.1.4. Pretoplogies. Recall that a quasi-topology τ is called a pretopology if the following condition holds:

(c) For any morphism V ⟶ X and cover 𝔘 = {u_{i} : U_{i} ⟶ X|i ∈ J} ∈ Cov_{τ} (X), there exist fiber products U_{i} ×_{X} V for all i ∈ J, and the set of canonical projections 𝔘 ×_{X} V := {p_{i} : U_{i} ×_{X} V ⟶ V |i ∈ J} belongs to Cov_{τ} (V).

Clearly any pretopology is continuous.

4.1.5. Presheaves and sheaves. A presheaf of sets on a category B is any functor B^{op} ⟶ Sets. A presheaf of sets on a quasi-site (A, τ) is a presheaf on A.

A presheaf of sets F on a quasi-site (A, τ) is called a sheaf if for any X ∈ ObA and any U ∈ Cov_{τ}, the natural morphism F (X) ⟶ lim(F |R_{U}) is an isomorphism. Here F |R_{U} is the presheaf of sets on the sieve R_{U} generated by U which assigns to any object Y ⟶ X of R_{U} the set F (Y) and to any morphism (Y ⟶ X) ⟶^{λ} (Y′ ⟶ X) of R_{U}, the map F (λ) : F (Y′) ⟶ F (Y).

4.1.5.1. Proposition. Let τ be a quasi-topology on the category A such that for any cover {U_{i} ⟶ U| i ∈ J} ∈ τ(U), all fiber products U_{i} ×_{U} U_{j}, i, j ∈ J, exist. A presheaf of sets F on the category A is a sheaf for the quasi-topology 𝔗_{τ} generated by τ iff for any X ∈ ObA and for any cover {U_{i} ⟶ U| i ∈ J}, the diagram of sets

F (U) ⟶ ∏_{i∈J} F (U_{i}) ⇉ ∏_{i,j∈J} F (U_{i} ×_{U} U_{j})

is exact.

4.1.5.2. Subcanonical quasi-topologies. Any quasi-topology 𝔗 on A such that all representable presheaves are sheaves on the quasi-site (A, τ), is called subcanonical.

4.2. Natural quasi-topologies on the category of affine schemes. Fix a base ℭ = (A^{∼}, C, Φ^{∼} : A^{∼} ⟶ End_{2}^{∼}C).

4.2.1. The fp quasi-topology. We call a morphism w : W ⟶ X of Aff_{A^{∼}} flat if Spec_{ℭ}(w) preserves kernels of coreflexive pairs of arrows and faithfully flat and if in addition it reflects isomorphisms. Composition of faithfully flat morphisms is a faithfully flat morphism. If morphisms X ⟶ Z ⟵ Y are faithfully flat, then the projections X ⟵ X ×_{Z} Y ⟶ Y are faithfully flat. Hence faithfully flat morhisms regarded as covers form a quasi-topology τ_{fp}^{ℭ} which we call the faithfully flat, shortly fp quasi-topology. Unlike the commutative case, the fp quasi-topology is not a pretopology in general.

4.2.1.1. The fp quasi-topology in the case of the left base. Let ℭ be the left base of a monoidal category A^{∼}, i.e. ℭ = (A^{∼}, A, (𝔏, a)) (cf. 3.4.2). Fix algebras R^{∼}, S^{∼} in A^{∼} and an algebra morphism ψ : R^{∼} ⟶ S^{∼}. A corresponding inverse image functor ψ^{∗} (a left inverse to the pull-back functor ψ_{∗}) is isomorphic to S^{∼} ⊗_{R^{∼}} −. The composition ψ^{∗}ψ_{∗} : S^{∼} − mod_{A} ⟶ S^{∼} − mod_{A} is isomorphic to the functor (S^{∼} ⊗_{R^{∼}} S^{∼}) ⊗_{S^{∼}} −. The functor ψ^{∗}ψ_{∗} has a canonical structure of a comonad δ_{ψ} := ψ^{∗}η_{ψ}ψ_{∗} : ψ^{∗}ψ_{∗} ⟶ (ψ^{∗}ψ_{∗})^{2} with the counit ϵ_{ψ} : ψ^{∗}ψ_{∗} ⟶ Id_{S^{∼}−mod}. Here η_{ψ} and ϵ_{ψ} are adjunction morphisms. This comonad structure induces a comultiplication

δ′_{ψ} : S^{∼} ⊗_{R^{∼}} S^{∼} ⟶ (S^{∼} ⊗_{R^{∼}} S^{∼}) ⊗_{S^{∼}} (S^{∼} ⊗_{R^{∼}} S^{∼}) ≃ S^{∼} ⊗_{R^{∼}} S^{∼} ⊗_{R^{∼}} S^{∼}

on the S^{∼}-bimodule S^{∼}⊗_{R^{∼}} S^{∼}. The pair (δ_{S^{∼}}, S^{∼}⊗_{R^{∼}} S^{∼}) is a coalgebra in the monoidal category S^{∼} − bimod^{∼} = (S^{∼} − bimod, ⊗_{S^{∼}}, S^{∼}) of S^{∼}-bimodules with the counit ϵ′_{ψ} : S^{∼} ⊗_{R^{∼}} S^{∼} ⟶ S^{∼} induced by the multiplication S^{∼} ⊗ S^{∼} ⟶ S^{∼}. The inverse image functor ψ^{∗} = S^{∼}⊗_{R^{∼}} : R^{∼}−mod ⟶ S^{∼}−mod is the composition of the canonical functor

R^{∼} − mod ⟶ (S^{∼} ⊗_{R^{∼}} S^{∼}) − comod, M ⟼ (ψ^{∗}(M), δ_{ψ}(M)) = (S^{∼} ⊗_{R^{∼}} M, δ′_{ψ}), (1)

and the forgetful functor

(S^{∼} ⊗_{R^{∼}} S^{∼}) − comod ⟶ S^{∼} − mod, (N, ν : N ⟶ (S^{∼} ⊗_{R^{∼}} S^{∼}) ⊗_{S^{∼}} N) ⟼ N (2)

4.2.1.2. Proposition. Suppose the category A has kernels of coreflexive pairs of arrows. Then the functor (1) is an equivalence of categories if

{Spec_{A^{∼}}(ψ) : Spec_{A^{∼}}(S^{∼}) ⟶ Spec_{A^{∼}}(R^{∼})}

is an fp cover.

Proof. The assertion follows from (the dual form of) the Barr-Beck theorem. Details are left to the reader.

4.2.2. The fpqc quasi-topology on affine schemes. We define fpqc covers as sets of flat morphisms U = {u_{i} : U_{i} ⟶ X|i ∈ J} such that for a finite subset I of J, the set of morphisms U = {Spec_{ℭ}(u_{i}) |i ∈ I} is conservative. (i.e. it reflects isomorphisms). Denote by τ_{fpqc}^{ℭ} the function which assigns to each object X of the category Aff_{A^{∼}} of affine schemes in A^{∼} the family of fpqc covers of X. One can see that τ_{fpqc}^{ℭ} is a quasi-topology.

4.2.3. The lqc quasi-topology. We define lqc covers as fpqc covers 𝔘 = {u_{i} : U_{i} ⟶ X|i ∈ J} such that all inverse image functors Spec_{ℭ}(u_{i})^{∗} are localizations. We denote by τ_{lqc}^{ℭ} the function X ⟼ {lqc covers of X}.

4.2.3.1. Proposition. Suppose the category A has countable colimits which are preserved by functors M ⊗ − for all M ∈ ObA and Φ is a functor A ⟶ End_{ω}C. Then the lqc quasi-topology is a pretopology.

Proof. It suffices to prove the assertion in the case when A^{∼} = End_{ω}^{∼}C and Φ^{∼} acting naturally on C. Let {u_{i} : U_{i} ⟶ X| i ∈ J} be a finite Zariski cover and f : Y ⟶ X an arbitrary morphism of Aff_{ω}/C. By 2.6.3.1, the canonical morphism p_{i} : U_{i} ×_{C} Y ⟶ Y is a localization for any i ∈ J. The conservativity of {u_{i}^{∗} : X ⟶ U_{i}| i ∈ J} implies that of {p_{i}^{∗} : Y ⟶ U_{i} ×_{C} Y | i ∈ J}.

4.2.3.2. The lpq quasi-topology on Aff_{A^{∼}}. A finite set of arrows

{Spec_{A^{∼}}(S_{i}^{∼}) ⟶ Spec_{A^{∼}}(R^{∼})| i ∈ J}

is an lpq cover iff it is conservative, all functors S_{i}^{∼}⊗_{R^{∼}} are exact, and the canonical morphism S_{i}^{∼} ⊗_{R^{∼}} S_{i}^{∼} ⟶ S_{i}^{∼} is an isomorphism for all i ∈ J.

4.3. Descent. Let (A, 𝔗) be a quasi-site (i.e. 𝔗 is a quasi-topology on the category A), and let F : A ⟶ B be a functor. We call the pair (𝔗, F) subcanonical if for any X ∈ ObB, the presheaf h_{X}^{F} := B(F −, X) is a sheaf. Thus the pair (𝔗, Id_{A}) is subcanonical iff the quasi-topology 𝔗 is subcanonical in the sense of 4.1.5.2.

4.3.1. Lemma. Suppose the functor F : A ⟶ B is a composition of a functor F′ : A ⟶ B′ and a fully faithful functor G : B′ ⟶ B. If (𝔗, F) is subcanonical, then the pair (𝔗, F′) is subcanonical. In particular, if (𝔗, F) is subcanonical and F is fully faithful, the quasi-topology 𝔗 is subcanonical.

Proof is left to the reader.

4.3.2. Proposition Suppose the category A has countable colimits which are preserved by functors M ⊗ − for all M ∈ ObA as well as by the functor Φ : A ⟶ End_{2}C (in particular it takes values in the subcategory End_{ω}C). Let τ denote the fp quasitopology τ_{fp}^{ℭ} on the category Aff_{A^{∼}} of affine schemes. And let F be the canonical functor Aff_{A^{∼}} ⟶ RCat/C. Then the pair (τ, F) is subcanonical. In particular, the quasi-topology τ = τ_{fp}^{ℭ} is subcanonical.

Proof. It suffices to prove the assertion in the case when A^{∼} = End_{ω}^{∼}C and Φ^{∼} being a natural action of A^{∼} on C.

(a) Let 𝔾 = (G, μ) be a monad on X such that the functor G preserves countable colimits, monomorphisms and kernels of pairs of morphisms and reflects isomorphisms. Then the square

\begin{matrix}1 & ⟶^{η} & 𝔾 \\ η ↓ & & ↓ ϕ \\ 𝔾 & ⟶^{ψ} & 𝔾 ⋆ 𝔾 \end{matrix}  (1)

is cartesian. Here 1 is the identical functor Id_{X} regarded as a monad on X; ϕ and ψ denote the canonical monad morphisms.

In fact, since the functor G preserves kernels of pairs of morphisms and reflects isomorphisms, the square

\begin{matrix}1 & ⟶^{η} & G \\ η ↓ & & ↓ Gη \\ G & ⟶^{ηG} & G ∘ G \end{matrix}  (2)

is cartesian. Note that the canonical morphism G ∘ G ⟶ G ⋆ G is (under the conditions on G) a monomorphism. Therefore the square (1) is cartesian.

(b) Let U denote the category 𝔾−mod_{X} of 𝔾-modules. The square (1) being cartesian implies that the square

\begin{matrix}U ×_{X} U & ⟶^{p} & U \\ q ↓ & & ↓ ϕ \\ U & ⟶^{ϕ} & X \end{matrix}  (3)

is cocartesian. Here ϕ denote the morphism having the forgetful functor as an inverse image functor.

(c) Let f, g : U ⇉ Y be morphisms such that f ∘ p = g ∘ q. This means that p^{∗}f^{∗} ≃ q^{∗}g^{∗}. It follows from (b) that for any L ∈ ObY, there exists an object M of X defined uniquely up to isomorphism and such that f^{∗}(L) ≃ ϕ^{∗}(M) ≃ g^{∗}(L). The map L ⟼ M extends to a functor γ^{∗} such that f^{∗} ≃ ϕ^{∗}γ^{∗} ≃ g^{∗}. The functor γ^{∗} is defined uniquely up to isomorphism and is right exact. The latter follows from the fact that ϕ^{∗}γ^{∗} has this property and the functor ϕ^{∗} preserves kernels of pairs of morphisms and monomorphisms and reflects isomorphisms. Therefore γ^{∗} is an inverse image functor of a uniquely defined morphism γ : Y ⟶ X.

4.3.3. Corollary. Let the assumptions of 4.3.2 hold. Then for any X = (X, f) ∈ ObRCat/C, the functor

h_{X}^{o} : Aff_{A^{∼}}^{op} ⟶ Sets, Y ⟼ (RCat/C)(Spec_{ℭ}(Y), X),

is a sheaf on (Aff_{A^{∼}}, 𝔗_{fp}).

Proof. The assertion follows from 4.3.1. and 4.3.2.

4.4. Coinduced topologies. Let 𝔗 be a quasi-topology on a category A. For any V ∈ ObA, denote by 𝔗(V) the set of sieves which contain a 𝔗-cover of V. Let F : A ⟶ B be a functor. For any V ∈ ObA, the functor F induces a functor F_{V} : A/V ⟶ B/F (V). Denote by 𝔗_{F} (F (V)) the family of all sieves S in B/F (V) such that F_{V}^{−1}(S) ∈ 𝔗(V), i.e. F_{V}^{−1}(S) contains a 𝔗-cover of V. For an arbitrary X ∈ ObB, denote by 𝔗_{F} (X) the family of all sieves S of B/X such that for any morphism v : F (V) ⟶ X, S^{v} ∈ 𝔗_{F} (F (V)).

