Noncommutative smooth spaces

Maxim Kontsevich, Alexander Rosenberg

0. CONVENTIONS AND NOTATIONS

We will work in the category Alg_{k} of associative unital algebras over a fixed base field k. If A ∈ Ob(Alg_{k}), we denote by 1_{A} ∈ A the unit in A and by m_{A} : A ⊗ A⟶A the product. For an algebra A, we denote by A^{opp} the opposite algebra, i.e. the same vector space as A endowed with the multiplication m_{A^{opp}}(a⊗b) := m_{A}(b⊗a). If A and B are two algebras, then A ⊗_{k} B is again an algebra. Also, A ⋆ B denotes the free product of A and B over k, the coproduct in the category Alg_{k}. By A-mod we denote the abelian category of left A-modules. Analogously, mod-A are right modules (the same as A^{opp}-modules) and A-mod-A are bimodules over k, or, equivalently, A ⊗_{k} A^{opp}-modules. We shall write ⊗ instead of ⊗_{k}. For a vector space V, we denote by Sym^{∗}(V) and ⊗^{∗}(V) resp. the free commutative associative (polynomial) and free associative (tensor) k-algebra respectively, generated by V.

1. SMOOTH ALGEBRAS

A typical example of an algebra for this text is a free finitely generated algebra k⟨x_{1}, …, x_{d}⟩, in contrast to usual noncommutative algebras close to commutative ones, like universal enveloping algebras, algebras of algebraic differential operators etc. We consider the free algebra as “the algebra of functions on the non-commutative affine space” which we denote N𝔸^{d}. Recently there were several attempts to understand algebraic geometry of this space. Gelfand and Retakh started the study of basic identities in the free skew field with d generators (see [GR]), which is can be considered as the “generic point” in N𝔸^{d}. One of us (see [K]) tried to develop “differential geometry” in the algebra of noncommutative formal power series, i.e. at the formal neighborhood of N𝔸^{d} at zero. Kapranov (see [Ka]) described a differential-geometric picture related with the completion of the free algebra with respect to the commutator filtration, i.e. with the algebra of functions on the infinitesimal neighborhood of the usual affine space 𝔸^{d} in N𝔸^{d}. Here we would like to study N𝔸^{d} as the whole space, without completions and localizations. The basic intrinsic property of the free algebra is smoothness.

1.1. Equivalent definitions

Few years ago J. Cuntz and D. Quillen gave the following

Definition ([CQ1], Definition 3.3 and Proposition 6.1). An algebra A is quasi-free (= formally smooth) iff it satisfies one of the following equivalent properties:

1) (Lifting property for nilpotent extensions) for any algebra B, a two-sided nilpotent ideal I ⊂ B (I = BIB, I^{n} = 0 for n ≫ 0), and for any algebra homomorphism f :

A⟶B/I, there exists an algebra homomorphism f̃: A⟶B such that f = pr_{B⟶B/I} ∘ f̃, where pr_{B⟶B/I} : B⟶B/I is the natural projection.

2) Ext_{A−mod−A}^{2}(A, M) = 0 for any bimodule M ∈ Ob(A-mod-A).

3) The A-bimodule Ω_{A}^{1} := Ker(m_{A} : A ⊗ A⟶A) (see 1.1.2) is projective.

The definition of formal smoothness via the lifting property 1) is analogous to the Grothendieck’s (actually Quillen’s) definition of formally smooth algebras in the commutative case. The equivalence of properties 1)-2)-3) is an easy exercise in homological algebra.

Properties 1),2) are not constructive: one can not check them directly. The property 3) looks better, although it is still not clear a priori how to work with it in complicated infinite-dimensional examples. J. Cuntz and D. Quillen have found another characterization of smooth algebras which is convenient for calculations in practice. In order to give this characterization we need an auxiliary definition.

Let T A be the algebra generated by symbols \underline{a}, \underline{Da}, where a ∈ A, subject to the following relations:

a) the map a ↦ \underline{a} is a homomorphism of unital k-algebras,

b) the map a ↦ \underline{Da} is k-linear,

c) \underline{D(a · b)} = \underline{a} · \underline{Db} + \underline{Da} · \underline{b}, (the Leibnitz rule).

Algebra T A is naturally ℤ_{≥0}-graded: deg(\underline{a}) = 0, deg(\underline{Da}) = +1. We identify A with the subalgebra of T A consisting of elements of degree zero.

1.1.2. Differential envelope

Before going further on we would like to notice that T A is isomorphic as an abstract ℤgraded algebra to the so-called differential envelope ΩA, the universal differential ℤ-graded super-algebra containing A.

By definition, ΩA is the universal ℤ-graded super-algebra containing A and endowed with an odd differential d of degree +1 (satisfying the Leibnitz rule in super sense d(a·b) = d(a)·b+(−1)^{deg(a)}a·d(b)) and such that d^{2} = 0. One can see immediately that d1 = 0. As a vector space, the n-th graded component (ΩA)^{n} for n ≥ 0 is isomorphic to A⊗(A/k·1_{A})^{⊗n}. The isomorphism is given by the following map

a_{0} ⊗ a_{1} ⊗ … ⊗ a_{n} ⟼ a_{0} · da_{1} · . . . · da_{n} ∈ (ΩA)^{n} .

The space of 1-differentials Ω_{A}^{1} := (ΩA)^{1} can be furthermore identified via the map

A^{⊗3}⟶Ω_{A}^{1}, a_{1} ⊗ a_{2} ⊗ a_{3} ⟼ a_{a} · da_{2} · a_{3}

with the quotient space A^{⊗3}/∂_{4}(A^{⊗4}), where

∂_{4}(a_{1} ⊗ a_{2} ⊗ a_{3} ⊗ a_{4}) = a_{1}a_{2} ⊗ a_{3} ⊗ a_{4} − a_{1} ⊗ a_{2}a_{3} ⊗ a_{4} + a_{1} ⊗ a_{2} ⊗ a_{3}a_{4}) .

The map ∂_{4} can be, evidently, included into one of standard complexes in homological algebra

… ⟶A^{⊗4}⟶A^{⊗3}⟶A^{⊗2}⟶A⟶0

This complex can be contracted using the unit element 1_{A} ∈ A, and thus has vanishing cohomology. Therefore Cokernel(∂_{4}) = Kernel(∂_{2}) and we get an equivalent description of Ω_{A}^{1} as the kernel of the multiplication map A ⊗ A⟶A.

