QUASICOHERENT SHEAVES IN COMMUTATIVE AND NONCOMMUTATIVE GEOMETRY.

DMITRI ORLOV

Intorduction

In this paper we try to survey a category of objects of noncommutative algebraic geometry. In the commutative case the notion of scheme is introduced in two steps. The first step is the definition of affine schemes, and the second one prescribes how to glue a scheme from affine parts. This way to proceed appeal to the topological properties of commuative schemes. It uses the functor Spec from the category of commutative algebras to the category of topological spaces. On the other hand, any scheme can be considered as a presheaf of sets on the category of affine schemes. We will exploit this simple fact in our treatment of the noncomutative case.

Similar to commutative algebraic geometry, the category of affine noncommutative schemes is opposite to the category of algebras (we work over a basic ring k). Any object which can be regarded as a noncommutative scheme has to give rise to presheaf of sets on the category of affine objects. In this paper we consider all presheaves of sets on the categories of affine schemes both in the commutative and in noncomutative cases. To each presheaf X we associate a category Qcoh_{r}(X) of (right) quasicoherent modules on it (2.6). Moreover we prove that any quasicoherent module is a sheaf in the canonical topology on X (3.5 and 4.9). In the commutative case we define a new topology on the category of affine schemes that is related to quasicoherent modules (3.12). The sheafification functor for this topology, (called here effective descent topology) preserves categories of quasicoherent modules (3.13).

Further, we find a “good” embeddding of the commutative case into the noncomutative one, using presheaves of groupoids on the category of noncommutative affine schemes (4.4). This embedding induces equivalence for categories of quasicoherent modules on any object X, when we consider it both in the commutative and in noncomutative worlds (4.7).

I would like to thank Alexander Rosenberg for usefull discussions and for his interest to this work. My special thanks are to Andrei Suslin who answered to many my questions.

The work was done while the author was visiting the Max-Plank-Institute für Mathematik in Bonn. I am grateful to MPIM for support.

1. Some categorical preliminaries. Sites.

1.1 We collect here some facts related to categories and sites. The main reference is [1] (see also [5]). Let C be a category. Denote by C the category of contravariant functors from C to the category of sets. An object of C is called a presheaf of sets. There is a natural functor h : C ⟶ C that sends an object R ∈ C to the functor h_{R}(·) := Hom(·, R) . The presheaf h_{R} is called representable. Henceforce we will write R instead h_{R} . By Ioneda Lemma h is a full embedding, moreover for any presheaf X ∈ C there is the natural isomorphism Hom_{C}ˆ (R, X) = X(R) .

the author was partly supported by RFFI grants 96-01-00149 and 97-01-00933.

For any object X ∈ C denote by C/X the category of pairs (R, ϕ) , where R ∈ C and ϕ ∈ X(R) with a canonical functor j_{X} : C/X ⟶ C that sends (R, ϕ) to R . Note that any presheaf X ∈ C is a colimit in C of representable objects by the sistem C/X .

1.2 Let u : C → C′ be a functor. Denote by û^{∗} a functor from C′ to C that takes a presheaf Y ∈ C′ to the composition Y u .

Since the category of sets has all small limits and colimits the functor û^{∗} admits left and right adjoint functors û_{!} and û_{∗} respectively (see [1], I, 5.1.). The functor û^{∗} preserves all small limits and colimits. The functor û_{∗} preserves all small limits. The functor û_{!} preserves all small colimits and the restriction of it on the subcategory C ⊂ C coincides with u . Moreover for any presheaf X ∈ C we have

u_{!}X(S) ≅ lim_{⟶_{R∈C/X}} Hom(S, uR).

Furthermore, the functor û_{!} is fully faithful if and only if the functor u is fully faithful(see [1], I.5.6) (We recall that a functor F : C ⟶ C′ is called fully faithful if it induces an isomorphism Hom_{C}(Y, X) →^{∼} Hom_{C′}(F Y, F X) for any pair Y, X of objects from C .)

1.3 A sieve in the category C is a full subcategory D ⊂ C that satisfies the following condition: any object of C for which there exists a morphism from it to an object of D contains in D . Let R be an object, a sieve in the category C/R will be called a sieve on R .

It is easy to see that a sieve on R is nothing more than a subpresheaves of R in the category C . Henceforce, we will consider sieve on R as a presheaf. The notion of sieve is the main tool for the introduction of a topology on a category.

1.4 Definition. A Grothendieck topology T on a category C is defined by giving for each object R ∈ C a set J(R) of sieves on R which is called covering sieves. They have to satisfy the following conditions: T1) For any R the maximal sieve C/R is in J(R) . T2) If T ∈ J(R) and f : S ⟶ R is a morphism of C , then the sieve f^{∗}(T) :=

{U ⟶^{α} S | fα ∈ T } is in J(S) .

T3) If T ∈ J(R) is a covering sieve and U is a sieve on R such that f^{∗}(U) ∈ J(S) for each

S ⟶^{f} R in T , then U ∈ J(R) .

1.5 Definition. A site Φ = (C, T ) is a category C with a Grothendieck topology T .

Let Φ = (C, T ) be a site, and R be an object of C . A family of morphisms F = (f_{α} : R_{α} ⟶ R), α ∈ I is called covering if the sieve T_{F} ↪ R , which is generated by the family F , is a covering seive on R .

1.6 Definition. Let C be a category with fiber products. A pretopology on C is defined by specifying for each object R of C , a set Cov(R) of families of morphisms to R such that P1) If (R_{α} → R) α ∈ I is in Cov(R) and S → R is a morphism of C , then the family

(R_{α} × S → S), α ∈ I is in Cov(S) .

P2) If (R_{α} → R) α ∈ I is in Cov(R) and (R_{β_{α}} → R_{α}) β_{α} ∈ J_{α} is in Cov(R_{α}) for each

α ∈ I , then the family (R_{γ} → R) γ ∈ ∐_{α∈I} J_{α} is in Cov(R) . P3) The family R ⟶^{id_{R}} R is in Cov(R) .

Any pretopology P on C generates a topology T such that a sieve is T -covering iff it contains some P -covering family.

1.7 A presheaf X ∈ C is called sheaf (resp. separate presheaf) if for any object R ∈ C and for any covering sieve S ∈ J(R) the canonical morphism

Hom_{C}ˆ (R, X) ⟶ Hom_{C}ˆ (S, X)

is a bijection (resp. an injection).

Let us define by Shv(C, T ) the category of all sheaves on a site Φ = (C, T ) . The category Shv(C, T ) is a full subcategory of C .

The inclusion functor i : Shv(C, T ) ⟶ C has a left adjoint functor \underline{a} : C ⟶ Shv(C, T ) that is known as the associated sheaf functor, or the sheafification functor. The functor \underline{a} preserves all finite limits and colimits. For any presheaf X we can define a presheaf LX by the rule

LX(R) = lim_{⟶}_{T ∈J(R)} Hom_{C}ˆ (T, X)  (1)

This correspondence gives a functor from C to itself which will be denoted by L . The composition of the functor L with itself is isomorphic to the functor i\underline{a} , i.e. we obtain an isomorphism \underline{a}X ≅ LLX . (see [1], III)

1.8 Let Φ = (C, T ) and Φ′ = (C′, T′) be two sites.

A functor u : C ⟶ C′ is called continue if for any sheaf Y ∈ Shv(C′, T′) the presheaf û^{∗}(Y) is a sheaf on site Φ . In this case û^{∗} induces a functor from Shv(C′, T′) to Shv(C, T ) , which will be denoted by u^{∗} . The functor u^{∗} has a left adjoint functor u_{!} = \underline{a}′û_{!}i .

