In this research we studied the tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories X⊗Y=X×Y, on the category Cork, (finite correspondences category) by understanding it to be the product of the underlying schemes on k. Although, to this is required to build a total tensor product on the category PST(k), where this construction will be useful to obtain generalizations on derived categories using pre-sheaves and contravariant and covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences. Some concrete applications are given through a result on field equations solution.
- algebraic variety
- additive pre-sheaves
- derived categories
- derived tensor products
- finite correspondences category
- 2010 Mathematics Subject classification: 13D09
This study is focused on the derived tensor product whose functors have images as cohomology groups that are representations of integrals of sheaves represented for its pre-sheaves in an order modulo k. This study is remounted to the K-theory on the sheaves cohomologies constructed through pre-sheaves defined by the tensor product on commutative rings. The intention of this study is to establish a methodology through commutative rings and their construction of a total tensor product 1on the category PST(k), considering extensions of the tensor products to obtain resolution in the projective sense of infinite sequences of modules of Étale sheaves. These sheaves are pre-sheaves of Abelian groups on the category of smooth separated schemes restricted to scheme X.
Likewise, the immediate application of the derived tensor products will be the determination of the tensor triangulated category of Étale motives to be equivalent to the derived category of discrete modules over the Galois group which says on the equivalence of functors of tensor triangulated categories2.
Then the mean result of derived tensor products will be in tensor triangulated category of effective motives and their subcategory of effective geometric motives Likewise, the motive M(X), of a scheme X, is an object of and belongs to if X, is smooth. However, this requires the use of cohomological properties of sheaves associated with homotopy invariant pre-sheaves with transfers for Zariski topology, Nisnevich and cdh topologies.
Finally, all this treatment goes in-walked to develop a motivic cohomology to establish a resolution in the field theory incorporating singularities in the complex Riemannian manifolds where singularities can be studied with deformation theory through operads, motives, and deformation quantization.
2. Fundaments of derived tensor products
We consider the Abelian category Ab, which is conformed by all functor images that are contravariant additive functors on small category of Likewise, is the category of all additive pre-sheaves on . Likewise, we can define this category as of points space:
Likewise, we have the Yoneda embedding as the mapping3:
which has correspondence rule
We need a generalization of the before categories and functors, therefore we give a ring R, originating the ring structure to be the Abelian category of the additive functors
being R-mod, the category of the modules that originate the ring structure. Then hX, is the functor
which is representable of the R-mod.
Likewise, the following lemma introduces the representable pre-sheaves and functors and their role to construct pre-sheaves that can be extended to pre-sheaves first using the projective objects of , and define the projective resolution to infinite complexes sequence.
Lemma 1.1. Every representable pre-sheaf is a projective object of , every projective object of is a direct summand of a direct sum of representable functors, and every in has a projective resolution.
Proof. We consider an analogue to (6) in the functor context:
Then each object is a projective object in Likewise, each is a quotient
then there exist a surjection such that
Then from the additive category until functional additive category modulus we have:
which proves the lemma.
Now suppose that with an additive symmetric monoidal structure is such that
This means that commutes with direct sum. Let and M, be R-modules; then is clear that:
We extend on in the same way, and this extends to tensor product of corresponding projectives. Then can be extended to a tensor product on all of .
Likewise, if then we have a pre-sheaf tensor product in the following way:
However, this does not correspond to since is not additive. However, this could be additive when one component F, or G, is element of . But if we want to get a tensor product on we need a more complicated or specialized construction. For this, we consider then of their representable pre-sheaves should be represented by . As a first step, we can extend to a tensor product
commuting with . Thus if of the above co-chain complexes as follows:
the chain complex is defined as the total complex of the double complex .
Then we can define a legitimate tensor product between two categories as follows:
Definition 1.1. Let be choosing projective resolutions
we define 4to be which means that the tensor product is total having that Then the tensor product to these pre-sheaves and the pre-sheaves is defined as:
The relation (17) means the chain homotopy equivalent of the is well defined up to chain homotopy equivalence, and analogous for
In particular, given that and are projective, we have
Likewise, the ring is an additive symmetric monoidal category.
We consider the following lemma.
Lemma 1.2. The functor is right adjoint to In particular is left exact and is right exact.
Proof. Let be
Then in (20) we have:
where the lemma is proved. ■
We consider the following examples.