If one considers analogous definitions in the purely commutative case, one can see immediately the difference between T A and ΩA. In the case of A equal to the algebra of functions on a smooth affine variety X, the commutative analogue of the algebra T A is the algebra of functions on the total space T X of the tangent bundle to X. On the contrary, the commutative analogue of ΩA is the algebra of differential forms on X, which is the same as the algebra of functions on supermanifold ΠT X, the total space of the odd tangent bundle to Z (as usual in super-mathematics, letter Π denotes the functor of changing the parity).

1.1.3. Another criterion of formal smoothness

Theorem. An algebra A is formally smooth iff it satisfies the property

4) there exists a derivation \underline{D} : T A⟶T A of degree +1 such that \underline{D}(\underline{a}) = \underline{Da} for all a ∈ A.

If A is an algebra generated by elements (a_{i})_{i∈I}, then in order to define \underline{D} one should define only elements \underline{D}(\underline{Da_{i}}) ∈ T A of degree +2 satisfying certain relations. In the case when the generating set I is finite and all relations between (a_{i}) follow from a finite number of relations we say that A is finitely presented. It is easy to see that in this case algebra T A is also finitely presented and in order to check that some candidates for \underline{D}(\underline{Da_{i}}) satisfy needed relations one should make only a finite calculation.

The theorem analogous to the one above holds in the commutative case (if the characteristic of the ground field k is zero). The derivation \underline{D} in commutative case is a vector field on the total space of the tangent bundle. One can easily see that choices of \underline{D} correspond to symmetric connections (i.e. connections with vanishing torsion) on the tangent bundle to Spec(A).

1.1.4. Homological properties of modules over smooth algebras

Theorem. ([CQ1], Proposition 5.1) If A is a formally smooth algebra, then A is hereditary, i.e. the category A-mod has homological dimension ≤ 1. In other words, any submodule of a projective module is projective. Equivalently, every A-module admits a projective resolution P_{−1}⟶P_{0} of length 2.

The proof is immediate, because

Ext_{A−mod}^{n}(E, F) = Ext_{A−mod−A}^{n}(A, Hom_{k−mod}(E, F)) ∀E, F ∈ Ob(A − mod) .

Definition. We call an algebra A smooth, if it is formally smooth and finitely generated.

Theorem. If A is a smooth algebra and E, F are A-modules finite-dimensional over k, then the vector spaces Ext_{A−mod}^{0}(E, F) = Hom_{A−mod}(E, F) and Ext_{A−mod}^{1}(E, F) are both finite-dimensional.

The statement of the theorem is evident for Ext^{0}. The space Ext^{1} coincides with the set of equivalence classes of structures of an A-module on E ⊕ F such that natural maps

F⟶E ⊕ F⟶E

are morphisms of A-modules, modulo the action of the vector space Hom_{k−mod}(E, F) considered as an abelian group. If A is finitely generated, then the set of A-module structures on E ⊕ F as above is a finite-dimensional vector space over k, and we get the statement of the theorem.

1.2. Examples of smooth algebras

E1) The free algebra k⟨x_{1}, …, x_{d}⟩ = ⊗^{∗}(k^{d}) d ≥ 0. We think of this algebra as the one corresponding to a noncommutative affine space N𝔸_{k}^{d} (see Sect. 3 where we define noncommutative spaces in general).

E2) The matrix algebra Mat(n × n, k).

E3) The algebra of upper-triangular matrices

A = {(a_{ij})_{1≤i,j≤n}| a_{ij} ∈ k, a_{ij} = 0 for i > j} ⊂ Mat(n × n, k) ,

E4) O(C), the algebra of functions on a smooth affine curve C over k.

E5) P aths(Γ), the algebra of finite paths in a finite oriented graph Γ. The basis of P aths(Γ) consists of sequences (v_{0}, e_{0,1}, v_{1}, e_{1,2}, …, e_{n−1,n}, v_{n}), n ≥ 0 of vertices v_{i} of Γ and oriented edges e_{i,i+1} connecting v_{i} with v_{i+1} (there could be several edges connecting two given vertices). The unit element in P aths(Γ) is equal to the sum over vertices v of Γ of paths of zero length (v). The multiplication in P aths(Γ) is given by concatenation of paths:

(v_{0}, e_{0,1}, …, v_{n}) · (v′_{0}, e′_{0,1}, …, v′_{n′}) =
(v_{0}, e_{0,1}, …, v_{n}, e′_{0,1}, …, v′_{n′})

if v_{n} = v′_{0}, and 0 otherwise.

The last example E5) contains E1) and E3) as particular cases. In E1) the corresponding graph has one vertex and d loops, and in E3) it is the chain of n − 1 consecutive oriented edges.

There are several constructions using which one can construct new smooth algebras from the old ones. In the following list A, A_{1}, A_{2} denote smooth algebras, and the result of the constructions is automatically a smooth algebra:

C1) A_{1} ⊕ A_{2}, the direct sum of two smooth algebras (or an arbitrary finite number of smooth algebras),

C2) A_{1} ⋆ A_{2}, the free product of smooth algebras.

C3) (Localization) If S is a subset of a formally smooth algebra A, then the algebra A[S^{−1}] obtained by formally adjoining the inverses of elements of S is formally smooth. In particular, if A is smooth and S ⊂ A is an arbitrary finitely generated multiplicative subset of A, then S^{−1}A is smooth. More generally, one can invert not only individual elements of A but square matrices with coefficients in A (i.e. S could be a subset of ∐_{n≥1} Mat(n × n, A)), or even rectangular matrices.

C4) End_{A−mod}(P), where P is a finitely-generated projective A-module. C5) (The total space of the tangent bundle) T A, the algebra defined in 1.1.

1.3. Representation spaces of smooth algebras

Let A be a smooth k-algebra. We associate with A an infinite sequence (Repr_{n}^{A}, n ≥ 1) of smooth affine varieties over k. The n-th variety Repr_{n}^{A} parameterizes homomorphisms from A to the standard matrix algebra Mat(n × n, k). The set of B-points of it, where B is a commutative algebra over k, is defined as

Repr_{n}^{A}(B) := Hom_{Alg_{k}}(A, Mat(n × n, B)) .

It is clear that since A is finitely-generated, the scheme Repr_{n}^{A} is an affine scheme of finite type over k. Moreover, from the definitions of smoothness in both commutative and noncommutative case, it follows that the scheme Repr_{n}^{A} is smooth. The affine algebraic group P GL_{k}(n) acts via conjugations on Repr_{n}^{A}.