A functor u : C ⟶ C′ is called cocontinue if for any object R ∈ C and for any covering sieve S ↪ u(R) the sieve on R , which is generated by all arrows Z ⟶ R such that u(Z) ⟶ u(R) factorized through S , is a covering sieve on R . Note that u is cocontinue iff the functor û_{∗} : C ⟶ C′ takes any sheaf on Φ to a sheaf on Φ′ . Denote by u_{∗} : Shv(C, T ) ⟶ Shv(C′, T′) the restriction of û_{∗} on the categories of sheaves. It has a left adjoint functor u^{∗} = \underline{a}û^{∗}i′ (see [1], III).

Let u : C′ ⟶ C be a functor and T is a topology on C . The induced topology on C is the largest topology such that the functor u is continue.

1.9 Fix an object X ∈ C and consider the category C/X with the canonical functor j_{X} : C/X ⟶ C . If T is a topology on C , then denote by T_{X} the induced topology on C/X and by Φ_{X} denote the site (C/X, T_{X}) . It is proved in [1], III.5. that a seive S of an object (Z ⟶ X) is covering in C/X iff the sieve ĵ_{X}_{!}(R) ↪ Z is covering in C . Therefore, the functor j_{X} : C/X ⟶ C is continue and cocontinue. This gives us three adjoint functors:

\begin{matrix}j_{X∗} : & Shv(C/X, T_{X}) & ⟶ & Shv(C, T ) \\ j_{X}^{∗} : & Shv(C, T ) & ⟶ & Shv(C/X, T_{X}) \\ j_{X!} : & Shv(C/X, T_{X}) & ⟶ & Shv(C, T ) \end{matrix}  (2)

The functor j_{X∗} is called the direct image functor. The functor j_{X}^{∗} is called the inverse image functor, or restriction functor to C/X . The functor j_{X!} is called extension by zero onto C .

Let m : Y ⟶ X be a morphism in C . It induces a functor j_{m} : C/Y ⟶ C/X . As above, there are three functors

j_{m}^{∗} : Shv(C/X, T_{X}) ⟶ Shv(C/Y, T_{Y}), j_{m∗}, j_{m!} : Shv(C/Y, T_{Y}) ⟶ Shv(C/X, T_{X})

such that j_{m∗} and j_{m!} are right and left adjoint to j_{m}^{∗} respectively.

1.10 Proposition. [1] The functor j_{X!} : Shv(C/X, T_{X}) ⟶ Shv(C, T ) has the following decomposition

Shv(C/X, T_{X}) ⟶^{e_{X}} Shv(C, T )/\underline{a}X ⟶ Shv(C, T ),

and the functor e_{X} is an equivalence.

Moreover, the canonical sheafification map m : X ⟶ \underline{a}X for any presheaf X ∈ C induces the equivalences j_{m}^{∗} and j_{m∗} = j_{m!} between the categories Shv(C/X, T_{X}) and Shv(C/\underline{a}X, T_{\underline{a}X}) .

1.11 A topology, for which every representable presheaf is a sheaf, is called subcanonical; the largest such topology is called canonical. The canonical topology will be denoted by can .

Let us describe covering families for the canonical topology.

We need to give the definition of equalizer. If f_{i} : X ⟶ Y, i = 1, 2 are two morphisms of a category C , the an equalizer is a morphism k : K → X which is final in a full subcategory of C/X consisting of morphisms k′ : K′ → X with f_{1}k′ = f_{2}k′ . When it exists, the equalizer of two morphisms is unique up to isomorphism.

Suppose that all fiber products exist in a category C . A family (X_{α} → X)_{α∈I} of maps in the category C is called a universal effectively epimorphic family if for all maps Z ⟶ X and for all objects Y ∈ C the following diagram of sets

Hom(Z, Y) ⟶ ∏_{α∈I} Hom(Z ×_{X} X_{α}, Y) \overrightarrow{⟶} ∏_{α,β∈I} Hom(Z ×_{X} (X_{α} ×_{X} X_{β}), Y)

is an equalizer.

It is clear that a family is covering for the canonical topology if and only if it is a universal effectively epimorphic family. If now X ∈ C is some object then a family (W_{α} ⟶ W)_{α∈I} is covering for the object (W → X) ∈ C/X in the canonical topology on C/X iff it is a universal effectively epimorphic family in C .

1.12 Proposition. Let X be a presheaf on the category C . Then the topology on C/X induced by the canonical topology on C with respect to the functor j_{X} : C/X ⟶ C coincides with the canonical topology on C/X .

It follows from Corollary 3.3. of [1] and from the description of covering families for the canonical topologies on C and C/X .

2. Ringed sites and quasicoherent sheaves

2.1 A category C with a presheaf of rings is called a ringed category, i.e. we fix an object A of the category C with ring structure. A site Φ = (C, T ) with a sheaf of rings A is called a ringed site.

2.2 Let Φ = (C, T ) be a ringed site with a sheaf of rings A . A presheaf of (right) A –modules is a presheaf M with a (right) module structure over ring object A .

If M is a sheaf it will be called a sheaf of (right) A –modules.

Denote by Mod_{r}(A, T ) the category of sheaves of (right) A –modules on the site Φ . The category Mod_{r}(A, T ) is an abelian category that verifies the conditions AB5 and AB3^{∗} and possesses a family of generators (see [1], II).

2.3 Let X be an object of C . The category C/X with the topology T_{X} , induced by the functor j_{X} : C/X ⟶ C , forms the site Φ_{X} = (C/X, T_{X}) . This site is ringed by the sheaf of rings A_{X} := j_{X}^{∗}A . We will denote the category of sheaves of (right) A_{X} –modules by Mod_{r}(A_{X}, T_{X}) (or by Mod_{r}(X, T_{X}) when the ring sheaf is fixed).

Let M be a sheaf of (right) A –modules. The sheaf j_{X}^{∗}M has the structure of (right) A_{X} – module. Furhter, if N is a sheaf of (right) A_{X} –modules then j_{X∗}N is a sheaf of (right) j_{X∗}A_{X} – modules. Using the canonical morphism A ⟶ j_{X∗}A , we obtain that j_{X∗}N has the structure of (right) A –module. Thus there is a couple of adjoint functors:

Mod_{r}(A_{X}, T_{X}) \overleftarrow{⟶}_{j_{X∗}}^{j_{X}^{∗}} Mod_{r}(A, T )

(for functors between categories of sheaves of modules we use the same notation as in 1.9 for functors between categories of sheaves of sets.)

The functor j_{X}^{∗} : Mod_{r}(A, T ) ⟶ Mod_{r}(A_{X}, T_{X}) has a left adjoint functor j_{X!} that is called the extension by zero. Note that, unlike the functors j_{X}^{∗} and j_{X∗} , the composition of the functor j_{X!} for sheaves of modules with the forgetful functor to the category of sheaves of sets does not coincide with the functor j_{X!} for sheaves of sets defined in 1.9 (see [1], IV.11.)