4.4.1. Proposition. The map X ⟼ 𝔗_{F} (X), X ∈ ObB, defines a topology, 𝔗_{F}, on the category B.

Proof. (a) Let f : Y ⟶ X be a morphism of B, and let R ∈ 𝔗_{F} (X). By definition of 𝔗_{F}, for any u : F (U) ⟶ Y, (R^{f})^{u} = R^{fu} ∈ 𝔗_{F} (F (U)). Hence R^{f} ∈ 𝔗_{F} (Y).

(b) Let R ∈ 𝔗_{F} (X), and let R_{1} be a sieve of B/X such that for any (Y ⟶^{f} X) ∈ ObR, the sieve R_{1}^{f} belongs to 𝔗_{F} (Y). We claim that R_{1} ∈ 𝔗_{F} (X).

In fact, let u : F (U) ⟶ X be an arbitrary morphism. We need to show that R_{1}^{u} ∈ 𝔗_{F} (F (U)), i.e. F^{−1}(R_{1}^{u}) ∈ 𝔗(U).

Since R ∈ 𝔗_{F} (X), R^{u} ∈ 𝔗_{F} (F (U)), i.e. F^{−1}(R^{u}) ∈ 𝔗(U). By assumption for any (V ⟶^{ξ} U) ∈ F^{−1}(R^{u}), F^{−1}((R_{1}^{u})^{F ξ}) ∈ 𝔗(V) which implies that (F^{−1}(R_{1}^{u}))^{ξ} ∈ 𝔗(V), hence F^{−1}(R^{u}) ∈ 𝔗(U).

The topology 𝔗_{F} will be called the topology coinduced by (𝔗, F). The following assertion gives a description of the coinduced topology 𝔗_{F} in the case when the topology 𝔗 is defined by a continuous base.

4.4.2. Lemma. Let τ be a continuous base of the topology 𝔗 (cf. A.1.5.1).

(a) A family of morphisms {u_{i} : U_{i} ⟶ X|i ∈ J} of B is a 𝔗_{F} -cover of X iff for any morphism v : F (V) ⟶ X, there exists a subset J^{v} of J and a set of commutative diagrams

\begin{matrix}F (U_{ij}) & ⟶^{u_{ij}} & U_{i} \\ F (p_{ij}) ↓ & & ↓ u_{i} \\ F (V) & ⟶^{v} & X & j ∈ J_{i} \end{matrix}  (1)

such that the set of arrows {p_{ij} : U_{ij} ⟶ V |i ∈ J^{v}, j ∈ J_{i}} belongs to τ.

(b) The topology 𝔗_{F} is the finest among the topologies 𝔗′ on B having the following property:

(#) For any V ∈ ObA and any cover {U_{i} ⟶ F (V)| i ∈ J} in 𝔗′, there exists a subset I of J and a set of commutative diagrams

\begin{matrix}F (U_{ij}) & ⟶^{u_{ij}} & U_{i} \\ F (p_{ij}) ↓ & & ↓ u_{i} \\ F (V) & ⟶^{id_{F (V )}} & F (V) & j ∈ J_{i} \end{matrix}  (1)

such that the set of arrows {p_{ij} : U_{ij} ⟶ V |i ∈ J, j ∈ J_{i}} belongs to τ.

Proof. The assertion follows from the definition of the topology 𝔗_{F} and A.1.5.2. Details are left to the reader.

4.4.3. The site (B_{F} , 𝔗_{F}). Fix a site (A, 𝔗) and a functor F : A ⟶ B. For an object X of B, denote by F_{X} the functor F/X ⟶ B, (V, F (V) ⟶^{h} X) ⟼ F (V) and by F ↓ X the natural cone F_{X} ⟶ X. Let B_{F} denote the full subcategory of the category B generated by all objects X of B such that the cone F ↓ X is initial. Note that the functor

B ⟶ F unct(A^{op}, Sets), X ⟼ B(F −, X),  (1)

induces a fully faithful functor from the category B_{F} to the category A^{∧} of functors from A^{op} to Sets. In fact, for any two objects X_{1}, X_{2} of the category B, any functor morphism ψ : B(F −, X_{1}) ⟶ B(F −, X_{2}) induces a cone morphism

F_{ψ} : F ↓ X_{1} ⟶ F ↓ X_{2}, (F (V) ⟶^{ξ} X_{1}) ⟼ (F (V) ⟶^{ψ(ξ)} X_{2})

If the cone F ↓ X_{1} is initial, F_{ψ} induces a uniquely defined morphism X_{1} ⟶ X_{2}.

4.4.3.1. Lemma. Suppose the pair (𝔗, F : A ⟶ B) is subcanonical. Then the topology induced on the subcategory B_{F} by the topology 𝔗_{F} is subcanonical.

Proof. (a) Note that a topology 𝔗′ on a category A′ is subcanonical iff for any X ∈ ObA′ and any R ∈ 𝔗′(X), the cone R is initial.

In fact, 𝔗′ is subcanonical iff for any Y ∈ ObA′ and any R ∈ 𝔗′(X), the canonical morphism A′(X, Y) ⟶ limA′(F_{X}|R, Y) is an isomorphism. Here F_{X}|R is the restriction to R of the forgetful functor F_{X} : A/X ⟶ A. This means that any cone R ⟶ Y decomposes into the canonical cone R = (F_{X}|R ⟶ X) and a unique morphism X ⟶ Y, hence the cone R is initial.

(b) Let R ∈ 𝔗_{F} (X), i.e. for any u : F (V) ⟶ X, R_{F}^{u} := F^{−1}(R^{u}) ∈ 𝔗(V). By assumption, B(F −, X) is a sheaf on (A, 𝔗). In particular, the canonical morphism B(F (V), X) ⟶ limB(F ∘ F_{V} |R_{F}^{u} , X) is an isomorphism. This means that the cone F (R_{F}^{u}) ⊂ B/F (V) is initial. If the cone F ↓ X is initial, this implies that the cone R is initial. The assertion follows now from (a).

4.4.4. The topology coinduced by a quasi-topology. Fix a quasi-site (A, τ). Applying the construction of a coinduced topology to the identical functor A ⟶ A, we obtain a topology, 𝔗^{τ}, coinduced by the quasi-topology 𝔗. It follows from 4.1.3.4 that if τ is continuous (e.g. it is a pretopology), then 𝔗^{τ} coincides with the coarsest topology 𝔗_{τ} generated by τ (cf. 4.1.3.3). If τ is subcanonical, then by 4.4.3.1, the topology 𝔗^{τ} is subcanonical.

4.4.5. Canonical topologies on the category Aff_{A^{∼}}. These are topologies coinduced by the canonical quasi-topologies introduced in 4.2.1, 4.2.2, and 4.2.3.

We denote by 𝔗_{fp}^{ℭ} the topology coinduced by τ_{fp}^{ℭ}.

We denote by 𝔗_{fpqc}^{ℭ} the topology on Aff_{A^{∼}} coinduced by fpqc covers and call it fpqc topology. Clearly τ_{fpqc}^{ℭ} is finer than the quasi-topology τ_{fp}^{ℭ} (see 4.2.1), hence the topology 𝔗_{fpqc}^{ℭ} is finer than the topology 𝔗_{fp}^{ℭ}.

We denote the topology coinduced by lqc covers by 𝔗_{lqc}. Under conditions of 4.2.3.1, lqc covers form a pretopology. Hence in this case, 𝔗_{lqc} coincides with the associated topology.

4.5. The category of (𝔗|F)-covers. Fix a quasi-site (A, 𝔗) and a functor F : A ⟶ B such that the pair (F, 𝔗) is subcanonical. Let X ∈ ObB. For any set U = {(U_{i}, F (U_{i}) ⟶^{u_{i}} X) | i ∈ J} of objects of the category F/X, denote by R_{U} the full subcategory of the category F/X objects of which are all objects of F/X having a morphism to some of the objects of U.

We define the category of Cov_{F |𝔗} of F |𝔗-covers as follows: objects of Cov_{F |𝔗} are pairs (U, X), where U is a set of objects of F/X such that X = colim(F |R_{U}). Here F |R_{U} denotes the restriction to R_{U} of the canonical functor F/X ⟶ B, (U, F (U) ⟶ X) ⟼ F (U). A morphism from (U, X) to (U′, X′) is given by a morphism f : X ⟶ X′ such that the sieve R_{U′}^{f} of F/X contains the sieve R_{U}. Here R_{U′}^{f} is the full subcategory of F/X generated by all objects (V, h) such that (V, f ∘ h) ∈ ObR_{U′}.

4.5.1. Locally affine (𝔗|F)-covers. Fix an object X of the category B and a pair of objects (U, F (U) ⟶^{u} X), (V, F (V) ⟶^{v} X) of the category F/X. Let D_{(U,u;V,v)}^{F} denote the

category objects of which are triples (ϕ, W, ψ), where W ∈ ObA and W ⟶^{ϕ} U, W ⟶^{ψ} V are morphisms of the subcategory A^{τ} (see 4.1.2.1) such that u ∘ F ϕ = v ∘ F ψ. Morphisms of triples are defined in an obvious way.

We call a cover (U, X), U = {(U_{i}, F (U_{i}) ⟶^{u_{i}} X) | i ∈ J}, locally affine if for any i, j ∈ J, there exists a set of objects U_{ij} of the category D_{(U_{i},u_{i};U_{j},u_{j})}^{F} such that

(#) the diagram D^{u_{i},u_{j}} ⟶ B has a colimit and the cone F |R_{U_{ij}} ⟶ colim(D^{u_{i},u_{j}}) is initial (i.e. colim(F |R_{U_{ij}}) exists and is naturally isomorphic to colim(D^{u_{i},u_{j}})).

Informally speaking, a locally affine (𝔗|F)-cover is generated by ”elements” of the quasi-site (A, 𝔗) with relations from (A, 𝔗).

We denote by 𝔏𝔞𝔠_{𝔗|F} the full subcategory of the category Cov_{𝔗|F} objects of which are locally affine (𝔗|F)-covers.

4.5.2. Zariski (𝔗|F)-covers. We call a cover (U, X), U = {(U_{i}, F (U_{i}) ⟶^{u_{i}} X) | i ∈ J}, a Zariski (𝔗|F)-cover if all arrows u_{i} are monomorphisms and for any i, j ∈ J, there exists a set of objects U_{ij} = {(u_{ij}^{iν}, U_{ij}^{ν}, u_{ij}^{iν})| ν ∈ J_{ij}} of the category D_{(U_{i},u_{i};U_{j},u_{j})}^{F} satisfying the condition (#) of 4.5.1 and having the property: for any ν ∈ J_{ij}, any pair of morphisms g_{1}, g_{2} : W ⟶ U_{ij}^{ν} such that u_{ij}^{iν} ∘g_{1} = u_{ij}^{iν} ∘g_{2} and u_{ij}^{jν} ∘g_{1} = u_{ij}^{jν} ∘g_{2} is trivial, i.e. g_{1} = g_{2}.

We denote by ℨ𝔞𝔯_{𝔗|F} the full subcategory of the category Cov_{𝔗|F} objects of which are Zariski (𝔗|F)-covers. It follows that ℨ𝔞𝔯_{𝔗|F} is a subcategory of the category 𝔏𝔞𝔠_{𝔗|F} of locally affine (𝔗|F)-covers.

4.5.3. Canonical functors. We shall write 𝔗 instead of (𝔗|Id_{A}). Thus we have the category Cov_{𝔗} of 𝔗-covers and its subcategories 𝔏𝔞𝔠_{𝔗} and ℨ𝔞𝔯_{𝔗} of resp. locally affine and Zariski 𝔗-covers. Since the pair (F, 𝔗) is subcanonical, the functor F induces a functor F_{Cov_{𝔗}} : Cov_{𝔗} ⟶ Cov_{(𝔗|F)}.

We have a canonical fully faithful functor A ⟶ ℨ𝔞𝔯_{𝔗}, U ⟼ (U ⟶^{id_{U}} U).

The composition of these two functors gives a canonical functor F_{𝔗} from the category

A to the category ℨ𝔞𝔯_{𝔗|F} which maps any object U of A to the cover {F (U) ⟶^{id_{F (U)}} F (U)}.

4.6. (𝔗|F)-spaces. Let (U, X), U = {(U_{i}, F (U_{i}) ⟶^{u_{i}} X) | i ∈ J}, be a cover. Suppose that for any i ∈ J, we are given a 𝔗-cover {U_{ij} ⟶^{u_{ij}} U_{i}| j ∈ J_{i}}. We call a set of

objects {(U_{ij}, F (U_{ij}) ⟶^{u_{i}∘F (u_{ij})} X) | i ∈ J, j ∈ J_{i}} a refinement, or a 𝔗-refinement of U. Since the pair (F, 𝔗) is subcanonical, any refinement of a cover is a cover.

We say that two covers (U, X) and (U′, X) are 𝔗-equivalent if they have a common refinement. It follows that 𝔗-equivalence is an equivalence relation. We define the category Es_{(𝔗|F)} of (𝔗|F)-spaces as follows. Objects of Es_{(𝔗|F)} are pairs (X, 𝔱), where 𝔱 is an equivalence class of covers of X. Morphisms from (X, 𝔱) to (X′, 𝔱′) are given by morphisms f : X ⟶ X′ such that for any (U′, X′) ∈ 𝔱′, there exists (U, X) ∈ 𝔱 such that f defines a morphism (U, X) ⟶ (U′, X′) of covers. One can see that if (X, 𝔱) ⟶^{f} (X′, 𝔱′) ⟶^{g} (X′′, 𝔱′′) are morphisms, then (X, 𝔱) ⟶^{gf} (X′′, 𝔱′′) is a morphism too.