Every element a ∈ A gives tautologically a matrix-valued function â ∈ O(Repr_{n}^{A}) ⊗ Mat(n × n, k) on Repr_{n}^{A}.

1.3.1. Examples

For A = k⟨x_{1}, …, x_{d}⟩, the representation scheme Repr_{n}^{A} coincides with the affine space 𝔸^{dn^{2}} over k.

For A = k ⊕ k = k⟨p⟩/(p^{2} = p), the scheme Repr_{n}^{A} is the disjoint union over all integers m, 0 ≤ m ≤ n of certain bundles over Grassmanians Gr(m, n). Fibers of these bundles are affine spaces parallel to the fibers of the cotangent bundles. The component of Repr_{n}^{A} corresponding to m (— the rank of the image of p ∈ A) has dimension 2m(n − m).

The example of A = k ⊕ k has an interesting feature. One can see that Repr_{n}^{A} carries a natural non-degenerate closed 2-form

ω = T race(p̂dp̂ ∧ dp̂)

where p̂ is the matrix-valued function on Repr_{n}^{A} corresponding to the generator p of A.

1.3.2. Noncommutative analogues of differential-geometric structures

We would like to propose the following general principle which could help to look for noncommutative versions of usual geometric notions (see also [K]).

A noncommutative structure of some kind on A should give an analogous “commutative” structure on all schemes Repr_{n}^{A}, n ≥ 1.

Here we make a list of several natural candidates:

Functions. Elements of the vector space A/[A, A] give rise to functions on Repr_{n}^{A}. Namely, any a ∈ A modulo commutators [A, A] gives the function T race(â). Thus, we get a homomorphism of vector spaces

A/[A, A]⟶O(Repr_{n}^{A}) .

It has an obvious extension to the homomorphism of algebras

Sym^{∗}(A/[A, A])⟶O(Repr_{n}^{A}) .

We will denote the element of O(Repr_{n}^{A}) corresponding to ϕ ∈ Sym^{∗}(A/[A, A]) by T race(ϕ).

In the following examples we can multiply our differential-geometric objects by an arbitrary element of the algebra Sym^{∗}(A/[A, A]).

Vector fields. Any derivation ξ ∈ Der(A) of A gives a vector field ξ̃ on Repr_{n}^{A}. Differential forms. Any element of ΩA (see 1.1.2) gives a matrix-valued differential

form on Repr_{n}^{A}, and thus a scalar-valued differential form after taking the trace in the matrix algebra. Again, using the vanishing of the trace of commutators one can see that the map ΩA⟶Ω^{∗}(Repr_{n}^{A}) goes through the quotient space

ΩA/[ΩA, ΩA]_{super}⟶Ω^{∗}(Repr_{n}^{A})

where the commutator in superalgebra ΩA is understood in super sense. Also, one can easily see that the differential d in Ω_{A} maps to the de Rham differential in usual forms Ω^{∗}(Repr_{n}^{A}). For example, algebra A = k ⊕ k has a closed 2-form. The definition of “noncommutative differential forms” as elements of ΩA/[ΩA, ΩA]_{super} is due to M. Karoubi.

De Rham cohomology. It is natural in the light of the previous discussion to define the de Rham cohomology of A as the cohomology of the complex ΩA/[ΩA, ΩA]. Using results of Cuntz and Quillen [CQ2], one can show that H_{dR}^{n}(A) defined in this way coincides with the Z/2ℤ-graded periodic cyclic homology HP_{n}(A) for n = 0, 1, and with the reduced periodic cyclic homology

H̃P_{n(mod 2)}(A) := HP_{n(mod 2)}(A)/HP_{n(mod 2)}(k)

for n ≥ 2. In other words, H_{dR}^{n}(A) coincides with HP_{n(mod 2)}(A) for n = 0, 1, 3, 5, … and has one dimension less than HP_{n(mod 2)}(A) for n = 2, 4, 6, . . ..

Volume element, polyvector fields. It looks plausible from examples that one can define noncommutative analogues of polyvector fields and volume elements, but we would like to postpone the discussion of these notions to the future. It seems that the divergence of a derivation of A with respect to a volume element belongs to the symmetric square of the vector space A/[A, A]. As an exercise to the reader we leave the following simple

Lemma. If A = k⟨x_{1}, … x_{d}⟩ then there exists a linear map

div : Der(A)⟶Sym^{2}(A/[A, A])

such that for any ξ ∈ Der(A) the divergence of the corresponding vector field ξ̃ on 𝔸^{dn^{2}} = Repr_{n}^{A} with respect to the standard volume element, is equal to T race(div(ξ)).

The last remark which we would like to make is that the correspondence (Smooth algebras) ⟶ (Representation schemes) gives a “justification” of the notation T A in 1.1. Namely, there is a natural isomorphism between Repr_{n}^{T A} and the total space of the tangent bundle T Repr_{n}^{A}.

1.3.3. Possible relations to matrix integrals and to M-theory

One can imagine that in the case k = ℂ, for a given “volume element” vol on A, a “function” f ∈ Sym^{∗}(A/[A, A]) and for a the set of real points γ_{n} in Hom(A, Mat(n × n, ℂ)) under some anti-holomorphic involution, one can take the integral of exp(T race(f))× vol and get a “matrix model”, an infinite sequence of numbers

I_{n} := ∫_{γ_{n}} e^{T race(f)}vol

parameterized by the dimension n = 1, 2, . . .. In mathematical physics such integrals were extensively studied. Typically, one integrates over the space of hermitean n × n matrices, or over real, or unitary matrices. In multi-matrix models the integration is taken over the set of, say, d-tuples (X_{1}, …, X_{d}) of hermitean matrices. It is believed that the asymptotic behavior of I_{n} as n⟶∞ is related with some kind of string theory.

Also, one of recently proposed matrix models, so called M-theory, is formulated in the same fashion. Roughly speaking, M-theory on the space-time manifold X = ℝ^{d} is a matrix theory corresponding to the free algebra with d generators, the noncommutative affine space. M-theory on curved spaces should correspond to nontrivial smooth noncommutative algebras.

1.3.4. Double tangent space and formal noncommutative structure

Here we would like to give some examples of natural non-classical structures on manifolds Repr_{n}^{A}. First of all, there is a natural vector bundle T_{(2)} on the square Repr_{n}^{A} ×_{k} Repr_{n}^{A} together with the identification of its pullback via the diagonal embedding Δ : Repr_{n}^{A}⟶Repr_{n}^{A} ×_{k} Repr_{n}^{A} with the tangent bundle T_{Repr_{n}^{A}}.