Let m : Y ⟶ X be a morphism in C . As above, the functor j_{m} : C/Y ⟶ C/X induces three functors

j_{m}^{∗} : Mod_{r}(A_{X}, T_{X}) ⟶ Mod_{r}(A_{Y} , T_{Y}), j_{m∗}, j_{m!} : Mod_{r}(A_{Y} , T_{Y}) ⟶ Mod_{r}(A_{X}, T_{X})

such that j_{m∗} and j_{m!} are right and left adjoint to j_{m}^{∗} . Moreover the functors j_{m}^{∗} and j_{m∗} commute with the same functors between the categories of sheaves of sets defined in 1.9 with respect to forgetful functors.

2.4 Proposition. Let Φ = (C, T ) be a site ringed by a sheaf A . Suppose m : X ⟶ \underline{a}X is the canonical sheafification map for a presheaf X ∈ C . Then the functors j_{m}^{∗} and j_{m∗} between the categories Mod_{r}(A_{X}, T_{X}) and Mod_{r}(A_{\underline{a}X}, T_{\underline{a}X}) are quasiinverse equivalences.

It follows from Proposition 1.10.

2.5 Assume that a presheaf X is representable, i.e. X ∈ C . Let M be a right A(X) –module. One can attach to it a presheaf of (right) A_{X} –modules M̃ on the category C/X . It is defined as

M̃(S) := M ⊗_{A(X)} A(S).

for any object S → X of C/X .

Fix an object X ∈ C and consider the category C/X .

2.6 Definition. A presheaf F of (right) A_{X} –modules on the category C/X will be called a (right) quasicoherent module on X if for any m ∈ X(R) the presheaf ĵ_{m}^{∗}(F) on C/R with the canonical structure of (right) A_{R} –module has a form M̃ , where M is a (right) A(R) –module.

Morphisms between quasicoherent modules are , by definition, morphisms between them considered as presheaves of (right) A_{X} –modules. Denoted by Qcoh_{r}(X) the category of (right) quasicoherent modules on X , which is a full subcategory of the category of presheaves of (right) A_{X} –modules.

Note that the definition of quasicoherent module on X ∈ C does not depend from the topology on C

2.7 Remark It follows immediately from the definition that each quasicoherent module on R ∈ C is isomorphic to M̃ for some A(R) –module M , i.e the category Qcoh_{r}(R) is equivalent to the category of (right) A(R) –modules. Hence a presheaf F of (right) A_{X} –modules is a quasicoherent module on X ∈ C iff for each (R ⟶^{s} X) ∈ C/X the presheaf ĵ_{s}^{∗}F is a quasicoherent module on R .

2.8 The category Qcoh_{r}(X) for any X ∈ C has all cokernels and all direct sums. Hence it has all small colimits (see, for example, [4] I.5).

Let m : Y ⟶ X be a morphism in C . The functor ĵ_{m}^{∗} : (C/X) ⟶ (C/Y) induces the inverse image functor from Qcoh_{r}(X) to Qcoh_{r}(Y) which will be denoted by m^{∗} . The functor m^{∗} is right exact, this means that it preserves cokernels.

2.9 Assume now that the topology T on C satisfies the following condition: every quasicoherent module on any object X ∈ C is a sheaf for the topology T_{X} on C/X . In this case the category Qcoh(X) is a full subcategory of Mod(A_{X}, T_{X}) . Denote by φ_{X} the inclusion Qcoh(X) ↪ Mod(A_{X}, T_{X}) .

2.10 Proposition. Let Φ = (C, T ) be a site ringed by a sheaf A . Suppose that all quasicoherent modules on any object X are sheaves for the topology T_{X} on C/X . Let m : X ⟶ \underline{a}X be the canonical sheafification map for a presheaf X ∈ C . Then the inverse image functor m^{∗} : Qcoh(\underline{a}X) ⟶ Qcoh(X) is fully faithful. Proof. There is the commutative diagram of functors

Qcoh(X) ⟶^{φ_{X}} Mod(O_{X}, can_{X})
m^{∗}↑  ↑j_{m}^{∗}
Qcoh(\underline{c}X) ⟶^{φ_{\underline{c}X}} Mod(O_{\underline{c}X}, can_{\underline{c}X})

where the horisontal arrows are full embeddings. By the Proposition 2.4, the functor j_{m}^{∗} is an equivalence. Hence the functor m^{∗} is fully faithful. □

2.11 Suppose that the inclusion functor φ_{X} : Qcoh_{r}(X) ⟶ Mod_{r}(A_{X}, T_{X}) has a right adjoint functor Q_{X} . The functor Q_{X} is called coherator.

Since φ_{X} is fully faithful the composition Q_{X}φ_{X} is isomorphic to the identity functor on Qcoh_{r}(X) . If F ⟶^{f} G is a morphism in Qcoh_{r}(X) then it has a kernel Ker(f) which can be defined as Q_{X}Ker(φ_{X}(f)) .

It is easy to check that if coherator exists then the category Qcoh_{r}(X) is abelian.

For a representable object X ∈ C , the coherator Q_{X} exists and it is clearly the functor sending a sheaf F of A_{X} –modules to the quasicoherent module { \atop F(X)} .

Let m : Y ⟶ X be a morphism in C . If there is a coherator Q_{X} then the functor m^{∗} has a right adjoint functor m_{∗} which is defined as the composition Q_{X}j_{m∗}φ_{Y} .

2.12 To any presheaf X ∈ C we associate the category of (right) quasicoherent modules Qcoh_{r}(X) . This construction can be extended on a larger class of object by two ways.

First, let us consider an immersiom i : Y ↪ X . One can define a category of pair Qcoh_{r}(X, Y) as a full subcategory of Qcoh_{r}(X) consisting of such modules F for which i^{∗}(F) ≡ 0 .

Any immersion i : Y ↪ X induces the factor object X/Y . It is a pointed object and it is defined as a coproduct

\begin{matrix}Y & ↪^{i} & X \\ ↓ & & ↓p \\ ∗ & ↪ & X/Y \end{matrix}

We can compare the categories Qcoh_{r}(X/Y, ∗) and Qcoh_{r}(X, Y) .

2.13 Proposition. A natural functor p^{∗} : Qcoh(X/Y, ∗) ⟶ Qcoh(X, Y) is an equivalence. Proof. Let us construct a quasiinverse functor p_{∗} : Qcoh_{r}(X, Y) ⟶ Qcoh_{r}(X/Y, ∗) . For any R ∈ C the set X/Y (R) = X(R)/Y (R) . Let s ∈ X/Y (R) be a section. Eaither there is a unique pull back of s to the section s′ ∈ X(R) or it goes through ∗ → X/Y . If now F is a quasicoherent module on X such that i^{∗}(F) ≡ 0 , then we put p_{∗}(F)(s) ≅ F(s′) or put it by zero if it is pointed section. It is easy to see that this define a functor that is quasiinverse to p^{∗} . □

2.14 Second, we can consider not only presheaves of sets but , say, presheaves of groupoids (We will not discuss more general case of presheaves of categories). By groupoid we mean a small category in which every arrow is invertible. All small groupoids generated the 2–category. If we identify all functors between groupoids which are equivalent then we obtain the 1–category. Denote this 1– category of all small groupoids by Grd . Any set can be considered as a groupoid, objects of which are elements of this set and morphisms are only identities. We can attach to each groupoid G two sets: the first Ob(G) is the set of all objects of G , the second π_{0}(G) is the seT of connected components of G . There are two natural morphisms Ob(G) ⟶ G and π_{0} : G ⟶ π_{0}(G) .