4.6.1. Proposition. Suppose the quasi-topology 𝔗 is continuous (for instance, 𝔗 is a pretopology). Then we have a natural functor

Cov_{(𝔗|F)} ⟶ Es_{(𝔗|F)}  (1)

which is equivalent to the localization of the category Cov_{(𝔗|F)} by the class of morphisms

of the form (U, X) ⟶^{id_{X}} (U′, X).

Proof is left to the reader.

4.6.2. Locally affine (𝔗|F)-spaces. We say that two locally affine (𝔗|F)-covers (U, X) and (U′, X) are 𝔗-equivalent if they have a common locally affine refinement. We define the category 𝔏𝔞𝔰_{(𝔗|F)} of locally affine (𝔗|F)-spaces as follows. Objects of 𝔏𝔞𝔰_{(𝔗|F)} are pairs (X, 𝔱), where 𝔱 is an equivalence class of locally affine (𝔗|F)-covers of X. Morphisms from (X, 𝔱) to (X′, 𝔱′) are given by morphisms f : X ⟶ X′ such that for any locally affine cover (U′, X′) ∈ 𝔱′, there exists a locally affine cover (U, X) ∈ 𝔱 such that f defines a morphism (U, X) ⟶ (U′, X′) of covers. The composition is defined as in Es_{(𝔗|F)}. We have an analog of Proposition 4.6.1:

4.6.2.1. Proposition. Suppose the quasi-topology 𝔗 is continuous. Then we have a natural functor

𝔏𝔞𝔠_{(𝔗|F)} ⟶ 𝔏𝔞𝔰_{(𝔗|F)}  (1)

which is equivalent to the localization of the category 𝔏𝔞𝔠_{(𝔗|F)} of locally affine covers by

the class of morphisms of the form (U, X) ⟶^{id_{X}} (U′, X). In particular, there is a canonical functor

F_{𝔗} : A ⟶ Es_{(𝔗|F)}.  (2)

4.6.2.2. Note. In the case of an arbitrary (i.e. not necessarily continuous) quasitopology 𝔗, there is a canonical functor

F_{𝔗} : A_{𝔗} ⟶ Es_{(𝔗|F)}  (3)

(cf. 4.1.2.2). In particular, there is a canonical functor

A^{𝔗} ⟶ Es_{(𝔗|F)}  (4)

(cf. 4.1.2.1).

4.6.3. (𝔗|F)-schemes. We define a (𝔗|F)-scheme as a pair (X, 𝔱), where 𝔱 is an equivalence class (with respect to refinements) of Zariski (𝔗|F)-covers. Morphisms from (X, 𝔱) to (X′, 𝔱′) are given by morphisms f : X ⟶ X′ such that for any Zariski (𝔗|F)- cover (U′, X′) ∈ 𝔱′, there exists a Zariski (𝔗|F)-cover (U, X) ∈ 𝔱 such that f defines a morphism (U, X) ⟶ (U′, X′) of covers. We denote the category of (𝔗|F)-schemes by 𝔖𝔠𝔥_{𝔗|F}.

We leave to the reader the formulation of the corresponding version of Propositions 4.6.1 and 4.6.2.1.

4.6.4. Quasi-separated locally affine (𝔗|F)-spaces and schemes. We call a locally affine (𝔗|F)-space (resp. (𝔗|F)-scheme) (X, 𝔱) quasi-separated if it has a (resp. Zariski) 𝔗-cover {F (U_{i}) ⟶ X| i ∈ J} of X such that for any i, j ∈ J, the set of objects U_{ij} in the condition (#) in 4.5.1 (resp. in the conditions of 4.5.2) is finite.

4.6.5. Semiseparated locally affine (𝔗|F)-spaces and schemes. We call a 𝔗cover {F (U_{i}) ⟶ X| i ∈ J} of a (𝔗|F)-space (X, 𝔱) semiseparated if for any i, j ∈ J, there exists a cartesian product of the form

F (U_{ij}) ⟶^{F (p_{ij})} F (U_{i})
F (q_{ij}) ↓ ↓
F (U_{j}) ⟶ X

A space (X, 𝔱) which has a semiseparated cover is called semiseparated. Clearly every semiseparated space is locally affine.

4.7. (𝔗|F)-spaces and 𝔗-spaces. Suppose the quasi-topology 𝔗 on the category A is subcanonical. This means exactly that the pair (𝔗, h^{A}), where h^{A} is the canonical functor A ⟶ F unc(A^{op}, Sets), U ⟼ A(−, U), is subcanonical. We will write 𝔗-spaces instead of (𝔗|h^{A})-spaces.

4.7.1. Proposition. Suppose the pair (𝔗, F) is subcanonical and 𝔗 is continuous (e.g. it is a pretopology). Then

(a) For any (𝔗|F)-space (X, 𝔱), the presheaf

Es_{𝔗|F} (F_{𝔗}−, (X, 𝔱)) : A^{op} ⟶ Sets

is a sheaf.

(b) The map

(X, τ) ⟼ Es_{𝔗|F} (F_{𝔗}(−), (X, τ))

determines a functor h_{𝔗|F} : Es_{𝔗|F} ⟶ Sheaves(A, 𝔗). The functor h_{𝔗|F} is faithful (resp. fully faithful) if F is faithful (resp. fully faithful).

(c) The functor h_{𝔗|F} maps locally affine (𝔗|F)-spaces to locally affine 𝔗-spaces and (𝔗|F)-schemes to 𝔗-schemes.

Proof. (a) This follows from fact that the pair (𝔗, F) is subcanonical.

(b) The assertion follows from (a) and definitions.

(c) Fix a (𝔗|F)-space (X, 𝔱). To the space (X, 𝔱) there corresponds the (𝔗|h^{𝔗})-space (F, 𝔱^{∼}), where h^{𝔗} is the canonical functor A ⟶ Sheaves(A, 𝔗), F denotes the sheaf Es_{𝔗|F} (F_{𝔗}−, (X, 𝔱)) (cf. (a)), and 𝔱^{∼} is the class of equivalent sieves of h^{𝔗}/F corresponding to 𝔱. The (𝔗|h^{𝔗})-space (F, 𝔱^{∼}) has the following property: for any (Z, f) ∈ Obh^{𝔗}/F and any U ∈ 𝔱^{∼}, the sieve R_{U}^{(Z,f)} contains a sieve R_{U′} for some 𝔗-cover U′ of Z. The rest of the argument is left to the reader.

4.7.2. Note that the functor h_{𝔗|F} of 4.7.1 maps quasi-separated (resp. semiseparated) locally affine (𝔗|F)-spaces to quasi-separated (resp. semiseparated) 𝔗-spaces. This correspondence induces the similar correspondence between quasi-separated (resp. semiseparated) (𝔗|F)- and 𝔗-schemes.

4.7.3. Representable sheaves. We shall say that a sheaf of sets 𝔉 on the site (A, 𝔗) is representable by a (𝔗|F)-space (X, 𝔱) if 𝔉 ≃ Es_{𝔗|F} (F_{𝔗}−, (X, 𝔱)). Since (X, 𝔱) is a colimit of the natural functor F_{𝔗}/(X, 𝔱) ⟶ Es_{𝔗|F}, the representing (𝔗|F)-space (if any) is defined uniquely up to isomorphism.

4.8. Closed immersions.

4.8.1. Representable morphisms. Let P be a class of morphisms of the category C satisfying the following conditions:

(a) A composition of a morphism from P with any isomorphism belongs to P.

(b) If f : X ⟶ Y is a morphism from P, then for any g : Z ⟶ Y, there exists a fibered product X ×_{Y} Z and the projection X ×_{Y} Z ⟶ Z belongs to P.

Let F, G be presheaves of sets. A morphism F ⟶ G is called representable by a morphism of P if for any h_{X} ⟶ G, the projection F ×_{G} h_{X} ⟶ h_{X} is of the form h_{u} for a morphism u ∈ P.

Denote by P^{∧} the class of all morphisms of C^{∧} representable by morphisms of P. Note that P^{∧} is invariant under the base change: if F ⟶ G belongs to P^{∧} and H ⟶ G is an arbitrary morphism, then the projection H ×_{G} F ⟶ H belongs to P^{∧}. Clearly a morphism h_{X} ⟶ h_{Y} belongs to P^{∧} iff it is of the form h_{w} with w ∈ P.

4.8.1.1. Lemma. Let P and Q be classes of morphisms of the category C satisfying the conditions (a), (b). Then

(i) The intersection P ∩ Q has the properties (a) and (b).

(ii) If P is closed under the composition, then P^{∧} has the same property.

Proof is left to the reader.

4.8.1.2. Standard examples. 1) The class 𝔐 = 𝔐(C) of all monomorphisms has property 4.8.1(b) and is closed under the composition.

2) Same holds for the class E^{u} = E^{u}(C) of universal epimorphisms. Recall that a morphism f : X ⟶ Y is called a universal epimorphism if for any morphism V ⟶ Y, there exists a fibered product X ×_{Y} V and the canonical projection X ×_{Y} V ⟶ V is an epimorphism.

4.8.2. Strict monomorphisms and closed immersions. For the definition of a strict monomorphism, see 1.6. Recall that if the category C has fibered coproducts, then strict monomorphisms can be defined as morphisms X ⟶ Y such that the diagram X ⟶ Y ⇉ Y ∐_{X} Y is exact.

The composition of a strict monomorphism with an isomorphism is a strict monomorphism. If the category C has fiber products, then the class 𝔐_{s} = 𝔐_{s}(C) of strict monomorphisms of the category C satisfies the condition 4.8.1(b) too.

In fact, consider the diagram

\begin{matrix}X ×_{Y} V & ⟶^{p_{2}} & V \\ p_{1} ↓ & & ↓ g \\ X & ⟶^{f} & Y & \overrightarrow{⟶} & Z \end{matrix}  (1)

where Y ⇉ Z is an arbitrary pair of arrows from the class Λ_{f} of arrows equalizing f. But then p_{2} is a universal arrow equalizing all pairs Λ_{f} ∘ g := {(u_{1}g, u_{2}g)| (u_{1}, u_{2}) ∈ Λ_{f}}.

4.8.2.1. Lemma. (a) If the composition, gf, of morphisms X ⟶^{f} Y ⟶^{g} Z is a strict monomorphism, then f is a strict monomorphism.

(b) Any retraction is a strict monomorphism.

Proof. (a) If gf is a kernel of a pair of arrows u_{1}, u_{2} : Z ⇉ V, then f is a kernel of the pair u_{1}g, u_{2}g : Y ⇉ V. The general case is left to the reader.

(b) Let p : X ⟶ Y is a retraction, i.e. there exists a morphism e : Y ⟶ X such that ep = id_{X}. Then p is a kernel of the pair id_{X}, pe : X ⇉ X.

In fact, if f : Y ⟶ X is a morphism equalizing the pair (id_{X}, pe), then f = p ∘ (ef); and this decomposition is unique because p is a monomorphism.

4.8.2.2. Closed immersions of presheaves of sets. And let F, G be presheaves of sets on C. We call a morphism F ⟶ G a closed immersion if it belongs to 𝔐_{s}^{∧}, i.e. if it is representable by a strict monomorphism. In particular a closed immersion h_{X} ⟶ h_{Y} of representable functors is of the form h_{u}, where u is a strict monomorphism.

4.8.3. Example. Let C be the category CAff/k of commutative affine schemes over Spec(k). Then strict monomorphisms are exactly closed immersions of affine schemes. Let X and Y be arbitrary schemes identified with the corresponding sheaves of sets on the category C = CAff/k. Then a morphism X ⟶ Y is a closed immersion in the sense of the definition 4.8.2.2 iff it is a closed immersion of schemes in the conventional sense.

This example shows in particular that a strict monomorphism of (pre)sheaves is not necessarily a closed immersion. For instance, if X ⟶ Y is a scheme morphism, the diagonal morphism f : X ⟶ X×_{Y} X is a kernel of the natural pair of arrows X×_{Y} X ⇉ X, hence is a strict monomorphism of sheaves of sets. But it is a closed immersion (in the sense of 4.8.2.2) only if the scheme morphism f is separated.

4.8.4. Example. Let k be an arbitrary associative ring and C the category Aff/k of affine schemes over k. A morphism Spec(f) : Spec(A) ⟶ Spec(B) is a strict monomorphism iff the diagram B ∏_{A} B ⇉ B ⟶^{f} A is exact. The latter means that A is the quotient of B by the two-sided ideal Ker(f).

5. Schemes and locally affine spaces.

5.0. Fix a base ℭ = (A^{∼}, C, Φ^{∼}). We assume that A is a category with countable colimits and that the tensor product preserves countable colimits, as well as the functor Φ : A ⟶ EndC. These conditions imply that the category Aff_{A^{∼}} of affine schemes over ℭ has fiber products (cf. 3.2.2).

In what follows we shall apply the formalism of Section 4 to the case when (A, 𝔗) is the site (Aff_{A^{∼}}, 𝔗_{fpqc}^{ℭ}) of affine schemes over the base ℭ with the fpqc topology, B is the category RCat/C of ’spaces’ over C, and F : A ⟶ B is the functor

Spec_{ℭ} : Aff_{A^{∼}} ⟶ RCat/C, X ⟼ (Qcoh_{X/ℭ}, f_{C}),  (1)

(cf. 3.5). In this case we shall write Es_{fpqc}/ℭ instead of Es_{𝔗|F}.

5.1. The category of fpqc locally affine spaces over ℭ. This is by definition the category 𝔏𝔞𝔰_{𝔗|F} of locally affine (𝔗|F)-spaces, where 𝔗 is the fpqc topology, 𝔗_{fpqc}^{ℭ}, on the category Aff_{A^{∼}} of affine schemes (see 4.4.5), and F is the embedding 5.0(1). We denote this category by 𝔏𝔞𝔰_{fpqc}/ℭ. There is a canonical functor 𝔏𝔞𝔰_{fpqc}/ℭ ⟶ RCat/C, (X, τ) ⟼ X, and a canonical functor Aff_{A^{∼}} ⟶ 𝔏𝔞𝔰_{fpqc}/ℭ which assigns to any ℭ-affine scheme X the object (X, 𝔗_{fpqc}(X)) of the category Es_{fpqc}/ℭ (cf. 4.6).