In what follows we will describe the bundle T_{(2)}. First of all, k-points of Repr_{n}(A) can be identified with equivalence classes of pairs (E, pr_{E}) where E is an A-module finitedimensional over k, and pr_{E} : A^{n}⟶E is an epimorphism to E from the standard ndimensional free A-module such that the set of n canonical generators of A^{n} maps to a basis of E over k. The fiber of T_{(2)} at the pair ((E, pr_{E}), (F, pr_{F})) is defined as

Hom_{A−mod}(Ker(pr_{E}), F) .

Let us prove that it is a finite-dimensional vector space. It is easy to see that it fits into an exact sequence

0⟶Hom_{A−mod}(E, F)⟶F^{n}⟶Hom_{A−mod}(Ker(pr_{E}), F)⟶Ext_{A−mod}^{1}(E, F)⟶0

All spaces except the one in question are finite-dimensional by the theorem 1.1.4. Also, it is easy to see that on the diagonal E = F, pr_{E} = pr_{F}, the space Hom_{A−mod}(Ker(pr_{E}), F) coincides with the tangent space to Repr_{n}^{A} at the point (E, pr_{E}).

Another structure on Repr_{n}^{A} is the formal noncommutative structure in the sense of Kapranov. There is a canonical sheaf O_{Repr_{n}^{A}}^{noncomm} of formal noncommutative functions on Repr_{n}^{A} with the quotient O_{Repr_{n}^{A}}. For example, if A = k⟨x_{1}, …, x_{d}⟩ and n = 1 then the global sections of the sheaf O_{Repr_{n}^{A}}^{noncomm} is the projective limit lim(A/I_{n}) where I_{n}, n ≥ 1 is a decreasing sequence of two-sided ideals

I_{n} = ∑ ; l≥1,(m_{1},...,m_{l}):∑ m_{i}=n A · A^{[m_{1}]} · A · A^{[m_{2}]} · . . . · A^{[m_{l}]} · A .

Here A^{[m]} for m ≥ 1 is defined as the linear span of the set of commutators of depth m:

[a_{0}, [a_{1}, [… [a_{m−1}, a_{m}] …] ∈ A, a_{0}, a_{1}, …, a_{m} ∈ A .

We refer the reader to [Ka] for the definition of the formal noncommutative structure and its differential-geometric meaning. We would like only to mention that Kapranov proposed a general construction of formal noncommutative structure on moduli spaces of objects in abelian categories, like the category of coherent sheaves on algebraic varieties etc. His construction admits a useful extension. Namely, if C is a k-linear triangulated category and E_{0} is a fixed object, then one can try to consider the “moduli space” of pairs (E, p), where E is an object of C and p : E_{0}⟶E is a morphism. The tangent complex for such a pair is RHom_{C}(E, Cone(p)). It carries a natural structure of an associative non-unital algebra, and the arguments from [Ka] are applicable there.

1.4. What we would like to do?

It is clear from all the previous discussion that the subject of smooth noncommutative geometry merits further development. All our previous constructions and examples are affine, in particular they give affine representation schemes. It would be desirable to give a definition of “smooth noncommutative schemes”, and also of “quasi-projective noncommutative schemes”, such that it contains finitely generated (or maybe finitely presented) smooth algebras as an affine case, and give rise to smooth schemes via certain functor of “representations to Mat(n × n, k)”. We expect that representation schemes carry bitangent bundles, Kapranov’s formal noncommutative structure, etc.

In the second part of this text we describe a general approach to noncommutative algebraic geometry based on flat topology. We choose (temporarily) the name “spaces” (or “noncommutative spaces”) instead of “schemes” for several reasons. Noncommutative schemes were introduced in earlier works of one of us (see [R2]), where analogues of Zariski topology were studied. Also, our formalism describes not only usual schemes but algebraic spaces too. As the reader will see, an amazing variety of basic constructions in usual algebraic geometry can be extended to the noncommutative setting. Our category of noncommutative spaces could be used in other situations. For example, quantum projective spaces arising from Sklyanin algebras also fit into our definition.

In the third part we study one particular space which we call “noncommutative projective space” and denote by Nℙ_{k}^{n}. We beleive that Nℙ_{k}^{n} is one of the principal examples of what should be called “smooth projective noncommutative variety”.

2. NONCOMMUTATIVE SPACES

2.1. Covers and refinements

Here we describe intermediate objects which are not yet spaces (there are no structure sheaves on them). These objects are machines producing abelian categories. Essentially all definitions can be made in general monoidal categories (even in non-additive categories), instead of the category of vector spaces over k with the monoidal structure given by the tensor product.

2.1.1. Category of finite covers Definition. Objects of category Covers_{k} are given by the following

DATA:

1) an associative algebra B ∈ Ob(Alg_{k}),

2) M ∈ Ob(B − mod − B), a bimodule over B,

3) m_{M} : M⟶M ⊗_{B} M, a homomorphism of bimodules,

4) e_{M} : M⟶B, also a homomorphism of bimodules, satisfying the following AXIOMS:

1) M is faithfully flat as the right B-module, i.e. the functor M⊗_{B} : B − mod⟶B − mod is exact and does not kill any non-zero morphism,

2) M with m_{M} and e_{M} is a coassociative coalgebra with counit in the monoidal category of B ⊗ B^{opp}-modules.

We will usually denote objects of the category Covers_{k} as pairs (B, M) skipping data m_{M} and e_{M}. Morally, we consider pairs (B, M) as “noncommutative stacks” together with a finite affine cover with affine pairwise intersections of members of the cover (see examples in the next subsection).

For every object (B, M) of Covers_{k} we have an abelian category QCoh(B, M) whose objects are pairs (E, m_{E}) where E is a B-module and m_{E} : E⟶M ⊗_{B} E is a homomorphism of B-modules which defines a coaction of coalgebra M on E. We call objects of QCoh(B, M) quasi-coherent sheaves on the “noncommutative stack” corresponding to (B, M). One can justify the name “sheaf” introducing an appropriate Grothendieck topology.