Let C_{gr} be the category of presheaves of groupoids on C , i.e the category of all contravariant functors from C to the category of small groupoids Grd . It contains the category C as a full subcategory. Any presheaf X ∈ C_{gr} induces two presheves of sets Ob(X) and π_{0}(X) defined as

Ob(X)(R) = Ob(X(R)), π_{0}(X)(R) = π_{0}(X(R)), R ∈ C

There are natural morphisms ob_{X} : Ob(X) ⟶ X and π_{0X} : X ⟶ π_{0}(X) between the presheaves of groupoids. It is easy to see that these correspondences gives two functors Ob and π_{0} from C_{gr} to C .

The morphism π_{0X} : X ⟶ π_{0}(X) splits for any X ∈ C_{gr} .(Actually, choosing for each connected component some representative, we obtain a splitting). The functor π_{0} : C_{gr} ⟶ C is simultaneously right and left adjoint to the canonical immersion C ⟶ C_{gr} .

If Φ = (C, T ) is a site then it is defined the category of sheaves of groupoids Shv_{gr}(C, T ) .

We say that the presheaf X ∈ C_{gr} is a sheaf of groupoids if for any groupoid G ∈ Grd the presheaf of sets:

R ↦ Hom_{Grd}(G, X(R))

is a sheaf.

2.15 Suppose now that site Φ = (C, T ) is ringed by a sheaf A . For any object X ∈ C_{gr} let us define a category C/X . The set of objects of C/X is the set of all s ∈ Ob(X(S)), S ∈ C . A morphism from s ∈ Ob(X(S)) to r ∈ Ob(X(R)) is a pair (α, f) , where f : S ⟶ R in C and α is an isomorphism from X(f)(s) to r in X(R) . There is a canonical functor j_{X} : C/X ⟶ C . This gives us the site Φ_{X} = (C/X, T_{X}) that is ringed by the sheaf A_{X} = j_{X}^{∗}A . We can define the category of sheaves of A_{X} –modules Mod_{r}(A_{X}, T_{X}) . Futhermore, repeating Definition 2.6, we define the category Qcoh_{r}(X) of quasicoherent modules on X . The morphisms ob_{X} and π_{0X} induce the inverse image functors ob_{X}^{∗} : Qcoh(X) ⟶ Qcoh(Ob(X)) and π_{0X}^{∗} : Qcoh(π_{0}(X)) ⟶ Qcoh(X) respectively.

3. Commutative case

3.1 Fix a commutative ring k . Denote by Comalg/k the category of commutative algebras over k and by Aff/k denote the category of affine schemes that is opposite to Comalg/k . We note that the category of affine shemes has all fiber products and they correspond to the tensor products in the category Comalg/k . The canonical embedding of the category affine schemes to the category of all schemes Sch/k preserves all fiber products and finite disjoint unions. But it does not preserve all coproducts. For example, the infinite disjoint union of {Spec(R_{α})}_{α∈I} in Sch/k does not coincide, in general, with Spec( ∏_{α∈I} R_{α}) .

3.2 In this section by C we mean the category of all affine schemes Aff/k . The canonical contravariant functor Aff/k ⟶ Comalg/k gives a presheaf of rings O on Aff/k with

O(Spec(R)) := R

for any k –algebra R . This presheaf is represented by the ring scheme Speck[t] . Therefore it is a sheaf for any subcannonical topology on C .

Further, the category C/X for any object X ∈ C is ringed by the presheaf O_{X} which is a sheaf for any subcanonical topology.

3.3 Denote by can the canonical topology on C = Aff/k and by \underline{c} : C ⟶ Shv(C, can) denote the associative sheaf functor defined in 1.7, which is a left adjoint to the inclusion functor i : Shv(C, can) ⟶ C . The covering families for the canonical topology are the universal effectively epimorphic families defined in 1.11. In our case there is a simple but useful description of the universal effectively epimorphic families in terms of modules.

3.4 Proposition. A family (f_{α} : Spec(R_{α}) → Spec(R))_{α∈I} is universal effectively epimorphic in Aff/k if and only if for any R –module M the following diagram

M ⟶ ∏_{α∈I} M ⊗_{R} R_{α} \overrightarrow{⟶} ∏_{α,β∈I} M ⊗_{R} (R_{α} ⊗ R_{β})  (3)

is an equalizer. Proof. ⇒ Take an R –module M and consider S = R⊕M as R –algebra with multiplicaition low given as (r_{1}, m_{1})(r_{2}, m_{2}) = (r_{1}r_{2}, r_{1}m_{2} + m_{2}r_{1}) . Denote by S_{α} an algebras S ⊗_{R} R_{α} . We have the family (g_{α} : Spec(S_{α}) → Spec(S))_{α∈I} , where g_{α} = 1_{S} ⊗ f_{α} . This family is effectively epimorphic. Therefore, considering morphisms to the scheme Spec(k[t]) , we obtain that

S ⟶ ∏_{α∈I} S ⊗_{R} R_{α} \overrightarrow{⟶} ∏_{α,β∈I} S ⊗_{R} (R_{α} ⊗ R_{β})  (4)

is an equalizer. And hence this is true for M as a direct summand. ⇐ Let (f_{α} : Spec(R_{α}) → Spec(R))_{α∈I} be a family. We know that for any morphism Spec(S) ⟶ Spec(R) the sequence (4) is exact. Further, for any algebra A the sequence of Hom′s from A to the sequence (4) in the category of commutative algebras is exact too. Hence (f_{α})_{α∈I} is an universal effectively epimorphic family. □

As above, we denote by Qcoh(X) the category of quasicoherent modules on X for any presheaf X ∈ C and by Mod(X, can_{X}) denote the category of sheaves of O_{X} –modules for the canonical topology on C/X .

3.5 Corollary. Let X be a presheaf on the category C = Aff/k . Any quasicoherent module on X is a sheaf in the canonical topology on the category C/X . Therefore the category Qcoh(X) is a full subcategory of Mod(X, can_{X}) .

This corollary follows from the previous proposition and 1.12.

Denote by φ_{X} the inclusion Qcoh(X) ↪ Mod(X, can_{X}) .

3.6 Remark Note that the category of all schemes Sch/k is embedding in C = (Aff/k) and for any scheme X ∈ Sch/k a quasicoherent sheaf in our definition one-to-one corresponds to the quasicoherent sheaf in the usual definition. (Consider the small Zariski site X_{Zar} and the natural inclusion of the categories X_{Zar} ⟶ Aff/X . Then the inverse image functor takes a quasicoherent sheaf in our definition to a quasicoherent sheaf in the usual definition and this is one-to-one correspondence). Futhermore, if a scheme X is quasicompact and quasiseparated then there exists a coherator Q_{X} : Mod(X, can_{X}) ⟶ Qcoh(X) . For the category of sheaves of modules on the small Zariski site it was proved in [2], II.3. Consequently, it is fulfilled in our situation too.