5.2. The category of schemes over ℭ. The category Sch/ℭ of schemes over ℭ is defined as the category Sch_{𝔗|F} of (𝔗|F)-schemes, where 𝔗 is the lqc topology 𝔗_{lqc}^{ℭ} (cf. 4.4.5).

5.2.1. Locally affine spaces and schemes over k. Let k be an associative ring and C the category k−mod of left k-modules. In this case, locally affine spaces (resp. schemes) over C will be called locally affine k-spaces (resp. k-schemes). Let (X, τ), X = (X, f), be an arbitrary locally affine k-space (resp. k-scheme); and let 𝔘 = {u_{i} : U_{i} ⟶ X | i ∈ J} be an fpqc affine cover (resp. Zariski affine cover) of (X, τ). This means that the category U_{i} is naturally equivalent to the category of R_{i}-modules, where R_{i} = B_{i}(O_{i}, O_{i})^{o}, O_{i} is the ’structure sheaf’ on U_{i} : O_{i} = u_{i}^{∗}(k) (see 2.5).

Thus any fpqc locally affine space or scheme over k − mod is locally (in the corresponding sense) the category of left modules over algebras over k.

5.3. Quasi-coherent presheaves.

5.3.1. Quasi-coherent presheaves on fpqc ℭ-spaces. The notion of a quasicoherent presheaf on an fpqc locally affine space over ℭ is a special case of the notion of a quasi-coherent presheaf in a fibered category (see 1.5 and 1.5). It is defined as follows. Let 𝔄 be the category objects of which are affine schemes over ℭ and morphisms are flat morphisms. Let F_{fpqc} be the fibered category with the base 𝔄 and such that the fiber over any affine scheme over ℭ Y is the category of left O_{Y} -modules. Consider the canonical functor 𝔍 : 𝔄 ⟶ 𝔏𝔞𝔰_{ℭ}. To any fpqc locally affine space (X, τ) over ℭ, there corresponds the fibered subcategory F_{fpqc}^{(X,τ)}. The category Qcoh_{(X,τ)} of quasi-coherent presheaves on (X, τ) is by definition the category of quasi-coherent presheaves of the fibered category F_{fpqc}^{(X,τ)} (cf. 1.5).

In a less formal way, this can be described as follows. Fix an fpqc locally affine ℭ-space (X, τ). Denote by C_{τ} the category objects of which are elements of τ-covers which and morphisms are given by elements of τ-covers too. A quasi-coherent presheaf M^{∼} assigns to any object U ⟶ X of C_{τ} an O_{U}-module M^{∼}(U) and to any morphism (U ⟶ X) ⟶^{u} (V ⟶ X) of C_{τ} an isomorphism ξ_{u} : u^{∗}(M^{∼}(V)) ⟶ M^{∼}(U).

5.3.1.1. Note that for any fpqc locally affine space (X, τ), X = (X, f), over ℭ, the canonical functor X ⟶ Qcoh_{(X,τ)}, M ⟼ M^{∼}, where M^{∼}(U ⟶^{u} X) = u^{∗}(M), is faithful. This follows from the fact that for any cover 𝔘, the set of inverse image functors {u^{∗}| (U ⟶^{u} X) ∈ 𝔘} is conservative.

5.3.2. Quasi-coherent presheaves on a scheme over ℭ. The category Qcoh_{(X,τ)} of quasi-coherent presheaves on a scheme (X, τ) is described in a similar way, with the fpqc quasi-topology replaced by the Zariski (i.e. lqc) quasi-topology. As in the fpqc case, the canonical functor X ⟶ Qcoh_{(X,τ)} is faithful. Details are left to the reader.

5.4. Coproduct of schemes over ℭ. Fix a base ℭ = (A^{∼}, C, Φ^{∼}).

5.4.1. Proposition. Let F = {f_{i} : X_{i} ⟶ C|i ∈ J} be a finite family of schemes over ℭ. Suppose the categories C and A have finite products. Then a coproduct of the family F in RCat/C exists and is a scheme over ℭ.

Proof. By 1.3.1, there exists a coproduct of F in RCat_{C}. If (u_{ij} : U_{ij} ⟶ X_{i}|j ∈ J_{i}) is an affine cover of (X_{i}, f_{i}), then (u_{ij} : U_{ij} ⟶ X_{i}|i ∈ J, j ∈ J_{i}) is an affine cover of ∏_{i∈J}(X_{i}, f_{i}).

5.4.2. Corollary. If the categories C and A have finite products, then the category Sch_{ℭ} of schemes over ℭ has finite coproducts.

5.5. Noncommutative locally affine spaces and schemes as sheaves of sets. Let (A, 𝔗) be the site (Aff_{A^{∼}}, 𝔗_{fpqc}^{ℭ}) of affine schemes over the base ℭ with the fpqc topology. Let h denote the canonical functor A ⟶ A^{∧}, X ⟼ A(−, X). To any presheaf of sets X on (A, 𝔗), we assign a fibered category E_{X} over the base h/𝔉: the fiber over an object (Spec_{ℭ}(R^{∼}), ξ) of the category h/X is the forgetful functor from the category R^{∼} − mod_{ℭ} of R^{∼}-modules to the category C (i.e. an object of the category RCat/C). Thus we have a functorial map which assigns to any presheaf X of sets on the site (A, 𝔗) a ’space’ Qcoh_{X/ℭ} over C of quasi-coherent sheaves on 𝔉 and to any sheaf morphism X ⟶ Y the corresponding morphism Qcoh_{X/ℭ} ⟶ Qcoh_{Y/ℭ} of ’spaces’ over ℭ. We denote the restriction of this map to the category Sh(A, 𝔗) of sheaves of sets on (A, 𝔗) by Qcoh_{ℭ}.

5.5.1. Proposition. The functor Qcoh_{ℭ} has a canonical lifting to a functor Qcoh_{ℭ}^{∼} : Sh(A, 𝔗) ⟶ Es_{fpqc}/ℭ. The functor Qcoh_{ℭ}^{∼} maps locally affine 𝔗_{fpqc}^{ℭ}-spaces (resp. 𝔗_{fpqc}^{ℭ}- schemes) to fpqc locally affine spaces (resp. schemes) over ℭ.

Proof. (a) To any sheaf X on (A, 𝔗), we assign canonically an object (X, 𝔱_{h}) of the category Es_{𝔗|h} (cf. the part (c) of the argument of 4.7.1). To any sieve of h/X corresponds a sieve of Spec_{ℭ}/Qcoh_{X/ℭ}, and this correspondence maps equivalent sieves to equivalent sieves.

(b) Suppose X has a locally affine (resp. Zariski) 𝔗-cover. The functor Qcoh_{ℭ}^{∼} maps this cover to an affine cover of Qcoh_{X/ℭ}^{∼} defining the structure of an fpqc locally affine space (resp. of a scheme) over the base ℭ. Details are left to the reader.

5.5.2. Proposition. The category of fpqc locally affine spaces (resp. schemes) over ℭ is naturally equivalent to the category of locally affine 𝔗_{fpqc}-spaces (resp. of 𝔗_{fpqc}- schemes.

Proof. By 4.7.1, for any subcanonical pair (𝔗, F), the functor h_{𝔗|F} : (X, 𝔱) ⟼ Es_{𝔗|F} (F_{𝔗}(−), (X, 𝔱)) induces functors between the corresponding categories of locally affine spaces and schemes: 𝔏𝔞𝔰_{𝔗|F} ⟶ 𝔏𝔞𝔰_{𝔗} and 𝔖𝔠𝔥_{𝔗|F} ⟶ 𝔖𝔠𝔥_{𝔗}. In the case when (A, 𝔗) is the site (Aff_{A^{∼}}, 𝔗_{fpqc}^{ℭ}), the functor Qcoh^{∼} : Sh(A, 𝔗) ⟶ Es_{fpqc}/ℭ of 5.5.1 induces the quasi-inverse functors.

6. Examples of noncommutative schemes.

6.1. Vector fiber of an object. Let A^{∼} be a monoidal category, and let F denote the forgetful functor

AlgA^{∼} ⟶ A, (R, μ) ⟼ R.

6.1.1. Lemma. Let E ∈ ObA be such that the functors E ⊗ − : A ⟶ A and − ⊗ E preserve countable coproducts. And let there exists a coproduct ⊕_{n≥0}E^{⊗n}. Then the functor A(E, F−) : AlgA^{∼} ⟶ Sets is corepresentable.

Proof. Denote by T (E) the algebra (⊕_{n≥0}E^{⊗n}, μ_{E}), where the multiplication μ_{E} is given by the identical morphisms E^{⊗n} ⊗ E^{⊗m} ⟶ E^{⊗(m+n)}. For any algebra (R, μ) in A^{∼}, the natural map A(E, R) ⟶ AlgA^{∼}(T (E), (R, μ)) is a functorial isomorphism.

6.1.2. Corollary. Assume that A has countable coproducts and ⊗ : A × A ⟶ A preserves countable coproducts in both arguments. Then the functor

F : AlgA^{∼} ⟶ A, (R, μ) ⟼ R

has a left adjoint.

6.1.3. Corollary. Let E_{1}, E_{2} ∈ ObA be such that the functors E_{i} ⊗ − and − ⊗ E_{i}, i = 1, 2, preserve countable coproducts. Assume that T (E_{1}), T (E_{2}), and T (E_{1} ⊕ E_{2}) exist. Then there is a natural algebra isomorphism T (E_{1} ⊕ E_{2}) ≃ T (E_{1}) ⋆ T (E_{2}).

6.1.4. Vector fibers. Let E ∈ ObA satisfy the conditions of 6.1.1. The vector fiber, V(E), associated with E is the affine scheme Spec_{A^{∼}}(T (E)).

It follows that if the objects E_{1}, E_{2} of A satisfy the assumptions of 6.1.3, then the product V(E_{1}) × V(E_{2}) exists and is isomorphic to V(E_{1} ⊕ E_{2}).

6.1.5. Proposition. Let E, E′ ∈ ObA satisfy the assumptions of 6.1.1, and let E′ ϕ : E ⟶ E′ be a strict epimorphism. If Φ(E′) is right exact, then the corresponding morphism V(E′) ⟶ V(E) is a closed immersion.

Proof. Since ϕ : E ⟶ E′ is a strict epimorphism and the functor E ⟼ T (E) is right exact (compatible with all colimits as any functor having a right adjoint), the corresponding algebra morphism T (ϕ) : T (E) ⟶ T (E′) is a strict epimorphism, hence the assertion.

6.2. Quasi-coherent modules on a vector fiber. Fix a base ℭ = (A^{∼}, C, Φ). Let an object E of A is such that Φ^{∼}(T (E)) is naturally isomorphic to T (Φ(E)) (which is the case if the functors E ⊗ − and Φ are compatible with countable coproducts). Then the category Qcoh_{V(E)/ℭ} of quasi-coherent V(E)/ℭ-modules is isomorphic to the category

of actions of Φ(E): its objects are pairs (M, Φ(M) ⟶^{ξ} M), and morphisms and their composition are defined in an obvious way.

6.3. Vector fiber associated with an admissible pair of objects. Fix a base ℭ = (A^{∼}, C, Φ).

6.3.1. Admissible pairs of objects. Finite objects. We call a pair (M, L) of objects of C ℭ-admissible if the functor

A ⟶ Sets, F ⟼ C(M, Φ(F)(L))  (1)

is corepresentable, i.e. there is an object L^{∧}M of A (defined uniquely up to isomorphism) and a functorial (in F) isomorphism C(M, Φ(F)(L)) ≃ A(L^{∧}M, F). We call an object L of the category C finite if (M, L) is an admissible pair for any M ∈ ObC.

6.3.1.1. Example. Let C = R − mod for an associative ring R, A^{∼} is the monoidal category of R-bimodules. Then finite objects of the category C are projective R-modules of finite type. If L is a projective R-module of finite type and M an arbitrary left Rmodule, then L^{∧}M ≃ M ⊗ L^{∨}, where L^{∨} denotes the dual to L (right) R-module: L^{∨} ≃ R − mod(L, R). In particular, if L = R, then L^{∧}M is isomorphic to the R-bimodule M ⊗ R_{r}, where R_{r} is R regarded as a right R-module.

6.3.1.2. Example. Let ℭ = (A, A^{∼}, Φ) be the left base corresponding to a monoidal category A^{∼} (cf. 2.3.2.). Let L, M ∈ ObA. By definition, the pair (M, L) is admissible iff the functor A ⟶ Sets, F ⟼ C(M, F ⊗ L)) is corepresentable. Suppose that L is a finite object, i.e. there exists an object L^{!} such that the functor L^{!} ⊗ − is a right adjoint to L ⊗ −. Then F ⊗ L ≃ Hom(L^{!}, F), hence

C(M, F ⊗ L)) ≃ C(M, Hom(L^{!}, F)) ≃ C(M ⊗ L^{!}, F)

functorially in F. In other words, L is finite object of the monoidal category A^{∼} iff it is a finite object (in the sense of 6.3.1) of the left base.

6.3.2. Proposition. Let (Ψ^{∼}, g^{∗}; λ) : ℭ′ = (A^{′∼}, C′, Φ^{′∼}) ⟶ ℭ = (A^{∼}, C, Φ^{∼}) be a morphism (cf. 3.6) such that the functor Ψ has a left adjoint, Ψ^{∗}. Then for any ℭ-admissible pair (L, M), the pair (g^{∗}(L), g^{∗}(M)) is ℭ′-admissible.