Definition. Morphisms f in the category Covers_{k} from (B_{1}, M_{1}) to (B_{2}, M_{2}) are given by the following

DATA:

1) f_{B} : B_{2}⟶B_{1}, a morphism of algebras,

2) f_{M} : M_{2}⟶M_{1}, a morphism of k-vector spaces satisfying the following AXIOMS:

1) the diagram where vertical arrows are structure morphisms of M_{i} as B_{i}-bimodules (i = 1, 2) is commutative:

B_{2} ⊗ M_{2} ⊗ B_{2} ⟶^{f_{B}⊗f_{M} ⊗f_{B}} B_{1} ⊗ M_{1} ⊗ B_{1}
↓  ↓
M_{2}  ⟶^{f_{M}}  M_{1}

2),3) two analogous diagrams including coproduct morphisms m_{M_{i}} and counit morphisms e_{M_{i}} respectively, are commutative.

Notice that the direction of the morphism f = (f_{B}, f_{M}) is opposite to the direction of the pullback morphisms of algebras and bimodules.

Every morphism f = (f_{B}, f_{M}) : (B_{1}, M_{1})⟶(B_{2}, M_{2}) of covers defines a functor f^{∗} : QCoh(B_{2}, M_{2})⟶QCoh(B_{1}, M_{1}), the pullback of quasi-coherent sheaves. This functor maps an object (E, m_{E}) to the object (B_{1} ⊗_{B_{2}} E, m′_{E}) where the coaction morphism

m′_{E} : B_{1} ⊗_{B_{2}} E⟶M_{1} ⊗_{B_{1}} B_{1} ⊗_{B_{2}} E = M_{1} ⊗_{B_{2}} E

is defined in a natural way using f_{B}, f_{M} and m_{E}.

2.1.2. Examples of finite covers

FC1) Let S/Spec(k) be a separated quasi-compact scheme, e.g. a quasi-projective scheme. We choose a finite cover (U_{i})_{i∈I} of S in Zariski topology by affine schemes. It follows from separatedness that pairwise intersections U_{i} ∩U_{j} are again affine. The algebra B and the bimodule M are defined as

B := O(⊔_{i}U_{i}), M := O(⊔_{i,j}(U_{i} ∩ U_{j}))

and the structure of coalgebra on M is the natural one. It follows from the usual descent theory that the category QCoh(B, M) is equivalent to the category of quasi-coherent sheaves on S.

FC1’) The same statement holds for affine covers of separated quasi-compact schemes in fpqc topology, for algebraic stacks etc.

FC2) Let A be an associative algebra and M := A with the natural structure of an A-bimodule, and a coalgebra in the category A-mod-A. Then QCoh(A, M) is equivalent via the tautological functor to the category A-mod.

FC3) Let A be an associative algebra, and B ⊃ A be a larger algebra such that B is faithfully flat as the right A-module. Then we define M as B ⊗_{A} B with the natural structure of a coalgebra in the category B-mod-B. We have a natural morphism of covers

f : (A, A)⟶(B, M)

Theorem. The morphism f as above defines an equivalence f^{∗} between the categories QCoh(B, M) and QCoh(A, A) = A-mod.

This theorem follows from the general Barr-Beck theorem. Recall that a comonad (or cotriple) in the category C consists of a functor T : C → C and two functor morphisms δ : T → T^{2} and ϵ : T → Id_{C} such that T δ∘δ = δT ∘δ and ϵT ∘δ = id_{T} = T ϵ∘δ. A coaction of the comonad T is a morphism c : X → T (X) such that T c∘c = δ(X)∘c and ϵ(X)∘c = id_{X}. We denote by T -comod the category of T -comodules, i.e. objects of C endowed with a coaction of T, with naturally defined morphisms: a morphism (X, c) → (X′, c′) is a morphism f : X → X′ such that T f ∘ c = c′ ∘ f. An arbitrary pair of adjoint functors F : C_{1}⟶C_{2}, G : C_{2}⟶C_{1} with adjunction morhisms ϵ : F G → Id_{C_{2}} and η : Id_{C_{1}} → GF determines a comonad (T, δ, ϵ), where T = F G, δ = F ηG, and there is an obvious functor C_{1}⟶T -comod, , Y ↦ (F (Y), F η(Y)). The following version of Barr-Beck’s theorem is sufficient for our needs.

Theorem. Let F : C_{1}⟶C_{2} be a functor between two categories having a right adjoint functor G : C_{2}⟶C_{1}. Assume that the following conditions hold:

(a) a pair of arrows f, g : X ⇉ Y has a kernel if its image by F has a kernel;

(b) if h : Z⟶X is such that f ∘ h = g ∘ h and F h is the kernel of the pair (F f, F g), then h is a kernel of (f, g).

Then the canonical functor C_{1}⟶T -comod is an equivalence of categories.

We apply this theorem to the case C_{1} = A-mod, C_{2} = B-mod and F = B⊗_{A}.

Note that the arbitrary pairs of morphisms in Barr-Beck’s theorem can be replaced by so called coreflexive pairs, i.e. the pairs f, g : X ⇉ Y such that there exists a morphism e : Y → X such that e ∘ f = id_{X} = e ∘ g. And even this condition can be weakened (cf. [MLM], IV.4).

2.1.3. Refinements of covers

Let (B_{1}, M_{1}) be a cover and i : B_{1}⟶B_{2} be an inclusion of algebras such that B_{2} is faithfully flat as a right B_{1}-module. We define M_{2} as B_{2} ⊗_{B_{1}} M_{1} ⊗_{B_{1}} B_{2} with the natural structure of a coalgebra in the category B_{2}-mod-B_{2}. It is easy to see that (B_{2}, M_{2}) is again a cover and we have a natural morphism of covers f : (B_{2}, M_{2})⟶(B_{1}, M_{1}).

Definition. A morphism of covers isomorphic to f as above is called a refinement morphism.

We denote by Ref the class of refinement morphisms. Analogously to the example FC3) from the previous subsection, the functors f^{∗} for f ∈ Ref are equivalences of categories of quasi-coherent sheaves. Here we apply Barr-Beck theorem to the functor which acts from QCoh(B_{1}, M_{1}) to B_{2}-mod and is given by formula F (E, m_{E}) = B_{2} ⊗_{B_{1}} E.

The class of refinements is closed under compositions. Also, any pullback of a refinement morphism is again a refinement morphism. Thus, we can easily describe the localization of the category of covers with respect to refinements.

2.2. Noncommutative spaces

In this section we give the definition of noncommutative spaces (in several steps). Morally one should think about noncommutative space X as about an abelian category QCoh(X) and two adjoint functors π_{∗} : QCoh(X)⟶k-mod, π^{∗} : k-mod⟶QCoh(X). The pullback π^{∗}(k^{1}) of the standard 1-dimensional k-module is the “structure sheaf” O_{X}.