3.7 Corollary. Let X be a presheaf on the category C = Aff/k and m : X ⟶ \underline{c}X be a morphism of sheafification in the canonical topology. Then the inverse image functor m^{∗} : Qcoh(\underline{c}X) ⟶ Qcoh(X) is fully faithful.

It immediately follows from Propositions 3.4 and 2.4.

3.8 Corollary. Let T ↪^{i} Spec(R) be a sieve on an object Spec(R) ∈ C . The sieve T is covering in the canonical topology if and only if the functor i^{∗} : Qcoh(Spec(R)) ⟶ Qcoh(T) is fully faithful Proof. We can assume that the sieve T is generated by the family (f_{α} : Spec(R_{α}) → Spec(R))_{α∈I} . It was said in 2.11 that functor i^{∗} has a right adjoint functor i_{∗} that takes a quasicoherent module F on T to M̃ on Spec(R) , where the module M is an equalizer of

∏_{α∈I} F(Spec(R_{α})) \overrightarrow{⟶} ∏_{α,β∈I} F(Spec(R_{α} ⊗ R_{β})).

Assume that i^{∗} is fully faithful. Then i_{∗}i^{∗}(M̃) ≅ M̃ for any R –module M . Hence the sequence (3) is exact, and the sieve T is covering in the canonical topology.

Inverse statement direct follows from the previous corollary. □

3.9 On the category Aff/k there is a very important subcanonical topology. It is a well-known flat topology. The covering families for the flat topology are such universal effectively epimorphic families (f_{α} : Spec(R_{α}) → Spec(R))_{α∈I} for which all morphisms f_{α} are flat, i.e. R_{α} are flat R –algebras for all α . The flat topology on the category Aff/k will be denoted by ft .

3.10 Now we introduce a new topology which is subcanonical but finer than the flat topology. We call it by the effective descent topology ( edt ).

Recall briefly the descent theory for modules over commutative ring. Let f : R ⟶ S be a homomorphism of commutative rings. Let M be an S –module. There are two S ⊗_{R} S –modules M ⊗_{R} S and S ⊗_{R} M . Any S ⊗_{R} S –homomorphism ϕ : M ⊗_{R} S → S ⊗_{R} M defines three S ⊗_{R} S ⊗_{R} S –homomorphisms

\begin{matrix}ϕ_{1} : & S ⊗_{R} M ⊗_{R} S & ⟶ & S ⊗_{R} S ⊗_{R} M \\ ϕ_{2} : & M ⊗_{R} S ⊗_{R} S & ⟶ & S ⊗_{R} S ⊗_{R} M \\ ϕ_{3} : & M ⊗_{R} S ⊗_{R} S & ⟶ & S ⊗_{R} M ⊗_{R} S \end{matrix}

by tensoring ϕ with 1_{S} in the first, second and third positions respectively. Moreover, in composition with the multiplication μ : S ⊗_{R} M → M , ϕ induces a homomorphism ϕ̄ : M → M of S –modules:

ϕ̄: M ⟶ M ⊗_{R} S ⟶^{ϕ} S ⊗_{R} M ⟶^{μ} M

By a descent datum on an S –module M we mean an S ⊗_{R} S –homomorphism ϕ : M ⊗_{R} S → S ⊗_{R} M such that ϕ_{2} = ϕ_{1}ϕ_{3} and ϕ̄ is the identity on M .

If (M, ϕ) and (M′, ϕ′) are two descent data, then a morphism g : (M, ϕ) → (M′, ϕ′) of descent data is a S –homomorphism g : M → M′ such that (1_{S} ⊗ g)ϕ = ϕ′(g ⊗ 1_{S}) .

Such way we obtain the category of descent data for any homomorphism of commutative rings f : R ⟶ S . Denote it by Dd(f) . There is a functor d_{f} : Mod(R) ⟶ Dd(f) that takes an R –module N to (N ⊗_{R} S, ϕ_{N}) , where ϕ_{N} defined by the rule ϕ_{N}(n ⊗ s_{1} ⊗ s_{2}) = s_{1} ⊗ n ⊗ s_{2}. (An R –morphism h : N ⟶ N′ goes to h ⊗ 1_{S} ).

A homomorphism f : R ⟶ S of commutative rings is called an effective descent morphism if the functor d_{f} : Mod(R) ⟶ Dd(f) is an equivalence.

Denote by (T_{f} ↪^{i} Spec(R)) ∈ C the sieve on Spec(R) which consists of all maps Spec(B) ⟶ Spec(R) which go through the map f^{op} : Spec(S) ⟶ Spec(R) . It is easy to see that the category Dd(f) is nothing more than the category Qcoh(T_{f}) of quasicoherent modules on T_{f} and the functor d_{f} coincides with the inverse image functor i^{∗} .

Now we say that F = (f_{α} : Spec(R_{α}) → Spec(R))_{α∈I} is an effective descent family if the functor i^{∗} : Qcoh(Spec(R)) ⟶ Qcoh(T_{F}) is an equivalence, where T_{F} ↪^{i} Spec(R) ∈ C is the sieve generated by the family F .

3.11 Proposition. The effective descent families satisfy conditions P1)–3) of Definition 1.6, i.e. they generate a pretopology on C = Aff/k . Proof. Note that by Corollary 3.8 each effective descent family is an universal effectively epimorphic family. Consider some effective descent family F = (Spec(R_{α}) → Spec(R))_{α∈I} . Let Spec(S) ⟶ Spec(R) be a morphism and let F̄ be a quasicoherent module on the sieve TF̄ that is generated by the family F̄ = (Spec(R_{α} ⊗_{R} S) → Spec(S))_{α∈I} . We can define a quasicoherent sheaf F on T_{F} by the rule F(·) := F̄(· ×_{Spec(R)} Spec(S)) (this is the direct image of F̄ with respect to the canonical map from TF̄ to T_{F} . Since F is an effective descent family there is an R –module M such that F = i_{F}^{∗} M̃ . We see that the module M can be defined as an equalizer of

∏_{α∈I} F̄(Spec(S ⊗_{R} R_{α})) \overrightarrow{⟶} ∏_{α,β∈I} F̄(Spec(S ⊗_{R} R_{α} ⊗_{R} R_{β}))

Hence the R –module M has a structure of S –module and F̄ = i∗ ; F̄ M̃ . Therefore, F̄ is an effective descent family.

Let F = (Spec(R_{α}) → Spec(R))_{α∈I} be an effective descent family and, moreover, there fixed an effective descent family G_{α} = (Spec(R_{β_{α}}) → Spec(R_{α}))_{β_{α}∈J_{α}} for each α ∈ I . Let G be the family

(Spec(R_{γ}) → Spec(R))_{γ∈ ∐_{α∈I} J_{α}} . We have the sieves i_{F} : T_{F} ↪ Spec(R) , i_{G_{α}} : T_{G_{α}} ↪ Spec(R_{α})

and the map of the sieves p : T_{G} ↪ T_{F} .

Let f_{α} : Spec(R_{α}) ⟶ T_{F} and g_{α} : T_{G_{α}} → T_{G} be the canonical morphisms in the category C . Consider the following diagram of functors.

Mod(T_{G_{α}}, can)  ⟵^{j_{gα}^{∗}} Mod(T_{G}, can)
≀↑j_{i_{Gα}}^{∗}  ≀↑j_{p}^{∗}
Mod(Spec(R_{α}), can) ⟵^{j_{fα}^{∗}} Mod(T_{F} , can)

The both vertical arrows are equivalence because all effective descent families are covering in the canonical topology.