Proof. In fact, we have functorial isomorphisms

A′(Ψ^{∗}(L^{∧}M), E′) ≃ A(L^{∧}M, Ψ(E′)) ≃
C(M, ΦΨ(E′)(L)) ≃ C′(g^{∗}(M), Φ′(E′)(g^{∗}(L)))

Hence the assertion.

6.3.3. Corollary. Let A^{∼} (resp. A^{′∼}) be the category of continuous (i.e. having a right adjoint) endofunctors of a category C (resp. C′). Let ℭ = (A^{∼}, C, Φ^{∼}) and ℭ′ = (A^{′∼}, C′, Φ^{′∼}) be the corresponding actions, and let g : C′ ⟶ C be a continuous morphism such that g_{∗} has a right adjoint. Then for any ℭ-admissible pair (L, M), the pair (g^{∗}(L), g^{∗}(M)) is ℭ′-admissible. In particular, (g^{∗}(L), g^{∗}(M)) is admissible for any affine morphism g.

Proof. 1) The morphism g induces a morphism (Ψ_{g}^{∼}, g) : ℭ′ = (A^{′∼}, C′, Φ^{′∼}) ⟶ ℭ = (A^{∼}, C, Φ^{∼}), and the functor Ψ_{g} maps every continuous functor F′ : C′ ⟶ C′ (– an object of A′) to g_{∗}F′g^{∗}. It has a left adjoint, Ψ_{g}^{∗}, which maps any continuous functor F : C ⟶ C to g^{∗}F g_{∗}. By 6.3.2, the pair (g^{∗}(L), g^{∗}(M)) is admissible and (by the argumen of 6.3.2) (g^{∗}(L))^{∧}(g^{∗}(M)) = Ψ_{g}^{∗}(L^{∧}M) = g^{∗}(L^{∧}M)g_{∗}.

2) One can give an independent (on 6.3.2) argument. For any continuous functor F : C ⟶ C, the functor g_{∗}F g^{∗} is continuous, and we have canonical isomorphisms

C′(g^{∗}(L), F g^{∗}(M)) ≃ C(L, g_{∗}F g^{∗}(M)) ≃ A(L^{∧}M, g_{∗}F g^{∗}(M)) ≃ A′(g^{∗}(L^{∧}M)g_{∗}, F)

Since these isomorphisms are functorial in F, it follows that the pair (g^{∗}(L), g^{∗}(M)) is admissible and g^{∗}(L)^{∧}g^{∗}(M) ≃ g^{∗}(L^{∧}M)g_{∗}.

6.3.4. Vector fibers. With any ℭ-admissible pair (M, L) of objects of C, we associate a vector fiber V(L^{∧}M).

6.3.4.1. Proposition. Let (L, M) and (L, M′) be admissible pairs of objects.

(a) The pair (L, M ⊕M′) is admissible and V(L^{∧}(M ⊕M′)) ≃ V(L^{∧}M)×V(L^{∧}M′).

(b) Let ϕ : M ⟶ M′ be a strict epimorphism. Then the corresponding morphism V(L^{∧}M′) ⟶ V(L^{∧}M) is a closed immersion.

Proof. (a) We have

C(M ⊕ M′, Φ(−)(L)) ≃ C(M, Φ(−)(L)) ∏ C(M′, Φ(−)(L)) ≃
A(L^{∧}M, −) ∏ A(L^{∧}M′, −) ≃ A(L^{∧}M ⊕ L^{∧}M′, −)

which shows that the pair (L, M ⊕M′) is admissible and L^{∧}(M ⊕M′) ≃ (L^{∧}M)⊕(L^{∧}M′). The assertion follows now from 6.1.4.

(b) The assertion follows from 6.1.5. Details are left to the reader.

6.3.5. Interpretation. For any affine A^{∼}-scheme X = Spec_{A^{∼}}(R), denote by f_{X}^{∗} the inverse image functor L ⟼ Φ^{∼}(R)(L) of the canonical morphism f_{X} : Qcoh_{X/ℭ} ⟶ C.

For any two objects M, L of the category C, consider the functor

H_{M,L} : Aff_{A^{∼}}^{op} ⟶ Sets, X ⟼ Qcoh_{X/ℭ}(f_{X}^{∗}(M), f_{X}^{∗}(L))  (1)

6.3.5.1. Proposition. If the pair (M, L) is admissible, then the functor H_{M,L} is representable by the affine scheme V(L^{∧}M).

Proof. In fact, for any X = Spec(R^{∼}), we have the following functorial isomorphisms:

H_{M,L}(X) := Qcoh_{X/ℭ}(f_{X}^{∗}(M), f_{X}^{∗}(L)) ≃ C(M, Φ(R)(L)) ≃
A(L^{∧}M, R) ≃ Alg_{A^{∼}}(T (L^{∧}M), R^{∼}) = Aff_{A^{∼}}(X, V(L^{∧}M))  (1)

Hence the assertion.

6.4. The group scheme GL_{V}. Fix objects V, W of the category C. We have a functor

Iso_{V,W} : Aff_{A^{∼}}^{op} ⟶ Sets, X ⟼ Aut_{Qcoh_{X/ℭ}}(f_{X}^{∗}(V), f_{X}^{∗}(W))  (1)

6.4.1. Proposition. Let V, W be objects of C such that the pairs (V, W) and (W, V) are admissible. Then the functor Iso_{V,W} is representable by an affine A^{∼}-scheme.

Proof. (a) Let (V, W) and (W, Z) be admissible pairs of objects of C. Consider the functor G_{V,W,Z} : Aff_{A^{∼}}^{op} ⟶ Sets which assigns to any affine scheme X = Spec_{A^{∼}}(R) the pair of morphisms f_{X}^{∗}(V) ⟶^{v} f_{X}^{∗}(W) ⟶^{w} f_{X}^{∗}(Z). The functor G_{V,W,Z} is representable by V(W^{∧}V) × V(Z^{∧}W).

(b) Let (V, W), (W, V), and (V, V) be admissible pairs of objects of C. Denote by Φ_{V,W} the subfunctor of the functor G = G_{V,W,V} which assigns to any affine scheme X the subset of all pairs (v, u) ∈ G_{V,W,V} (X) such that u ∘ v = id. We claim that the functor Φ_{V,W} is representable by an affine scheme.

Consider two functorial maps

α_{X}, β_{X} : G_{V,W,V} (X) ⟶ Qcoh_{X/ℭ}(f_{X}^{∗}(V), f_{X}^{∗}(V))

defined by α_{X}(v, u) = u ∘ v and β_{X}(v, u) = id_{f_{X}^{∗}(V)}. Since the pair (V, V) is admissible,

Qcoh_{X/ℭ}(f_{X}^{∗}(V), f_{X}^{∗}(V)) ≃ Aff_{A^{∼}}(X, V(V^{∧}V))

Let α′, β′ denote the corresponding morphisms

V(W^{∧}V ⊕ V^{∧}W) ≃ V(W^{∧}V) × V(V^{∧}W) ⟶ V(V^{∧}V)

The functor Φ_{V,W} is representable by the kernel of the pair of morphisms α′, β′.

(c) Let (V, W), (W, V), (V, V), and (W, W) be admissible pairs of objects of C. And let X be an affine A^{∼}-scheme. The set Iso(f_{X}^{∗}(V), f_{X}^{∗}(W)) is naturally isomorphic to the set of pairs (u, v), where u : f_{X}^{∗}(V) ⟶ f_{X}^{∗}(W) and v : f_{X}^{∗}(W) ⟶ f_{X}^{∗}(V) are quasicoherent module morphisms such that u ∘ v = id and v ∘ u = id. So Iso(f_{X}^{∗}(V), f_{X}^{∗}(W)) is identified with the fiber product of the morphisms

Φ_{V,W} (X) ⟶^{ϕ} G_{V,W,V} (X) ⟵^{ψ} Φ_{W,V} (X)

where ϕ = ϕ_{V,W} is the natural embedding, ψ is the composition of the natural embedding ϕ_{W,V} : Φ_{W,V} ⟶ G_{W,V,W} and the functorial isomorphism

G_{W,V,W} ⟶ G_{V,W,V} , (u, v) ⟼ (v, u)

Thus the map Iso_{V,W} is extended to the functor which is the kernel of the pair of functor morphisms (ϕ, ψ). Since both source and target of the arrows ϕ, ψ, the functors Φ_{V,W} and G_{V,W,V}, are representable, the functor Iso_{V,W} is representable too.

6.4.2. Corollary. Let V be an object of C such that the pair (V, V) is admissible. The functor Aut_{V} is representable by an affine A^{∼}-scheme in groups.

6.5. Grassmannians. Fix a pair (L, M) of objects of the category C. For any affine A^{∼}-scheme X = Spec(R), denote by Grass_{L,M}(X) the set of all isomorphy classes of morphisms v : f_{X}^{∗}(M) ⟶ L′ which are locally split over C in the following sense: there exists an lqc cover 𝔘 of X and for any (U ⟶^{u} X) ∈ 𝔘, a morphism ϕ_{u} : L ⟶ M such that the composition u^{∗}(v) ∘ u^{∗}f_{X}^{∗}(ϕ_{u}) : u^{∗}f_{X}^{∗}(L) ⟶ u^{∗}(L′) is an isomorphism.

6.5.1. Proposition. The map X ⟼ Grass_{L,M}(X) is naturally extended to a functor (Aff_{A^{∼}})^{op} ⟶ Sets.

Proof. For any morphism t : Y ⟶ X and any element [v] of Grass_{L,M}(X), v : f_{X}^{∗}(M) ⟶ L′, the equivalence class of the morphism t^{∗}(v) : t^{∗}f_{X}^{∗}(M) ⟶ t^{∗}(L′) belongs to Grass_{L,M}(Y).

In fact, by definition of Grass_{L,M}(X, f), there exists a cover 𝔘 of X such that for any U = (U ⟶^{u} X) ∈ 𝔘, there is a morphism ϕ_{U} : L ⟶ M for which u^{∗}(v) ∘ u^{∗}f_{X}^{∗}(ϕ_{U}) is an isomorphism. This implies that t^{∗}(v) splits in the desired way over the cover t^{∗}𝔘 = {U ×_{X} Y ⟶ Y | U ∈ 𝔘}.

6.5.2. Theorem. Let (L, M) and (L, L) be admissible pairs of objects of the category C. Assume that A has countable coproducts and for any F ∈ ObA, the functor F ⊗ −, preserves them. Then the functor Grass_{L,M} : (Aff_{A^{∼}})^{op} ⟶ Sets is representable by a scheme.

Proof. It follows from the definition that the functor Grass_{L,M} induces a sheaf of sets on the category Aff_{A^{∼}} of affine schemes over ℭ which we also denote by Grass_{L,M}. It remains to show the existence of an affine cover defining the structure of a scheme on Grass_{L,M}. Actually the affine cover constructed below is semiseparated.

Fix a morphism ϕ : L ⟶ M. For any X ∈ Ob(Aff_{A^{∼}}), consider the subset F_{ϕ;L,M}(X) of Grass_{L,M}(X) formed by the isomorphy classes of all morphisms v : f_{X}^{∗}(M) ⟶ L′ such that v ∘ f_{X}^{∗}(ϕ) is an isomorphism.

6.5.2.1. Lemma. The map X ⟼ F_{ϕ;L,M}(X) is naturally extended to a functor F_{ϕ;L,M} : (Aff_{A^{∼}})^{op} ⟶ Sets which is a subfunctor of Grass_{L,M}.

The functor F_{ϕ;L,M} is representable by an affine scheme.

Proof. (i) In fact, if v : f_{X}^{∗}(M) ⟶ L′ belongs to F_{ϕ;L,M}(X), i.e. v ∘ f_{X}^{∗}(ϕ) is an isomorphism, then for any morphism h : Y ⟶ X, the composition h^{∗}(v) ∘ h^{∗}f_{X}^{∗}(ϕ) is an isomorphism. Clearly F_{ϕ;L,M} is a subfunctor of Grass_{L,M}.

(ii) Note that one can identify F_{ϕ;L,M}(X) with the set of epimorphisms v : f^{∗}(M) ⟶ f^{∗}(L) such that v ∘ f^{∗}(ϕ) = id_{f^{∗}(L)}. In fact, if v′ : f^{∗}(M) ⟶ L′ is such that w := v′ ∘ f^{∗}(ϕ) : f^{∗}(L) ⟶ L′ is an isomorphism, then v = w^{−1} ∘ v′ has the required property.

(iii) There are two maps

α_{X}, β_{X} : X(f_{X}^{∗}(M), f_{X}^{∗}(L)) ⇉ C(f^{∗}(L), f^{∗}(L))

defined by α_{X} : v ⟼ v ∘ f_{X}^{∗}(ϕ), β_{X} : v ⟼ id_{f_{X}^{∗}(L)}. The maps α_{X} and β_{X} are functorial in X, hence they define morphisms resp. α and β from the functor X ⟼ X(f_{X}^{∗}(M), f_{X}^{∗}(L)) to the functor X ⟼ X(f_{X}^{∗}(L), f_{X}^{∗}(L)). The first functor is representable by V(L^{∧}M) and the second one is representable by V(L^{∧}L). Let α′ and β′ be morphisms from V(L^{∧}M) to V(L^{∧}L) corresponding to resp. α and β. The functor F_{ϕ;L,M} : X ⟼ F_{ϕ;L,M}(X) is the kernel of the pair (α, β), hence it is representable by the kernel, F_{ϕ;L,M}, of the pair (α′, β′) of affine scheme morphisms.

6.5.2.2. Each of the functor morphisms F_{ϕ;L,M} ⟶ Grass_{L,M} is representable; i.e. for any affine A^{∼}-scheme X and any morphism h_{X} ⟶ Grass_{L,M}, the functor

Y ⟼ F_{ϕ;L,M}(Y) ×_{Grass_{L,M} (Y)} h_{X}(Y)

is representable by an affine subscheme of X.