2.2.1. Covers of noncommutative spaces

We define covers of noncommutative spaces adding some data and an axiom to the definition of the category of covers:

Definition. Objects of category Covers_{k}^{sp} are pairs (C, s_{C}) where C = (B, M) is a cover and s_{C} : B ⊗ B⟶M a homomorphism of bimodules which is an epimorphism and also a morphism of coalgebras. Morphisms in Covers_{k}^{sp} are morphisms of covers compatible with structure epimorphisms from B ⊗ B.

In other words, we demand that M should be a cyclic bimodule generated by an element m := s_{C}(1_{B} ⊗ 1_{B}) which behaves well with respect to the coalgebra structure. Such M is given by a left ideal Ker(s_{C}) in B ⊗ B^{opp} satisfying a complicated system of axioms.

We will call objects of Covers_{k}^{sp} space covers. Notice that all our examples of covers (except stacks) are automatically space covers.

The morphism s_{C} give rise to a canonical morphism from (C, s_{C}) to the refinement (B, B ⊗ B) of the “point” object Spec(k) := ((k, k), id) (the final object in Covers_{k}^{sp}). The pullback of the standard 1-dimensional module k^{1} under this morphism we call the structure sheaf O_{C}. It is represented by 1-dimensional free B-module B^{1} = B with the coaction of M arising from s_{C}.

Also, for any refinement f : (B_{2}, M_{2})⟶(B_{1}, M_{1}) from cover to a space cover there exists a unique structure of space cover on (B_{2}, M_{2}) such that f became a morphism of space covers.

We define equivalence classes of noncommutative spaces as equivalence classes of space covers modulo the relation generated by refinements. In order define the “right” category of noncommutative spaces we need to make an additional work.

2.2.2. Equivalence relation between morphisms of space covers

Two different morphisms of space covers could give isomorphic functors of pullbacks on categories of quasi-coherent sheaves. We would like to identify such morphisms.

Let (C, s_{C}) and (C′, s_{C′}) be two space covers, and B resp. B′ denote corresponding algebras.

Definition. Two morphisms f, g : (C, s_{C})⟶(C′, s_{C′}) are equivalent iff for any element ∑_{α} x_{α} ⊗ y_{α} ∈ Ker(s_{C′}) ⊂ B′ ⊗ B′ the following equations hold:

∑_{α} f(x_{α}) · g(y_{α}) = ∑_{α} g(x_{α}) · f(y_{α}) = 0 ∈ B

Theorem. The relation described above is an equivalence relation. There is a canonical identification of pullback functors for equivalent morphisms.

We will not give here the proof of this theorem. It follows from general categorical considerations and from an explicit calculation of certain products in the category of space covers (see theorem in 2.3).

The equivalence of morphisms is compatible with composition. Thus, we can form a quotient category { \atop Covers}_{k}^{sp}.

2.2.3. The category of noncommutative spaces

Definition. The category Spaces_{k} of noncommutative spaces over k is defined as the localization { \atop Covers}_{k}^{sp}[Ref^{−1}] of the category of space covers with respect to the class of refinements.

We call morphisms of the category Spaces_{k} “maps”. Let us spell explicitly how maps look like in terms of covers. A map from (B, M) to (B_{1}, M_{1}) is given by a refinement (B′, M′)⟶(B_{1}, M_{1}) and a morphism of space covers f′ : (B′, M′)⟶(B_{1}, M_{1}). Two such data ((B′, M′), f′) and ((B′′, M′′), f′′) give the same map iff for some (equivalently, for any) given common refinement (B′′′, M′′′) (e.g. the cartesian product of (B′, M′) and (B′′, M′′) over (B, M)) two corresponding cover morphisms from (B′′′, M′′′) to (B_{1}, M_{1}) are equivalent.

Theorem. The category of separated quasi-compact schemes over k is equivalent to a full subcategory of Spaces_{k}. The category Alg_{k}^{opp} is also equivalent to a full subcategory of Spaces_{k}. Category Spaces_{k} has finite limits.

Theorem. Construction (B, M)⟶QCoh(B, M) extends to a functor from the category Spaces_{k} to the category of abelian k-linear categories. For separated quasi-compact schemes, it gives usual quasi-coherent sheaves of commutative algebraic geometry. For associative algebras, it gives categories of left modules.

One can consider noncommutative stacks by eliminating the condition that the structure homomorphism s_{S} : B ⊗ B⟶M is surjective. We will not develop here the more general theory of stacks.

2.3. Affine covers

We call affine spaces noncommutative spaces corresponding to the cover of the type (B, B) where B is an algebra. Abusing notations, we will denote the space (B, B) by Spec(B), although here we do not use points of the spectrum ([R3], [R1], [R4]) at all. The algebra B can be identified with

Hom_{QCoh(Spec(B))}(O_{Spec(B)}, O_{Spec(B)}) .

Also, B coincides as a set with the set of maps from (B, B) to the (non)-commutative affine line 𝔸_{k}^{1} = N𝔸_{k}^{1} := Spec(k[t]). In general, d-dimensional noncommutative affine space N𝔸_{k}^{d} is defined as Spec(k⟨x_{1}, …, x_{d}⟩). The algebra structure on O(S) = Maps(S, N𝔸_{k}^{1}) is induced by the algebra structure on N𝔸_{k}^{1} considered as an object of Spaces_{k}.

Let π : Spec(B)⟶S be a cover of a space by an affine space. The category of spaces admits finite limits. Thus, we can form an infinite sequence of spaces S^{(n)}, n ≥ 1 taking the cartesian product over S of n copies of Spec(B). It is easy to see that all spaces S^{(n)} are affine. The algebra of functions on S^{(n)} is generated by n copies i_{1}(B), …, i_{n}(B) of the algebra B subject to the set of relations which we will describe now.

Relations: Let z = ∑_{α} a_{α} ⊗ b_{α} ∈ B ⊗ B be an arbitrary element of the kernel of the structure map s_{S} : B ⊗ B⟶M. Then for any k ≠, l, 1 ≤ k, l ≤ n the element ∑_{α} i_{k}(a_{α}) · i_{l}(a_{α}) is equal to zero in O(S^{(n)}).

One can also give a description of spaces S^{(n)} in terms of categories of quasi-coherent sheaves.

Theorem. The category QCoh(S^{(n)}) is canonically equivalent to the category of n-tuples (E_{i})_{i=1,...,n} of objects of QCoh(Spec(B)) ≃ B-mod endowed with an identfication of n objects π_{∗}(E_{i}), i = 1, …, n in the category QCoh(S).