Let F be a quasicoherent module on T_{G} . The sheaves of modules F_{α} := j_{f_{α}}^{∗}j_{p∗}(F) are quasicoherent for all α , because j_{i_{Gα}}^{∗} (F_{α}) ≅ j_{g_{α}}^{∗}(F) and G_{α} are effective descent families. Since the sieve T_{F} is generated by the family (f_{α})_{α∈I} the sheaf of modules j_{p∗}(F) is quasicoherent. By assumption, F is an effective descent family. Hence there is an R -module M such that j_{p∗}(F) ≅ i_{F}^{∗} (M̃) and, consequently, F ≅ i_{G}^{∗}(M̃) . □

3.12 The effective descent topology on the category C = Aff/K ( edt ) is a topology for which covering sieves are all sieves generated by the effective descent families. It follows from Corollary 3.8 that edt is subcanonical. Moreover it is finer than the flat topology. It was proved in a lot of places that a faithful flat homomorphism f : R ⟶ S of commutative ring is an effective descent morphism (see, for example, [3], [6], [7]).

Denote by \underline{e} : C ⟶ Shv(C, edt) the sheafification functor for the effective descent topology.

3.13 Theorem. Let X ∈ C be a presheaf and m : X ⟶ \underline{e}X be the sheafification morphism in the effective descent topology. Then the inverse image functor m^{∗} : Qcoh(\underline{e}X) ⟶ Qcoh(X) is an equivalence Proof. The functor i\underline{e} is isomorphic to the composition L ∘ L , where the functor L is defined by formula (1) in 1.7, i.e. the morphism m is the composition

X ⟶^{l_{X}} LX ⟶^{l_{LX}} \underline{e}X

Hence it is sufficient to prove that the functor l_{X}^{∗} : Qcoh(LX) ⟶ Qcoh(X) is an equivalence for all X ∈ C .

By Corollary 3.5, the category Qcoh(X) is a full subcategory of Mod(X, can_{X}) . The functor l_{X}^{∗} is induced by the functor j_{l_{X}}^{∗} : Mod(LX, can_{LX}) ⟶ Mod(X, can_{X}) which is an equivalence.

Hence, to prove the theorem we must show that for any coherent sheaf F on X the sheaf of O_{LX} – modules F′ = j_{l_{X}∗}F is quasicoherent. By Remark 2.7 we need to check that for each morphism s : Spec(R) ⟶ LX the sheaf j_{s}^{∗}F′ is quasicoherent. By definition of the functor L we have

LX(R) = lim_{⟶}_{T ∈J(R)} Hom_{C}ˆ (T, X)

Therefore there exists a covering sieve i : T ↪ Spec(R) such that the following diagram commutes

T  ⟶^{s′} X
i↓  ↓l_{X}
Spec(R) ⟶^{s} LX

Since j_{l_{X}}^{∗} and j_{l_{X}∗} are quasi-inverse by Proposition 2.4 there is an isomorphism j_{i}^{∗}j_{s}^{∗}F′ ≅ j_{s′}^{∗}F of quasicoherent modules on T . As T is a covering sieve the functor j_{i}^{∗} induces an equivalence i^{∗} : Qcoh(Spec(R)) ⟶^{∼} Qcoh(T) . Therefore, the sheaf j_{s}^{∗}F′ is quasicoherent on Spec(R) for all s : Spec(R) ⟶ LX . Thus F′ = j_{l_{X}∗}F is a quasicoherent module on LX . □

4. Noncomutative case

4.1 Fix a ring k . Denote by Alg/k the category of algebras over k . If f_{i} : R ⟶ S_{i}, i = 1, 2 be two homomorphisms in Alg/k then a coproduct S_{1} ∐_{R} S_{2} always exists. We can take a presentation by generators and defining relations for both R –algebras S_{1} and S_{2} and write all these presentations together. The coproduct will be denoted by S_{1} ⋆_{R} S_{2} .

Denote by NAff/k the category that is opposite to Alg/k . We will call it by the category of noncommutative affine schemes. The object of NAff/k , which corresponds to an algebra R , will be denoted by Sp(R) . The category of noncommutative affine shemes has all fiber products and the product Sp(S_{1}) ×_{Sp(R)} Sp(S_{2}) is isomorphic to Sp(S_{1} ⋆_{R} S_{2})

4.2 Let N : Aff/k ⟶ NAff/k be the natural inclusion functor that takes Spec(R) to Sp(R) . The functor N is fully faithful and has a right adjoint functor Com : NAff/k ⟶ Aff/k . The functor Com sends an object Sp(R) to Spec(R_{c}) . Here R_{c} is the maximal commutative factor-algebra of R , i.e. R_{c} = R/I , where I is the ideal generated by all relations of the form < ab − ba > .

There is a functor N̂_{!} : (Aff/k) ⟶ N(Aff/k) that is an extension of N onto the category of presheaves (see 1.2). By 1.2 the functor N̂_{!} has the following description

N̂_{!}(X)(Sp(S)) = lim_{⟶_{Spec(A)→X}} Hom(Sp(S), Sp(A))

The functor N̂_{!} is also fully faithful by 1.2 and it has a right adjoint functor N̂^{∗} , restriction of which on NAff/k coincides with the functor Com .

We mean the functor N̂_{!} as a “course” embedding of the “commutative geometry” into the “noncommutative geometry”. To get a “fine” embedding we need to consider presheaves of groupoids.

4.3 Now for each object Sp(S) ∈ NAff/k we define a category I_{S} . The objects of I_{S} are couples (Spec(A), m) with Spec(A) ∈ Aff/k and m : Sp(S) ⟶ Sp(A) . A morphism from (Spec(A), m) to (Spec(A′), m′) is a morphism ξ : Spec(A) → Spec(A′) such that m′ = ξm .

Any morphism f : Sp(S′) ⟶ Sp(S) defines a functor I_{f} : I_{S} ⟶ I_{S′} . Moreover there is a functor pr_{S} : I_{S} ⟶ Aff/k that takes an object (Spec(A), m) to the Spec(A) . It is easy to see that the functor N̂_{!} can be given by the following formula

N̂_{!}(X)(Sp(S)) = lim_{⟶_{I_{S}}} X(pr_{S}(·))  (5)

4.4 Consider the category I_{S}/X of objects pr_{S} –over X . It has as objects all triples (Spec(A), m, a) with m : Sp(S) → Sp(A) and a : Spec(A) → X and as morphisms all morphisms in I_{S} that are compartible with maps to X . Denote by Gr(I_{S}/X) the groupoid associated to the category I_{S} (i.e. we invert all arrows). Every morphism g : Sp(S′) ⟶ Sp(S) gives a functor Gr(I_{S}/X) ⟶ Gr(I_{S′}/X) . On the other hand, every f : X′ ⟶ X in (Aff/k) induces a functor Gr(I_{S}/X′) ⟶ Gr(I_{S}/X) .