In fact, any morphism h_{X} ⟶ Grass_{L,M} is uniquely determined by an element of Grass_{L,M}(X), i.e. by the equivalence class [v] of a locally split epimorphism v : f_{X}^{∗}(M) ⟶ f_{X}^{∗}(L). The corresponding map h_{X}(Y) ⟶ Grass_{L,M}(Y) sends any morphism t : Y ⟶ X into [t^{∗}(v)]. The fiber product F_{ϕ;L,M}(Y) ×_{Grass_{L,M} (Y)} h_{X}(Y) consists of all pairs (w, t), where t ∈ h_{X}(Y) and [w : g^{∗}(M) ⟶ g^{∗}(L)] are such that w ∘ f_{X}^{∗}(ϕ) = id_{f_{X}^{∗}(L)} and w = t^{∗}(v). Since v and ϕ here are fixed, the fiber product F_{ϕ;L,M}(Y) ×_{Grass_{L,M} (Y)} h_{X}(Y) can be identified with the set of all morphisms t : Y ⟶ X such that t^{∗}(v ∘ f_{X}^{∗}(ϕ)) = id_{t^{∗}f_{X}^{∗}(L)}. In other words, the fiber product F_{ϕ;L,M}(Y) ×_{Grass_{L,M} (Y)} h_{X}(Y) is identified with the kernel of the pair of morphisms β_{Y}, α_{Y} : h_{X}(Y) ⟶ Qcoh_{Y/ℭ}(f_{Y}^{∗}(L), f_{Y}^{∗}(L)), defined by β_{Y} : t ⟼ id_{f_{Y}^{∗}(L)}, α_{Y} : t ⟼ t^{∗}(v ∘ f_{X}^{∗}(ϕ)). The morphisms β_{Y}, α_{Y} are functorial in Y, and Qcoh_{Y/ℭ}(f_{Y}^{∗}(L), f_{Y}^{∗}(L)) ≃ h_{V(L^{∧}L)}(Y). Hence the morphisms β = (β_{Y}), α = (α_{Y}) define a pair of morphisms α′, β′ : X ⇉ V(L^{∧}L), and the functor Y ⟼ F_{ϕ;L,M}(Y) ×_{Grass_{L,M} (Y)} h_{X}(Y) is representable by the kernel of the pair (α′, β′).

6.5.2.3. The functor morphisms F_{ϕ;L,M} ⟶ Grass_{L,M} form a cover of Grass_{L,M} in the topology 𝔗_{lqc}.

It suffices to show that for any affine scheme X and any morphism ψ : h_{X} ⟶ Grass_{L,M}, the canonical morphisms F_{ϕ;L,M} ×_{Grass_{L,M}} X ⟶ X form an affine cover of X. In fact, the morphism ψ corresponds to an element [f_{X}^{∗}(M) ⟶^{v} L′] of the set Grass_{L,M}(X). By the definition of the Grassmannian, there exists a cover 𝔘 of X and for any (U ⟶^{u} X) ∈ 𝔘, a morphism ϕ_{u} : L ⟶ M such that the composition u^{∗}(v) ∘ u^{∗}f_{X}^{∗}(ϕ_{u}) : u^{∗}f_{X}^{∗}(L) ⟶ u^{∗}(L′) is an isomorphism. Thus, for any (U ⟶^{u} X) ∈ 𝔘, we have a commutative diagram

\begin{matrix}h_{U} & ⟶^{ϕ_{u}} & F_{ϕ_{u};L,M} \\ ↓ & & ↓ \\ h_{X} & ⟶^{ψ} & Grass_{L,M} \end{matrix}  (1)

Here the morphism ϕ_{u} corresponds to the element [u^{∗}(v)] ∈ F_{ϕ_{u};L,M}(U).

6.5.2.4. It follows from the argument of 6.5.2.2 that the cover

{F_{ϕ;L,M} ⟶ Grass_{L,M}| ϕ : L ⟶ M}

is semiseparated. In particular, it defines a structure of a semiseparated scheme on the sheaf Grass_{L,M}.

6.5.3. Fundamental element. The ℭ-scheme which represents the functor

X ⟼ Grass_{L,M}(X)

will be called L-th Grassmannian of M and denoted by Grass_{L,M} := (Qcoh_{Grass_{L,M} /ℭ}, π). Here π is the morphism Qcoh_{Grass_{L,M} /ℭ} ⟶ C. There is a canonical element [𝔳] = [π^{∗}(M) ⟶^{𝔳} L(1)] such that the map

h_{Grass_{L,M}} (X) ⟶ Grass_{L,M}(X), t ⟼ t^{∗}(𝔳)

is a functor isomorphism. We call [𝔳] the fundamental element on Grass_{L,M}.

6.5.4. Projective fibers. Suppose L is such a finite object that L^{∧}M ≃ M for any M ∈ ObC. Then we will write P(M) instead of Grass_{L,M} and call P(M) the projective fiber over C defined by M. In this case we shall write the fundamental element as

[π^{∗}(M) ⟶^{𝔳} O(1)].

6.5.4.1. Example. Let ℭ = (A, A^{∼}, 𝔏^{∼}) be the left base. Then P(M) = Grass_{1,M}, where 1 is the unit object of the monoidal category A^{∼}.

6.5.5. Functoriality. The map M ⟼ Grass_{L,M} is not functorial with respect to M. It is functorial, however, with respect to the class 𝔈𝔭𝔦_{H} of L-epimorphisms. The latter are defined as morphisms ψ : M′ ⟶ M such that the corresponding map C(L, ψ) : C(L, M′) ⟶ C(L, M) is surjective.

If ψ is a coretraction, i.e. there exists ϕ : M ⟶ M′ such that ψ ∘ ϕ = id_{M}, then ψ is an L-epimorphism for any object L of the category C. If L is a projective object, then any epimorphism is an L-epimorphism.

6.5.5.1. Proposition. Let L ∈ ObC. To any L-epimorphism ψ : M′ ⟶ M, there corresponds a functor morphism Grass_{L}(ψ) : Grass_{L,M} ⟶ Grass_{L,M′}.

The morphism Grass_{L,M}(ψ) : Grass_{L,M′} ⟶ Grass_{L,M} is a closed immersion.

Proof. (a) Let X be a ℭ-scheme and v : f_{X}^{∗}(M) ⟶ L′ a locally split morphism; i.e. there exists a cover 𝔘 of X and for any (U ⟶^{u} X) ∈ 𝔘, a morphism ϕ_{u} : L ⟶ M such that u^{∗}(v) ∘ u^{∗}f_{X}^{∗}(ϕ_{u}) is an isomorphism. Since ψ : M′ ⟶ M is an L-epimorphism, morphisms ϕ_{u} are ’lifted’ to M′; i.e. there exists ϕ′_{u} : L ⟶ M′ such that ϕ_{u} = ψ ∘ ϕ′_{u}. This gives a functorial map Grass_{L}(ψ) : Grass_{L,M}(X) ⟶ Grass_{L,M′}(X).

(b) The argument is similar to that of [GrD], 9.7.9.

6.6. Flag varieties.

6.6.1. Fix a category C and an object M of C. Let L = (L_{i})_{1≤i≤p} be an increasing sequence of objects of C such that the pairs (M, L_{i}) and (L_{i}, L_{i}) are admissible for all i. For any X ∈ ObRCat_{ℭ}, denote by F l_{L,M}(X) the subset of ∏_{1≤i≤p} Grass_{L_{i},M}(X) formed

by ([v_{1}], ..., [v_{p}]), v_{i} : f_{X}^{∗}(M) ⟶ f_{X}^{∗}(L_{i}), such that Ker(v_{i+1}) ⊆ Ker(v_{i}), 1 ≤ i ≤ p − 1. Elements of F l_{L,M}(X) will be called flags of the type L in M.

6.6.2. Lemma. The map X ⟼ F l_{L,M}(X) is naturally extended to a contravariant functor, F l_{L,M} : Aff_{A^{∼}}^{op} ⟶ Sets.

Proof. The extension of the map X ⟼ F l_{L,M}(X) to morphisms is uniquely determined by the condition that the canonical embeddings F l_{L,M}(X) ⟶ ∏_{1≤i≤p} Grass_{L_{i},M}(X)

define a functorial morphism

i_{L} = i_{L,M} : F l_{L,M} ⟶ ∏_{1≤i≤p} Grass_{L_{i},M}

Details are left to the reader.

6.6.3. Proposition. The functor F l_{L,M} : X ⟼ F l_{L,M}(X) is a scheme and the morphism

i_{L,M} : F l_{L,M} ⟶ ∏_{1≤i≤p} Grass_{L_{i},M}  (2)

is a closed immersion.

Proof is left to the reader.

6.6.4. Let H = (H_{i}|1 ≤ i ≤ p) be a set of objects of the category C. We call a morphism ψ : M ⟶ M′ an H-epimorphism if it is an H_{i}-epimorphism for all i (cf. 6.5.5).

6.6.5. Proposition. Fix objects M, M′ of the category C. Let L = (L_{i})_{1≤i≤p} be an increasing sequence of objects of C such that the pairs (M, L_{i}), (M′, L_{i}), and (L_{i}, L_{i}) are admissible for all i. To any L-epimorphism ψ : M ⟶ M′, there corresponds a closed immersion F l_{H}(ψ) : F l_{L,M′} ⟶ F l_{L,M}.

Proof. In fact, there is a commutative diagram

F l_{L,M′}(X) ⟶^{i_{M′} (X)} ∏_{1≤i≤p} Grass_{L_{i},M′}(X)
i_{v}(X) ↓  ↓
F l_{L,M}(X) ⟶^{i_{M} (X)} ∏_{1≤i≤p} Grass_{L_{i},M}(X)

functorial in X, which induces a commutative diagram

F l_{L,M′}  ⟶^{i_{M′}} ∏_{1≤i≤p} Grass_{L_{i},E′}
F l_{L}(v) ↓  ↓ ∏_{1≤i≤p} Grass_{L_{i},v}
F l_{L,M}  ⟶^{i_{M}}  ∏_{1≤i≤p} Grass_{L_{i},v}
Since i_{M}, i_{M′} and Grass_{L_{i},v} are closed immersions, F l_{L,v} is a closed immersion.

6.6.6. Let L′ = (L_{i_{k}})_{1≤k≤q} be an increasing subsequence of L, where (i_{k})_{1≤k≤q} is a strictly increasing sequence of integers in the interval [1, p]. There is a canonical morphism

p_{M}^{L′,L}(X) : F l_{L,M}(X) ⟶ F l_{L′,M}(X)

which maps any flag ([v_{i}]) of type L into the flag ([v_{i_{k}}]) of type L′, hence a canonical morphism

p_{M}^{L′,L} : F l_{L,M} ⟶ F l_{L′,M}

There is a commutative diagram

F l_{L,M}(X) ⟶^{i_{M′} (X)} ∏_{1≤i≤p} Grass_{L_{i},M}(X)
i_{v}(X) ↓  ↓
F l_{L′,M}(X) ⟶^{i_{M} (X)} ∏_{1≤k≤q} Grass_{L_{i_{k}} ,M}(X)

functorial in X, which induces a commutative diagram

F l_{L,M}  ⟶^{i_{E′}} ∏_{1≤i≤p} Grass_{L_{i},E′}
F l_{L,v} ↓  ↓ ∏_{1≤i≤p} Grass_{L_{i},v}
F l_{L′,M}  ⟶^{i_{M}} ∏_{1≤k≤q} Grass_{L_{i_{k}} ,v}

where the right vertical arrow is the canonical projection.

The following examples are studied in a bigger detail in [R3] and the last one, quantum flag variety, plays a central role in [LR3]. We reproduce them here to make a picture more complete.

6.7. Projective spectrum and a quasi-affine space related to a graded algebra. Let k be a commutative ring, Γ a commutative directly ordered group with the unit element 0. And let R be an associative Γ-graded k-algebra.

6.7.1. The cone of R. For any γ ∈ Γ, set R_{>γ} := ⊕_{σ>γ}R_{σ}. For any R-module M and any γ ∈ Γ, denote by M_{γ} the subset of all elements of M annihilated by R_{>γ}. Denote by T_{+} the full subcategory of the category R − mod generated by all R-modules M such that M = sup{M_{γ}|γ ∈ Γ}. One can see that T_{+} is a Serre subcategory of the category R − mod. Thus we have the quotient category Cone_{Γ}(R) := R − mod/T_{+} and a canonical continuous morphism (localization) R − mod/T_{+} ⟶ R − mod. Composing it with the canonical morphism (global section functor) R − mod ⟶ R_{0} − mod, we obtain an R_{0}-’space’ Cone_{Γ}(R) := (R − mod/T_{+}, π) called the quasi-affine space (or affine cone) of the algebra R.

6.7.2. Projective spectrum of R. Let F be the natural functor from the category 𝔤𝔯_{Γ}R − mod of Γ-graded R-modules to the category R − mod. And let 𝔗_{+} denote the preimage of the Serre subcategory T_{+} in 𝔤𝔯_{Γ}R − mod. Since the functor F is exact, 𝔗_{+} is a Serre subcategory of 𝔤𝔯_{Γ}R − mod, hence we have a quotient category P roj_{Γ}(R) = 𝔤𝔯_{Γ}R − mod/𝔗_{+} and the following canonical continuous morphisms:

The morphism π′ : Proj_{Γ}(R) ⟶ 𝔤𝔯R−mod having a localization at 𝔗_{+} as an inverse image functor.