It is easy to see that the collection of spaces S^{(n)} (and natural maps between them) form a contravariant functor from the category of finite non-empty sets to the category of affine spaces. For any space U, the set of maps Maps(U, S^{(2)}) is a subset of the square of the set Maps(U, S^{(1)}) and forms there a graph of an equivalence relation. A map from U to S can be described locally (i.e. after passing to a cover) as a map to S^{(1)} modulo the equivalence relation coming from maps to S^{(2)}.

2.4. Cohomology of quasi-coherent sheaves

As we mentioned in the previous section, for every space S, there exists a distinguished object O_{S} in the category QCoh(S). It is easy to see that the functor E ↦ Hom(O_{S}, E) is the right adjoint to the inverse image functor f_{S}^{∗}; QCoh(point)⟶QCoh(S) for the canonical map f_{S} : S⟶point = Spec(k). We denote the vector space Hom(O_{S}, E) as Γ(E) (the space of global sections of E). The functor Γ is left exact.

We would like to define now the derived functor for Γ. First of all, it makes sense because of the following

Theorem. Category QCoh(S) has enough injective objects.

The proof is the following. Let us chose an affine cover π : Spec(B)⟶S. The functor π^{∗} has a right adjoint π_{∗}. The category B-mod (canonically ≃ QCoh(Spec(B))) has enough injective objects. Now, for any quasi-coherent sheaf E on S let us chose an embedding of π^{∗}E into an injective B-module I. Due to the fact that the inverse image functor π^{∗} is exact, π_{∗}I is again an injective object. Since the functor π^{∗} is faithful, the natural homomorphisms E⟶π_{∗}π^{∗}E is a monomorphism. And the image π_{∗}π^{∗}E⟶π_{∗}I of the embedding π^{∗}E⟶I is a monomorphism too, because the direct image functor π_{∗} is left exact (as any functor having a left adjoint).

Thus, one can proceed and define derived functor RΓ (and more generally RHom, more generally derived functor of any left exact functor) using injective resolutions.

Theorem. For any cover (B, M) of S and for any object (E, m_{E}) ∈ QCoh(B, M), the cohomology of S with coefficients in E can be calculated via the Čech complex

E⟶M ⊗_{B} E⟶M ⊗_{B} M ⊗_{B} E⟶ …

One can extend the previous result to the relative case. For every morphism f : (C_{1}, s_{C_{1}})⟶(C_{2}, s_{C_{2}}) in category Covers_{k}^{sp} the functor f^{∗} admits a right adjoint f_{∗}. Since the direct image functor f_{∗} is left exact, there is a derived functor for f_{∗}.

2.5. The definition of coherent sheaves

In commutative algebraic geometry the abelian category of coherent modules are usually considered in the case of noetherian rings. In the noncommutative setting, our principal examples are free algebras in several indeterminates which are far from being noetherian. Nevertheless, one can define a reasonable abelian category of modules over smooth algebras as well.

Definition. A module M over an algebra A is called coherent iff it is finitely presented, i.e. there exists an exact sequence

F_{−1}⟶F_{0}⟶M⟶0

where F_{−1}, F_{0} are free finitely generated A-modules.

We denote Coh(A) the full subcategory of A − mod objects of which are coherent modules.

Lemma. For any hereditary algebra A (e.g. for a formally smooth A), the category Coh(A) is an abelian category.

It is enough to check that the kernel of any morphism ϕ : M⟶N of coherent Amodules is coherent. The proof is the following. In hereditary abelian categories any object in the bounded derived category is isomorphic to its cohomology. Thus, Ker(ϕ) is a direct summand in D^{b}(A − mod) of a perfect complex. Using telescope construction we see that Ker(ϕ) is quasi-isomorphic in an infinite complex bounded from above of finitely generated projective modules. From this it follows that Ker(ϕ) is coherent.

The property of a module over arbitrary algebra to be (or not to be) finitely presented is preserved under faithfully flat extensions of the algebra. This implies that for spaces one can define the notion of a finitely presented quasi-coherent sheaf passing to an arbitrary affine cover. The property of being finitely presented is independent of the cover.

For any noncommutative space S admitting a cover by a smooth or noetherian affine space, the category of finitely-presented quasi-coherent sheaves is abelian. In such a case we call this the category Coh(S) of coherent sheaves.

2.6. Formally smooth, formally non-ramified and formally étale morphisms.

We call an affine scheme morphism Spec(A) → Spec(B) a thickenning if the corresponding algebra morphism B → A is an epimorphism and its kernel is a nilpotent ideal.

Following the pattern of commutative algebraic geometry [EGA, IV.17], we give the following

Definitions. 1) Let f : X → Y be a morphism. We call f formally smooth (resp. formally non-ramified, resp. formally étale) if, for any affine scheme Spec(A), any thickenning Spec(T) → Spec(A), and any morphism Spec(A) → Y, the canonical morphism

Hom_{Y} (Spec(A), X) → Hom_{Y} (Spec(T), X)

defined by the immersion Spec(T) → Spec(A) is surjective (resp. injective, resp. bijective).

2) A k-space X will be called formally smooth if the canonical morphism X → Spec(k) is formally smooth.

Remarks. (i) If X is affine, then the definition of the formal smoothness given here is in accordance with the one in 1.1.

(ii) To check that f is formally smooth (resp. formally non-ramified, resp. formally étale), it suffices to do it in the case when the square of the kernel J of the algebra epimorphism A → T is zero.

(iii) The properties of f defined in 2.6 are the properties of the represented by f functor Y′ ↦ Hom_{Y} (Y′, X) from the category dual to the category of Y -spaces to the category Sets. They make sense for any functor (Spaces_{k}/Y)^{op} → Sets, representable or not.

(iv) It follows from the definition that f is formally étale iff it is both formally smooth and formally non-ramified. One can show that Spec(A) is étale (i.e. Spec(A) → Spec(k) is étale) iff the algebra A is separable. The latter means that A is a projective A-bimodule, or equivalently, A has dimension zero with respect to Hochschild cohomology. An example of a separable algebra is the algebra Mat(n × n, k) of n × n matrices.

3. NONCOMMUTATIVE PROJECTIVE SPACE

3.1. Definition á la Grothendieck.

For any integer d ≥ 1, we define the noncommutative projective space Nℙ_{k}^{d−1} as a space representing a certain contravariant functor from the category of affine spaces to the category Sets.