Let us define for an object X ∈ (Aff/k) a presheaf of groupoids \overline{N}X ∈ N(Aff/k)_{gr} by rule

\overline{N}X(Sp(S)) := Gr(I_{S}/X)  (6)

This correspondence gives us a functor \overline{N} from (Aff/k) to the category of presheaves of groupoids N(Aff/k)_{gr} . It immediately follows from formula (5) that π_{0}(\overline{N}X) is isomorphic to Ñ_{!}X . Hence we have the commutative diagram of functors

[figure: \overline{N}; π_{0}; N(Aff/k); N(Aff/k)_{gr}; (Aff/k); N̂_{!}]

4.5 Proposition. The functor \overline{N} : (Aff/k) ⟶ N(Aff/k)_{gr} has the following properties:

1) there is an isomorphism \overline{N}(Spec(A)) ≅ N(Spec(A)) = Sp(A) , i.e. the restriction of the functor \overline{N} on the category Aff/k is isomorphic to N ;

2) there is an isomorphism \overline{N}X(Sp(A)) ≅ X(Spec(A)) for any object Spec(A) ∈ Aff/k ;

3) \overline{N} is fully faithfull.

Proof. The groupoid associated to a category with final or initial object is equivalent to the one-point set. This implies that Gr(I_{S}/Spec(A)) is equivalent to the set Hom_{NAff}(Sp(S), Sp(A)) for any algebra S and commutative algebra A . Consequently, \overline{N}(Spec(A)) ≅ Sp(A) and 1) holds. By the same reason, the groupoid Gr(I_{A}/X) is equivalent to the set X(Spec(A) for any commutative algebra A . This proves the condition 2).

3) Since the functor N̂_{!} = π_{0}\overline{N} is faithful by 1.2 the functor \overline{N} is faithful too. Furhtermore, the composition N̂^{∗}π_{0}\overline{N} is equivalent to the identity functor on (Aff/k) .

Let f : \overline{N}Y ⟶ \overline{N}X be a morphism. It is defined by the morphisms f_{S} : Gr(I_{S}/Y) ⟶ Gr(I_{S}/X) for all S ∈ Alg/k . All these morphisms are compartible. Let t = (Spec(A), m, a) be an element of Gr(I_{S}/Y) , where m : Sp(S) ⟶ Sp(A) and a ∈ Y (Spec(A)) . The morphism f_{S} takes this element to t′ = (Spec(A), m, a′) ∈ Gr(I_{S}/X) , where a′ ∈ X(Spec(A) . Since f_{S} and f_{A} are compartible we have a′ = f_{A}(a) . Hence if we know the morphism f_{A} then we can find the element a′ . Thus every morphism f_{S} are unique defined by the set of morphisms f_{A} with commutative A . This implies that the functor \overline{N} is full. □ .

4.6 Let X be an object of (Aff/k) . We have two categories Aff/X and NAff/\overline{N}X , where the latter is defined in 2.15. There is a functor from Aff/X to NAff/\overline{N}X that sends an element a ∈ X(Spec(A)) to the element (Spec(A), id, a) ∈ Gr(I_{Spec(A)}/X) (see 4.4). Denote this functor by \overline{n}_{X} . This functor induces the inverse image functor \overline{n}_{X}^{∗} : Qcoh_{r}(\overline{N}X) ⟶ Qcoh(X) .

The correspondence to a presheaf of sets X ∈ (Aff/k) the presheaf of groupoids \overline{N}X ∈ N(Aff/k)_{gr} is considered by us as a “fine” embeding of the “commutative case” in the “noncommutative” one, because the following statement holds

4.7 Theorem. Let X be an object of (Aff/k) . The functor \overline{n}_{X}^{∗} : Qcoh_{r}(\overline{N}X) ⟶ Qcoh(X), induced by the natural functor \overline{n}_{X} : Aff/X ⟶ NAff/\overline{N}X , is an equivalence. Proof. We construct a quasi-inverse functor for \overline{n}_{X} . For any object X ∈ Aff/k we can define a presheaf of categories CX on NAff/k , i.e. contravariant functor fron NAff/k to the category of all small categories by rule ( compaire with (6)):

CX(Sp(S)) := I_{S}/X

There is a functor κ_{X} from NAff/CX to Aff/X that sends an element t = (Spec(A), m, a) ∈ I_{S}/X to the element a ∈ X(Spec(A)) , where m : Sp(S) ⟶ Spec(A) .

Let F be a quasicoherent sheaf on X . We can define a presheaf G on the category NAff/CX by rule

G(t) = F(a) ⊗_{(A,m)} S

(More formaly we can do it by the following way. The functor κ_{X} induces the functor κ̂_{X}^{∗} : (Aff/X) ⟶ (NAff/CX) (see 1.2). Then the presheaf G can be defined as κ̂_{X}^{∗}(F) ⊗_{κ̂_{X}^{∗}(O_{X})} O_{CX} .)

Further, a morphism α : t → t′ in I_{S}/X is, by definition, a morphism α : Spec(A) ⟶ Spec(A′) such that αm = m′ and a′α = a . Any such morphism gives us an isomorphism between G(t) and G(t′) , because

G(t) = F(a) ⊗_{(A,m)} S ≅ F(a′) ⊗_{(A′,α)} A ⊗_{(A,m)} S ≅ F(a′) ⊗_{(A′,m′)} S = G(t′)

Therefore, this presheaf G defines a presheaf H on \overline{N}X which is, evidently, quasicoherent. We define a functor \overline{n}_{∗X} from Qcoh(X) to Qcoh_{r}(\overline{N}X) as such functor that takes object F to H . It is easy to see that the functor \overline{n}_{∗X} is quasi-inverse of \overline{n}_{X}^{∗} . □

4.8 Futher, we will often write C instead of NAff/k . Let Φ = (C, can) be a site ringed by the sheaf O . Let X be an object of C and j_{X} : C/X ⟶ C is the canonical functor. Denote by can_{X} the canonical topology on C/X . It is induced by the canonical topology on C by 1.12. We have a ringed site Φ_{X} = (C/X, can_{X}) with the ring sheaf O_{X} . In the commutative case we know that any quasicoherent module on X is a sheaf for the canonical topology on C/X (see 3.5). Moreover, it easy follows from the definition of the canonical topology. The same statement in the noncommutative case is also true, but the proof is not so direct.

4.9 Proposition. Let X be a presheaf on the category C = NAff/k . Then each quasicoherent module M on X is a sheaf for the canonical topology on NAff/X .

Proof. Let A be a k -algebra and let (f_{α} : A ⟶ B_{α}), α ∈ I be a family of morphisms of algebras such that the opposite family of maps f_{α}^{op} in NAff/k is covering in the canonical topology. To prove the proposition we have to show that for any (right) A -module M a homomorphism ϵ_{M} in the diagram

M ⟶^{ϵ_{M}} ∏_{α∈I} M ⊗_{A} B_{α} \overrightarrow{⟶} ∏_{α,β∈I} M ⊗_{A} (B_{α} ⋆_{A} B_{β})

is an equalizer.

Denote by N a A –bimodule A ⊗_{ℤ} M . Define a ring S as A ⊕ N with multiplication low (a_{1}, n_{1})(a_{2}, n_{2}) = (a_{1}a_{2}, a_{1}n_{2} + n_{1}a_{2}) . By Lemma 4.10 the canonical map ϕ : M ⟶ N is an embedding. Denote by i the embedding N to S and by ψ_{α} the canonical maps M ⊗_{A} B_{α} ⟶ S ⋆_{A} B_{α} for any α ∈ I . Let ψ be the product of ψ_{α} by all α ∈ I .