The morphism φ : 𝔤𝔯_{Γ}R − mod ⟶ R_{0} − mod, with the direct image functor

φ_{∗} : 𝔤𝔯_{Γ}R − mod ⟶ R_{0} − mod, M = ⊕_{γ∈Γ}M_{γ} ⟼ M_{0}.  (1)

Note by passing that the inverse image functor φ^{∗} : V ⟼ R ⊗_{R_{0}} V is fully faithful. This follows from the fact that the adjunction morphism

η : Id_{R_{0}−mod} ⟶ φ_{∗} ∘ φ^{∗}, η(V) : V ⟶ (R ⊗_{R_{0}} V)_{0} = R_{0} ⊗_{R_{0}} V

is an isomorphism. Set π^{Γ} = φ ∘ π′ : P roj_{Γ}(R) ⟶ R_{0} − mod. We call Proj_{Γ}(R) := (P roj_{Γ}(R), π^{Γ}) the projective spectrum of R. The direct image of π^{Γ} is regarded as the global sections functor.

6.8. Affine covers. Let Γ and R be as in Subsection 6.7. Fix a Γ-graded associative k-algebra R. If R_{>0} is an ideal of R (for instance, Γ = ℤ and R_{n} = 0 if n < 0), then the canonical morphism Cone_{Γ} ⟶ R − mod can be regarded as the complement to the closed subspace of R − mod corresponding to the ideal R_{n>0} (cf. 1.6.1), i.e. as a Zariski open immersion of R_{0} − mod-’spaces’. However, unlike the commutative case, Cone_{Γ}(R) is not, in general, a subscheme of the affine scheme R − mod ⟶ R_{0} − mod.

6.8.1. Lemma. Let S = {S_{i}|i ∈ J} be a family of left homogeneous Ore subsets of the algebra R. And let, for each i ∈ J, 𝕊_{i} be the Serre subcategory of R − mod generated by all modules M such that any element of M is annihilated by some element of S_{i}. And let S_{i} be the preimage of 𝕊_{i} in 𝔤𝔯_{Γ}R − mod.

The following conditions are equivalent:

(a) The Serre subcategories {𝕊_{i}|i ∈ J} provide a cover of the ’quasi-affine space’ Cone_{Γ}(R); i.e. ⋂_{i∈J} 𝕊_{i} = T_{+}.

(b) The family of Ore sets S = {S_{i}|i ∈ J} has the properties:

(i) For any γ ∈ Γ and any i ∈ J, S_{i} ∩ R_{>γ} ≠ ∅.

(ii) If m is a left ideal of R such that m ∩ S_{i} ≠ ∅ for all i ∈ J, then R_{>γ} ⊆ m for some γ ∈ Γ.

Proof. (a)⇔(b). The condition (i) is equivalent to the inclusion T_{+} ⊆ 𝕊_{i} for all i ∈ J. The condition (ii) says that T_{+} contains the intersection ⋂_{i∈J} 𝕊_{i}. Therefore T_{+} coincides with ⋂_{i∈J} 𝕊_{i}.

6.8.1.1. Remark. In [VW], a ℤ_{+}-graded noetherian ring R such that there exists a family of left and right Ore sets S = {S_{i}|i ∈ J} satisfying the equivalent conditions of Lemma 6.8.1 is called schematic. Quite a few algebras of interest are schematic. We refer to [VW] for examples.

6.8.2. Proposition. Any family S = {S_{i}|i ∈ J} of left homogeneous Ore subsets of R satisfying the conditions (i), (ii) of Lemma 6.8.1 determines an affine cover {S_{i}|i ∈ J} of Cone_{Γ}(R).

Proof. The composition of π_{∗} and the direct image of u_{i} : Cone_{Γ}(R)/S_{i} ⟶ Cone_{Γ}(R) equals to the composition of

π′_{∗} ∘ u_{i∗} = (π′ ∘ u_{i})_{∗} : Cone_{Γ}(R)/S′_{i} ⟶ R − mod  (1)

and the functor φ_{∗} : 𝔤𝔯_{Γ}R − mod ⟶ R_{0} − mod (cf. (1) in 6.7). Note that the category Cone_{Γ}(R)/S′_{i} is naturally identified with the category 𝔤𝔯_{Γ}R − mod/S_{i}; so that the functor (1) becomes a right adjoint to the localization

Q_{i} : R − mod ⟶ R − mod/S_{i}.  (2)

Since (2) is a localization at a left Ore set S_{i}, the quotient category R − mod/S_{i} is equivalent to the category S_{i}^{−1}R−mod of S_{i}^{−1}R-modules. Thus R−mod/S_{i} can be replaced by S_{i}^{−1}R − mod. And the localization Q_{i} can be identified with the tensoring S_{i}^{−1}R⊗_{R}. Therefore a right adjoint functor to Q_{i} is exact. Since the functor φ_{∗} : R − mod ⟶ R_{0} − mod is exact, we obtain the exactness of π_{∗} ∘ u_{i∗}.

6.8.3. Lemma. Let

\begin{matrix}U ⟶^{g} X \\ u ↘ ↙ f \\ C \end{matrix}

be a commutative diagram of continuous morphisms.

(a) If u_{∗} : U ⟶ C is exact (resp. u is almost affine) and the direct image functor f_{∗} of f is faithful, then g_{∗} : U ⟶ X is exact (resp. g is almost affine).

(b) Suppose f^{∗} is a localization. If u_{∗} has a right adjoint, u^{!} (resp. u is affine), then g_{∗} has a right adjoint (resp. g is affine).

Proof. (a) Since u_{∗} ≃ f_{∗}g_{∗} and f_{∗} is faithful, the compatibility of u_{∗} with limits of a certain type of diagram implies the same property of g_{∗}. Clearly the faithfulness of u_{∗} implies that of g_{∗}, hence if u is almost affine (i.e. u_{∗} is exact and faithful), then g is almost affine.

(b) The condition ’f^{∗} is a localization’ means exactly that the direct image functor, f_{∗} is fully faithful. Suppose the direct image functor u_{∗} of u has a right adjoint, u^{!}. We claim that u^{!}f_{∗} is a right adjoint of the functor g_{∗}. In fact, we have functorial isomorphisms: U(−, u^{!}f_{∗}(M)) ≃ C(u_{∗}(−), f_{∗}(M)) ≃ C(f_{∗}g_{∗}(−), f_{∗}(M)). Since the functor f_{∗} is fully faithful, C(f_{∗}g_{∗}(−), f_{∗}(M)) ≃ C(g_{∗}(−), M), hence the assertion

6.8.3.1. Corollary. Under conditions of 6.8.2, the morphisms S_{i}^{−1}R − mod ⟶ Cone_{Γ}(R) are affine.

6.8.4. Affine covers of Proj. Under the conditions of Proposition 6.8.2, the family of Ore sets S = {S_{i}|i ∈ J} determines an affine cover of P roj_{Γ}(R) := 𝔤𝔯_{Γ}R − mod/T_{+} ⟶ 𝔤𝔯_{Γ}R − mod. It follows from 6.8.3 that the morphisms 𝔤𝔯_{Γ}S_{i}^{−1}R − mod ⟶ P roj_{Γ}(R) are affine (cf. 6.8.3.1).

6.8.5. Example: noncommutative skew projective spaces. Let A be an arbitrary associative k-algebra. And let q denote a matrix [q_{ij}]_{i,j∈J} with entrees in k such that q_{ij}q_{ji} = 1 for all i, j ∈ J. In particular, q_{ii} = 1 for all i ∈ J. To this data there corresponds a skew (or q−)polynomial algebra A_{q}[x], where x denotes the set of indeterminates {x_{i}|i ∈ J}. The defining relations are:

x_{i}x_{j} = q_{ij}x_{j}x_{i} for all i, j ∈ J,  (1)
x_{i}r = rx_{i} for all i ∈ J and r ∈ R  (2)

Let J = {0, 1, …, r}. Set Γ := ℤ^{r+1}; and let γ_{i}, i = 0, 1, …, r, denote the canonical generators of Γ. We provide Γ with a standard lexicographic preorder. Assigning to each x_{i} the parity γ_{i}, we turn the skew polynomial algebra R := A_{q}[x] into a Γ-graded algebra with R_{0}=A. For any i ∈ J, set S_{i} := {x_{i}^{n}|n ≥ 1}. The family S = {S_{i}|i ∈ J} satisfies the conditions (i), (ii) of Lemma 6.8.1. Therefore S determines, by Proposition 6.8, affine covers of the spaces Proj_{Γ}(R) and Cone_{Γ}(R). These covers have the usual ’classical’ properties:

(a) The category Cone_{Γ}(R)/𝕊′_{i} is equivalent to the category A_{q}[x, x_{i}^{−1}] − mod.

(b) Let Γ_{i} denote the quotient group Γ/ℤγ_{i} ≃ ℤ^{r}. We have:

P roj_{Γ}(R)/S′_{i} = 𝔤𝔯_{Γ}R − mod/S_{i} ≃ 𝔤𝔯_{Γ}R[x_{i}, x_{i}^{−1}] − mod.  (3)

The category 𝔤𝔯_{Γ}R[x_{i}, x_{i}^{−1}] − mod in (3) is naturally equivalent to the category 𝔤𝔯_{Γ_{i}}A_{q_{i}}[x/x_{i}]−mod of left Γ_{i}-graded modules over the skew polynomial algebra A_{q_{i}}[x/x_{i}]. Here x/x_{i} denotes {x_{j}/x_{i}|j ∈ J, j ≠ i}, and q_{i} denotes the matrix [q_{ni}q_{nm}q_{ni}^{−1}]_{n,m∈J−{i}} (cf. [R], Example I.7.2.2.4).

Note that A_{q_{i}}[x/x_{i}] is the Γ_{i}-component of the algebra A_{q}[x, x_{i}^{−1}] of the ’functions on Cone_{Γ}(R)/𝕊′′_{i}.

(c) One can see that the category P roj_{Γ}(R)/S′_{i} is naturally identified with the category 𝔤𝔯_{Γ_{i}}A_{q_{i}}[x/x_{i}]−mod and Cone_{Γ}(R)/𝕊′_{i} with 𝔤𝔯_{Γ_{i}}A_{q}[x, x_{i}^{−1}]−mod. And the canonical functor P roj_{Γ}(R)/S′_{i} ⟶ Cone_{Γ}(R)/𝕊′_{i} is induced by the tensoring by the algebra A_{q}[x, x_{i}^{−1}] over its Γ_{i}-component A_{q_{i}}[x/x_{i}] = A_{q}[x, x_{i}^{−1}]_{0}.

Note however, that the natural morphisms u_{i} : 𝔤𝔯_{Γ_{i}}A_{q_{i}}[x/x_{i}] − mod ⟶ P roj_{Γ}(R) do not form an affine cover of Proj_{Γ}(R) if r ≥ 1: the composition of u_{i∗} with the direct image of the projection π : P roj_{Γ}(R) ⟶ A−mod is not faithful, hence π ∘u_{i} is not affine.

6.8.5.1. The projective q-’space’ P_{q}^{r}. Let again R = A_{q}[x], x = (x_{0}, x_{1}, …, x_{r}). But take Γ = ℤ with the natural order; and set the parity of each x_{i} equal to 1. One can repeat with Cone_{ℤ}(R) and P_{q}^{r} := Proj_{ℤ}(R) the same pattern as with Cone_{Γ}(R) and P_{Γ}^{r} := Proj_{Γ}(R). Only this time the quotient groups Γ_{i} will be trivial, and we obtain a picture very similar to the classical one: P_{q}^{r} covered by r +1 affine spaces A_{q_{i}}[x/x_{i}]−mod, i = 0, 1, …, r. The details are left to the reader.

Note that the categories P^{r} and P_{Γ}^{r} := P roj_{Γ}(R) are not equivalent if r ≥ 1.

6.9. Flag varieties of quantized enveloping algebras. Let 𝔤 be a reductive Lie algebra over ℂ and U(𝔤) the enveloping algebra of 𝔤. Let 𝔓 denote the group of integral weights of 𝔤 and 𝔓_{+} the semigroup of nonnegative integral weights. Let R = ⊕_{λ∈𝔓_{+}}R_{λ}, where R_{λ} is the vector space of the (canonical) irreducible finite dimensional representation with the highest weight λ. The module R is a 𝔓 -graded algebra with the multiplication determined by the projections R_{λ} ⊗ R_{ν} ⟶ R_{λ+ν}, for all λ, ν ∈ 𝔓_{+}. It is well known that the algebra R is isomorphic to the algebra of regular functions on the base affine space of 𝔤. Recall that Y = G/U, where G is a connected simply connected algebraic group with the Lie algebra 𝔤, and U is its maximal unipotent subgroup.

The Cone_{Γ}(R) is equivalent to the category of quasi-coherent sheaves on the base affine space Y of the Lie algebra 𝔤. The category P roj_{Γ}(R) is equivalent to the category of quasi-coherent sheaves on the flag variety of 𝔤.

Let now 𝔤 be a semisimple Lie algebra over a field k of zero characteristic and U_{q}(𝔤) the quantized enveloping algebra of 𝔤. Define the 𝔓 -graded algebra R = ⊕_{λ∈𝔓_{+}}R_{λ} the same way as above. This time, however, the algebra R is not commutative. Following the classical example (and identifying spaces with categories of quasi-coherent sheaves on them), we call Cone_{Γ}(R) the quantum base affine space and Proj(R) the quantum flag variety of 𝔤.