Let A be an algebra. The set Map(Spec(A), Nℙ_{k}^{d−1}) should be functorially identified with the set of quotient modules F of the standard free d-dimensional A-module A^{d}, such that locally in flat topology F is isomorphic to the free 1-dimensional A-module. The last condition implies that F is projective.

From this definition, it is not clear whether Nℙ_{k}^{d−1} exists. We shall sketch a prove of its existence by producing an explicit affine cover.

3.2. Explicit cover.

We will construct a cover of Nℙ_{k}^{d−1} analogous to the Jouanolou’s cover of the projective space in commutative algebraic geometry. Recall that this is a cover by the affine quadric:

{(x_{1}, …, x_{d}; y_{1}, …, y_{d})| ∑_{i=1}^{d} x_{i}y_{i} = 1}⟶{(x_{1} : …: x_{d})} = ℙ^{d}(k) .

Let B be the associative algebra generated by 2d variables x_{1}, …, x_{d}, y_{1}, …, y_{d} satisfying the relation ∑_{i=1}^{d} y_{i}x_{i} = 1. The space Spec(B) represents the following functor on affine spaces: Maps(Spec(A), Spec(B)) is the set of quotient modules F of A^{d} together with an isomorphism i : A^{1} ≃ F of A-modules, and a splitting s of A^{n} into the direct sum of F and a complementary module F′. Also, for each n ≥ 1 we can consider functor whose value on Spec(A) is the set of collections (F, i_{1}, s_{1}, …, i_{n}, s_{n})), where F is a quotient module of A^{d} and for each k = 1, …, n, the triple (F, i_{k}, s_{k}) belongs to Maps(Spec(A), Spec(B)). It is easy to see that this functor is representable by an affine space whose algebra of functions we denote by B^{(n)}.

Thus we have constructed a contravariant functor from the category of non-empty finite sets to the category of affine spaces. We can construct (as in 2.3) a new contravariant functor from spaces to sets by passing locally to sets of equivalence classes of maps to Spec(B) modulo an equivalence relation coming from maps to Spec(B^{(2)}). This new functor coincides, more or less by definition, with the functor described in the previous section 3.1.

One can check by direct calculations that the algebra B^{(2)} is generated by two copies i_{1}(B), i_{2}(B) modulo relations of the type described in 2.3. The bimodule M in this case is (B ⊗ B)/I where I is the left B ⊗ B^{opp}-ideal generated by the following set of elements:

e_{j} := −1 ⊗ x_{j} + ∑_{i=1}^{d} x_{j}y_{i} ⊗ x_{i}, j = 1, …, d .

This shows that the functor defined in 3.1 is representable by (B, M).

Analogously, one can imitate the Grothendieck’s definition of projective space for coherent sheaves and introduce relative projective space ℙ(E) for any noncommutative space S and a finitely generated quasi-coherent sheaf E on S.

3.3. Derived category of quasi-coherent sheaves

Here we will only state the results without proofs (proofs will appear among other things in [KR]). We denote by Q_{d} the quiver which has two vertices {v_{0}, v_{1}} and d oriented edges all going from v_{0} to v_{1}.

Theorem. For any d ≥ 1 the category QCoh(Nℙ_{k}^{d−1}) has cohomological dimension 1. The bounded derived category D^{b}(QCoh((Nℙ_{k}^{d−1})) is equivalent to the bounded derived category of representations of the quiver Q_{d}.

Theorem. The algebra B described in 3.2 and the projective space Nℙ_{k}^{d−1} are formally smooth. The bounded derived category of coherent sheaves on Nℙ_{k}^{d−1} is of finite type (i.e. for any two objects E, F we have ∑_{i} rk(RHom^{i}(E, F)) < +∞), and it is equivalent to the bounded derived category of finite-dimensional representations of Q_{d}. The group K_{0}(Coh(Nℙ_{k}^{d−1})) is free abelian group with 2 generators.

The main ingredient in the proof of both theorems is a noncommutative analogue of Beilinson’s resolution of the sheaf of functions on the diagonal in ℙ_{k}^{d−1} × ℙ_{k}^{d−1}. The noncommutative resolution is much shorter than the commutative one, and has length 2. There are two coherent sheaves O and O(1) (the universal quotient module F from 3.1) which generate in an appropriate sense the whole derived category on Nℙ_{k}^{d−1}.

There are two surprising particular cases: d = 1 for which the noncommutative projective space is more complicated than the usual one (which is a point), and the case d = 2. It is well-known that the category D^{b}(Coh(ℙ_{k}^{1})) is equivalent to D^{b}(Q_{2} −mod_{finite}). Thus, in the case d = 2 we have three different abelian categories with the same derived category: the noncommutative projective line, the commutative projective line, and the quiver Q_{2}.

3.4. Grassmanians

One can define Grassmanians NGr_{k}(d′, d) for d′, d ≥ 1 in the same fashion as the projective space, changing the requirement that the quotient module E of O^{d} should be locally isomorphic to a free d′-dimensional module, instead of a free 1-dimensional module. We claim that in contrast to the commutative case we do not get a new space. It follows from the fact that there exists a non-zero algebra A such that A^{1} is isomorphic A^{2} (and thus to A^{d′}) as A-module. For example, one can take A equal to End(V) where V is an infinite-dimensional vector space. On the faithfully flat extension which is obtained by taking free product with such algebra A, we identify conditions on E for all Grassmanians. Thus, NGr_{k}(d′, d) coincides with Nℙ_{k}^{d−1} = NGr_{k}(1, d).

We expect that various remarkable identities in noncommutative linear algebra in free skew field discovered by Gelfand and Retakh (see [GR]) can be interpreted as identities between morphisms of coherent sheaves on Nℙ_{k}^{d−1} and on similar spaces.

3.5. Representation spaces

For a noncommutative space S we can try to define representation schemes Repr_{n}^{S}. At least, it is clear how sets of k-points should look like:

Reps_{n}^{S}(k) = Maps(Spec(Mat(n × n, k)), S) .

It follows from the definition of the projective space in 3.1 and from the discussion in 3.4 that the set of k-points of Nℙ_{k}^{d−1} is equal to the set of all non-zero quotient spaces of k^{d}. We leave to the reader as an exercise the description of maps from Spec(Mat(n × n, k)) to Nℙ_{k}^{d−1} for the case n ≥ 2.

Let us express again our hopes: representation schemes Repr_{n}^{S} for (formally) smooth non-affine spaces S should carry all structures described in Part 1 (the double tangent bundle, Kapranov’s formal noncommutative functions etc.).

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