Consider the following commutative diagram

\begin{matrix}M & ⟶^{ϵ_{M}} & ∏_{α∈I} M ⊗_{A} B_{α} & \overrightarrow{⟶} & ∏_{α,β∈I} M ⊗_{A} (B_{α} ⋆_{A} B_{β}) \\ iϕ↓ & & ↓ψ & & ↓ \\ S & ⟶^{e_{S}} & ∏_{α∈I} S ⋆_{A} B_{α} & \overrightarrow{⟶} & ∏_{α,β∈I}(S ⋆_{A} B_{α}) ⋆_{S} (S ⋆_{A} B_{β}) \end{matrix}  (7)

Since the family (f_{α}^{op}) is covering the map e_{S} is an equalizer. The maps ϕ and e_{S} are embeddings, hence the map ϵ_{M} is an embedding too. Thus we proved that any quasicoherent module is a separated sheaf.

Assume now that the following conditions hold

1. the maps ψ_{α} are embeddings for any α ,

2. intersection of e_{S}(S) and ψ(∏_{α} M ⊗_{A} B_{α}) coincides with e_{S}iϕ(M) . In this case we obtain that ϵ_{M} is an equalizer.

Thus, to finish the proof we need to show that our assumptions hold.

1) Denote an A -bimodule B_{α}/A by \overline{B}_{α} . Consider a graded ring R_{α} for any α ∈ I , defined as

R_{α} = B_{α} ⊕ (B_{α} ⊗_{A} N ⊗_{A} B_{α}) ⊕ (B_{α} ⊗_{A} N ⊗_{A} \overline{B}_{α} ⊗_{A} N ⊗_{A} B_{α}) ⊕ ⋯

with composition low

\begin{matrix}(b_{0} ⊗ n_{1} ⊗ \overline{b}_{1} ⊗ ⋯ ⊗ \overline{b}_{i−1} ⊗ n_{i} ⊗ b_{i})(b′_{i} ⊗ n_{i+1} ⊗ \overline{b}_{i+1} ⊗ ⋯ ⊗ \overline{b}_{i+j−1} ⊗ n_{i+j} ⊗ b_{i+j}) = \\ (b_{0} ⊗ n_{1} ⊗ \overline{b}_{1} ⊗ ⋯ ⊗ \overline{b}_{i−1} ⊗ n_{i} ⊗ \overline{b_{i}b′_{i}} ⊗ n_{i+1} ⊗ \overline{b}_{i+1} ⊗ ⋯ ⊗ \overline{b}_{i+j−1} ⊗ n_{i+j} ⊗ b_{i+j}) \end{matrix}

There are the ring homomorphisms B_{α} ⟶ R_{α} and S ⟶ R_{α} . Therefore , there is the canonical ring homomorphism S ⋆_{A} B_{α} ⟶ R_{α} . By Lemma 4.10 the map M ⊗_{A} B_{α} ⟶ B_{α} ⊗_{A} N ⊗_{A} B_{α} ≅ B_{α} ⊗_{ℤ} M ⊗_{A} B_{α} is an embedding. Hence the map M ⊗_{A} B_{α} ⟶ R_{α} is an embedding too. It is the composition of ψ_{α} with the canonical map S ⋆_{A} B_{α} ⟶ R_{α} . Thus, we obtain that ψ_{α} is an embedding for any α ∈ I .

2) The map i is the embedding of N into S . To check the second condition it is sufficient to show that intersection of e_{S}i(N) and ψ(∏_{α∈I} M ⊗_{A} B_{α}) coincides with e_{S}iϕ(M) . Note that the maps e_{S}i and ψ go through the canonical map ∏_{α∈I} B_{α} ⊗_{A} N ⊗_{A} B_{α} ⟶ ∏_{α∈I} S ⊗_{A} B_{α} On the other hand, there is a composition map

∏_{α∈I} S ⊗_{A} B_{α} ⟶ R_{α} ⟶ ∏_{α∈I} B_{α} ⊗_{A} N ⊗_{A} B_{α}

which splits previous one. Hence ∏_{α∈I} B_{α} ⊗_{A} N ⊗_{A} B_{α} is a direct summand of ∏_{α∈I} S ⊗_{A} B_{α} .

Denote by ρ and θ the maps N ⟶ ∏_{α∈I} B_{α} ⊗_{A} N ⊗_{A} B_{α} and ∏_{α∈I} M ⊗_{A} B_{α} ⟶ ∏_{α∈I} B_{α} ⊗_{A} N ⊗_{A} B_{α} respectively. There is the commutative diagram

0  0
↓  ↓
0 ⟶  M  ⟶^{ϵ_{M}}  ∏_{α∈I} M ⊗_{A} B_{α}
ϕ↓  ↓θ
0 ⟶ N = A ⊗_{ℤ} M ⟶^{ρ} ∏_{α∈I} B_{α} ⊗_{ℤ} M ⊗_{A} B_{α}

By Lemma 4.10 the embeddings ϕ and θ split as maps of abelian groups. Moreover, it is easy to see that these splits are commute. Therefore, the intersection of ρ(N) and θ(∏_{α∈I} M ⊗_{A} B_{α}) coincides with ρϕ(M) . This proves condition two and the proposition. □

4.10 Lemma. Let R be a ring and M be a right R –module. Then the canonical map M ⟶ R ⊗_{ℤ} M is an embedding of rights R –modules and it splits as a morphism of abelian groups.

Proof. It is clear that the composition of maps of abelian groups

M ⟶ R ⊗_{ℤ} M ⟶^{∼} M ⊗_{ℤ} R ⟶ M

is the identity. (Here the last map is the action of the ring R on the right module M .) Hence M ⟶ R ⊗_{ℤ} M is an embedding of abelian groups and, consequently, it is an embeddidg of right R -modules. □

4.11 Corollary. Let X be a presheaf on the category C = NAff/k . The category Qcoh_{r}(X) is a full subcategory of the category Mod_{r}(X, can_{X}) . If m : X ⟶ \underline{c}X is the morphism of sheafification in the canonical topology, then the inverse image functor m^{∗} : Qcoh_{r}(\underline{c}X) ⟶ Qcoh_{r}(X) is fully faithful.

This corollary follows from Propositions 4.9 and 2.4.

References

[1] M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas, SGA4, Lecture Notes in Math, v.269.

[2] P. Berthelot, A. Grothendieck, L. Illusie, Théorie des intersections et théoreme de Riemann-Roch, SGA6, Lect. Notes in Math., v.225, Springer, Heidelberg, 1971.

[3] A. Grothendieck, Revetements étale et groups fondamental, SGA1, Lecture Notes in Math, v.224, Springer, Heidelberg, 1971.

[4] C. Faith, Algebra: Rings, Modules and Categories I, Springer, 1973.

[5] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic, Springer, 1992.

[6] M. A. Knus, M. Ojanguren, Théorie de la Descente et Algèbres d’ Azumaya, Lecture Notes in Math, v.389, Springer, Heidelberg, 1974.

[7] J. Murre, Lectures on an Introduction to Grothendieck’s Theory of the Fundamental Group, Lecture Notes, Tata Institute of Fundamental Research, Bombay, 1967.

Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin str., GSP-1, Moscow 117966, Russia

   E-mail address: orlov@mi.ras.ru