6.9.1. An affine cover of the flag variety. Let W be the Weyl group of the Lie algebra 𝔤. Fix a w ∈ W. For any λ ∈ 𝔓_{+}, choose a nonzero w-extremal vector e_{wλ}^{λ} generating the one dimensional vector subspace of R_{λ} formed by the vectors of the weight wλ. Set S_{w} := {k^{∗}e_{wλ}^{λ}|λ ∈ 𝔓_{+}}. It follows from the Weyl character formula that e_{wλ}^{λ}e_{wμ}^{μ} ∈ k^{∗}e_{w(λ+μ)}^{λ+μ}. Hence S_{w} is a multiplicative set. It was proved by Joseph [Jo] that S_{w} is a left and right Ore subset in R. The Ore sets {S_{w}|w ∈ W } determine a locally affine cover of the quantum base affine space Cone_{Γ}(R) and the quantum flag variety Proj_{Γ}(R) of 𝔤. This cover enjoys properties similar to the properties (a)–(c) of the canonical cover of a ’projective space’ and its cone (cf. 6.8.5). Namely, Cone_{Γ}(R)/𝕊_{w} is naturally equivalent to S_{w}^{−1}R − mod and P roj_{Γ}(R)/S_{w} is naturally equivalent to (S_{w}^{−1}R)_{0} − mod.

7. Abelianization.

Fix a base ℭ = (A^{∼}, C, Φ). And let β = (β_{L,M} : L ⊗ M ⟶ M ⊗ L|L, M ∈ ObA) be a symmetry of the monoidal category A^{∼} = (A, ⊗, 1).

7.1. Abelianization of affine schemes. Denote by CAlg_{β}A^{∼} the full subcategory of the category AlgA^{∼} generated by β-commutative algebras.

7.1.1. Lemma. The inclusion functor CAlg_{β}A^{∼} ⟶ AlgA^{∼} has a left adjoint, 𝔞_{β}.

Proof. The functor 𝔞_{β} assigns to any algebra R^{∼} = (R, m) the cokernel R_{β} of the pair m, m ∘ β_{R,R} : R ⊗ R ⟶ R with the unique multiplication m_{β} : R_{β} ⊗ R_{β} ⟶ R_{β} such that the canonical epimorphism R ⟶ R_{β} is an algebra morphism.

We call 𝔞_{β} the β-abelianization functor.

Denote by Aff_{A^{∼}}^{β} the full subcategory of the category Aff_{A^{∼}} of affine schemes generated by β-commutative affine schemes, i.e. affine schemes isomorphic to Spec(R^{∼}) for a β-commutative algebra R^{∼}.

7.1.2. Proposition. (a) The inclusion functor Aff_{A^{∼}}^{β} ⟶ Aff_{A^{∼}} has a right adjoint.

(b) The functor 𝔄𝔟_{β} preserves arbitrary coproducts and limits.

(c) For any pair S^{∼} ⟵ R^{∼} ⟶ T^{∼} of β-commutative algebra morphisms, the fiber product of the corresponding β-commutative affine schemes is isomorphic to the affine scheme Spec(S^{∼} ⊗_{R^{∼}}^{β} T^{∼}).

(d) For any affine A^{∼}-scheme X, the adjunction morphism J_{β} ∘ 𝔄𝔟_{β}(X) ⟶ X is a closed immersion.

Proof. (a) The assertion follows from 7.1.1.

(b) The functor 𝔄𝔟_{β} preserves limits as any functor having a left adjoint.

The product of any family of β-commutative algebras (taken in the category AlgA^{∼} of all algebras in A^{∼}) is a β-commutative algebra, i.e. the β-abelianization functor 𝔞_{β} : AlgA^{∼} ⟶ CAlgA^{∼} is compatible with arbitrary products. By duality, 𝔄𝔟_{β} preserves arbitrary coproducts.

(c) The pull-forward of a pair S^{∼} ⟵ R^{∼} ⟶ T^{∼} of β-commutative algebra morphisms is isomorphic to 𝔞_{β}(S^{∼} ⋆_{R^{∼}} T^{∼}), and the latter is isomorphic to S^{∼} ⊗_{R^{∼}} T^{∼}. Hence the assertion.

(d) This follows from the fact that the canonical morphism from an algebra R^{∼} = (R, μ) to its β-abelianization, R_{β}^{∼}, is a cokernel of the pair of morphisms μ, μ ∘ β_{R,R} : R ⊗ R ⟶ R (cf. 5.5.1).

7.2. β-commutative schemes. The category Sch_{A^{∼}}^{β} of β-commutative A^{∼}-schemes is defined the same way as the category of A^{∼}-schemes with the only difference: the category Aff_{A^{∼}} of affine schemes is replaced by its subcategory Aff_{A^{∼}}^{β} of β-commutative schemes.

7.3. The β-abelianization of schemes. Let F be a functor Aff_{A^{∼}}^{op} ⟶ Sets. The β-abelianization of the functor F, F^{β}, is the composition of F with (the opposite to) the inclusion functor Aff_{A^{∼}}^{β} ⟶ Aff_{A^{∼}}.

Let X be a scheme. Suppose that the β-abelianization of the functor h_{X} : Sch_{A^{∼}}^{op} ⟶ Sets, Y ⟼ Sch_{A^{∼}}(Y, X), is representable by a β-commutative scheme X_{β}. The scheme X_{β} will be called a β-abelianization of X.

By 7.1.2, the β-abelianization of a functor represented by an affine scheme X is represented by the abelianization of X.

7.3.1. Proposition. Let X_{β} be a β-abelianization of a A^{∼}-scheme X. The canonical morphism X_{β} ⟶ X is a closed immersion.

Proof. The assertion follows from 7.1.2.

7.4. Examples. The following examples show that all noncommutative varieties discussed in Section 6 have β-abelianizations.

7.4.1. β-abelianization of vector fibers. Let E ∈ ObA and be as in 6.1.1. Then the tensor algebra T (E), hence the vector fiber V(E) = Spec(T (E)), are defined. The βabelianization V_{β}(E) of V(E) is Spec(S_{β}(E)), where S_{β}(E) is the β-symmetric algebra of E. It follows that the β-abelianization of an admissible pair (L, M) of objects of the category C is represented by the β-commutative affine scheme V_{β}(L^{∧}M) := Spec(S_{β}(L^{∧}M)).

7.4.1.1. Affine q-’spaces’. Let k be a commutative ring, and let q be a matrix (q_{ij})_{0≤i,j≤r} with entrees in k such that q_{ij}q_{ji} = 1 for all 1 ≤ i, j ≤ r. The matrix q defines a symmetry, β_{q}, of the monoidal category A^{∼} = 𝔤𝔯_{ℤ^{r+1}}k − mod^{∼} of ℤ^{r+1}-graded k-modules. Let γ_{i} denote the i-th standard generator of the group ℤ^{r+1}, and let P_{i} be of a free k module of rank 1 generated by an element of degree γ_{i}. Set E = ⊕_{0≤i,j≤r}P_{i}. The β_{q}-abelianization of V(E) is the q-’space’ Spec(k_{q}[x]) (cf. 6.8.5).

7.4.1.2. Projective q-’spaces’. Let E ∈ Ob𝔤𝔯_{ℤ^{r+1}}k − mod be the same as in 7.4.1.1. The β_{q}-abelianization of the projective fiber P(E) := Proj(T (E)) is isomorphic to the q-projective space P_{q}^{r} := Proj(k_{q}[x]) of 6.8.5.1.

7.4.2. The β-commutative group scheme GL_{β}(V).

7.4.2.1. Proposition. Let V be an object of C such that the pair (V, V) is admissible. The functor

Aut_{V}^{β} : (Aff_{A^{∼}}^{β})^{op} ⟶ Sets, (X) ⟼ Aut(f_{X}^{∗}(V))

is representable by an affine β-commutative A^{∼}-scheme in groups.

Proof. The assertion follows from 6.4.1 and 7.3.1.

7.4.3. β-Grassmannians. Let L, M be objects of the category C.

7.4.3.1. Proposition. Under conditions of 6.5.2, the β-abelianization of the functor Grass_{L,M} is representable by a β-commutative scheme Grass_{L,M}^{β}.

Proof. The assertion can be proved by following step by step the argument of 6.5.3. The details are left to the reader.

The β-commutative scheme Grass_{L,M}^{β} will be called the L-th β-Grassmannian of M.

7.4.3.2. The usual Grassmannians. Let A^{∼} be the monoidal category of quasicoherent sheaves on a scheme S = (S, O), β the standard symmetry. Let ℭ = (A, A^{∼}). Fix a quasi-coherent sheaf M, and take L equal to the direct sum of n copies of O_{S}. Then Grass_{M,L}^{β} is the usual n-the Grassmannian of the quasi-coherent sheaf M in the sense of [GrD], 9.7: Grass_{M,L}^{β} = Grass_{n}(M).

7.4.4. β-flag varieties. Let M ∈ ObC, and let L = (L_{i})_{1≤i≤p} be an increasing sequence of objects of C such that the pairs (M, L_{i}) and (L_{i}, L_{i}) are admissible for all i.

7.4.4.1. Proposition. Under conditions of 6.6.3, the β-abelianization of the functor F l_{L,M} is representable by a β-commutative scheme F l_{L,M}^{β}.

Proof. The assertion follows from the corresponding fact about Grassmannians (cf. 7.4.3.1).

7.4.4.2. The usual flag varieties. Let A^{∼} be the monoidal category of quasicoherent sheaves on a scheme S = (S, O) with the standard symmetry β. And let ℭ = (A, A^{∼}). Fix a quasi-coherent sheaf M on S. Fix an increasing sequence m = (m_{i})_{1≤i≤p} of nonnegative integers. Set L = (L_{i})_{1≤i≤p}, where L_{i} = O_{S}^{m_{i}} – the direct sum of m_{i} copies of the structure sheaf O_{S}. Then Fl_{L,M}^{β} is the usual flag variety of the type m in the quasi-coherent sheaf M in the sense of [GrD], 9.9.

7.5. β-nilpotent schemes. For simplicity, we assume that all categories in this subsection are abelian. To any pair of closed subschemes Y and Z of a ℭ-scheme (X), one can assign their Gabriel product, Y • Z, defined as follows. We identify Qcoh_{Y} and Qcoh_{Z} with full subcategories of Qcoh_{X}. Then Qcoh_{Y •Z} is the full subcategory of X generated by all M ∈ ObX such that there exists an exact sequence 0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0 with M′ ∈ ObY and M′′ ∈ ObZ. The subcategory Qcoh_{Y •Z} defines a closed subscheme ([R], III.6.2.1). In particular, for any closed subscheme Y of a ℭ-scheme X has thickennings Y^{(n)}, n ≥ 1 defined by Y^{(1)} = Y, and Y^{(n+1)} = Y^{(n)} • Y.

An A^{∼}-scheme X will be called a β-nilpotent A^{∼}-scheme of degree ≤ n if there exists a closed β-commutative subscheme Y of X such that X coincides with the n-th thickenning of Y. We denote by Sch_{A^{∼}}^{β,n} the full subcategory of the category Sch_{A^{∼}} generated by β-nilpotent schemes of degree ≤ n. Clearly Sch_{A^{∼}}^{β} = Sch_{A^{∼}}^{β,1} and Sch_{A^{∼}}^{β,n} ⊆ Sch_{A^{∼}}^{β,m} if n ≤ m. We denote by Aff_{A^{∼}}^{β,n} the category of affine β-nilpotent schemes of degree ≤ n: Aff_{A^{∼}}^{β,n} = Aff_{A^{∼}} ∩ Sch_{A^{∼}}^{β,n}.

For any functor F : Sch_{A^{∼}}^{op} ⟶ Sets, denote by F^{β,n} the composition of F with the (opposite to the) inclusion functor Sch_{A^{∼}}^{β,n} ⟶ Sch_{A}^{∼}. We call the functor F^{β,n} the β-nilpotezation of the functor F of degree n, or simply β-nilpotezation of F if n = ∞.

Let X be a A^{∼}-scheme. Suppose that the β-nilpotezation of degree n of the functor h_{X} : Sch_{A^{∼}}^{op} ⟶ Sets, Y ⟼ Sch_{A^{∼}}(Y, X), is representable by a A^{∼}-scheme X_{β,n}. The scheme X_{β,n} will be called a β-nilpotezation of X of degree n.

7.5.1. Proposition. (a) If a β-abelianization of a A^{∼}-scheme X exists, then there exist β-nilpotezations of X all degrees.

(b) The canonical morphism X_{β,n} ⟶ X is a closed immersion.

(c) For all n, the functor 𝔑_{β,n} of β-nilpotezation of degree n, 𝔑_{β,n} : X ⟼ X_{β,n} preserves arbitrary coproducts and limits taken in Sch_{A^{∼}}.

Proof. (a) & (b) Let X_{β} be a β-abelianization of a A^{∼}-scheme X, and let X_{β} ⟶ X be the canonical immersion. The n-th neighborhood of X_{β} in X is exactly the β-nilpotezation of X of degree n, X_{β}^{(n)} = X_{β,n}.

Since X_{β} ⟶ X is a closed immersion, X_{β,n} = X_{β}^{(n)} ⟶ X is a closed immersion. The latter implies that X_{β,n} is a subscheme of the A^{∼}-scheme X.

(c) The compatibility with limits follows from general properties of representable functors. The compatibility the functor 𝔑_{β,n} with coproducts is follows from the description of coproducts.

7.5.2. Proposition. (a) For any affine A^{∼}-scheme X, there exist β-nilpotezation of X of degree n for all n.

(b) All β-nilpotezations of an affine A^{∼}-scheme are affine. In particular, the inclusion functor Aff_{A^{∼}}^{β,n} ⟶ Aff_{A^{∼}} has a right adjoint, 𝔑_{β,n} : Aff_{A^{∼}} ⟶ Aff_{A^{∼}}^{β,n}.

Proof. (a) The assertion (a) follows from 7.1.2, 7.3.1, and 7.5.1.

(b) Let X be an affine A^{∼}-scheme. By 7.5.1, the canonical morphism X_{β,n} ⟶ X is a closed immersion. The assertion now follows from the fact that any closed subscheme of an affine A^{∼}-scheme is affine.

7.5.3. Examples. Any of noncommutative schemes discussed in Section 6 has βnilpotezations of all degrees. This follows (thanks to 7.5.1(a)) from the fact that all these schemes have β-abelianizations.

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