Example 1.1. We consider the category of free R-modules over a commutative ring . This category is equivalent to the category of all R-modules where pre-sheaf associated to M, is and and are the familiar and
Here, for any two modules we have:
Example 2.1. Let be the category of R-modules M, such that:
where is a fraction field6 and is the torsion submodule of Then associated to is which is pre-sheaf. Here and are and
Example 3.1. Let R, be a simplicial commutative ring and be a category cofibrant replacement. Here, the pre-sheaf associated to M, which is the Kähler 1-differentials module, is and here and are and . Here the category is of the cotangent complexes of R.
Proposition 1.1. If and are in then there is a natural mapping
compatible with the monoidal pairing
Proof. We have as defined in (18):
If , then
We consider the Universal mapping which is commutative:
Then (31) is compatible with the monoidal pairing. ■
If the (projective) objects hX, are flat, that is to say, is an exact functor then is called a balanced functor . Here agrees with the usual left derived functor But here we do not know when the hX, are flat. This is true in Example 1.1. But it is not true in Then we need to extend to a total tensor product on the category of bounded above co-chain complexes (15). This would be the usual derived functor if were balanced , and our construction is parallel. Likewise, if C, is a complex in there is a quasi-isomorphism with P, a complex of projective objects. Any such complex P, is called a projective resolution of C, and any other projective resolution of C, is chain homotopic to P .
Likewise, if D, is any complex in and
is a projective resolution, we define
Now, how do we understand the extensions of these tensor products in chain homotopy equivalence?
Since P, and Q, are bounded above, each
is a finite sum, and is bounded above. Then, since P, and Q, are defined up to chain homotopy, the complex is independent (up to chain homotopy equivalence) of the choice of P, and Q. Then there exists a mapping
which extends the mapping
of Definition 1.1.
We consider the following lemma to obtain in the extension (36) a derived triangulated category that will be useful in the context of derived tensor categories whose pre-sheaves are Étale pre-sheaves.
The importance of a triangulated category together with the additional structure as the given by pre-sheaves lies in obtaining distinguished triangles of categories that generate the long exact sequences of homology that can be described through of short exact sequences of Abelian categories. Likewise, the immediate examples are the derived categories of Abelian category and the stable homotopy category of spectra or more generally, the homotopy category of a stable ∞-category. In both cases is carried a structure of triangulated category.
3. Derived triangulated categories with structure by pre-sheaves and
We enounce the following proposition.
Proposition 3.1. The derived category equipped with structure is a tensor-triangulated category.
Proof. We consider a projective object , where , is a projective category defined as the points set
We consider the application defined by the mapping:
where the objects are those that are determined by
Then we have
via the chain homotopy. For other side
which is risked from structure when in , which then is true from the lemma 2.1.■
Now, for bounded complexes of pre-sheaves we can give the following definitions.
Definition 3.1. Let and be bounded complexes of pre-sheaves. There is a canonical mapping:
which was foresee in the Definition 1.1. By right exactness of and given in Lemma 1.1, it suffices to construct a natural mapping of pre-sheaves
For in is the monoidal product in followed by the diagonal mapping of triangle:
that is to say,
satisfies the triangle7:
With all these dispositions and generalities, now we can specialize to the case when8
and is the tensor product
Then we have the Yoneda embedding:
We denote as for the tensor product on or
and for . Then there is a natural mapping
Here is the tensor product induced to But, before we will keep using the product which we can define as:
The above can be generalized through the following lemma.
Lemma 3.1. The pre-sheaf is a direct summand of . In particular, it is projective object of PST. Likewise, for the following sequence of pre-sheaves with transfers, the exactness is explicit9:
Then, it is sufficient to demonstrate that , preserve quasi-isomorphisms.
Definition 3.2. A pre-sheaf with transfers is a contravariant additive functor:
to describe the functor category on the field k, whose objects are pre-sheaves with transfer and whose morphisms are natural transformations.
Likewise, analogously we can define to the tensor product their extension to
Likewise, we have the definition.
Definition 3.3. If F, and G, are pre-sheaves of R-modules with transfers, we write:
the Étale sheaf associated to
If C, and D, are bounded above complexes of pre-sheaves with transfers, we shall write for and
where P, and Q, are complexes of representable sheaves with transfers, and Then there is a natural mapping
Lemma 3.2. If F, and are Étale sheaves of R-modules with transfers, and F, is locally constant, the mapping:
induces an isomorphism
Remember that a pre-sheaf is defined as:
Definition 3.4. A pre-sheaf of Abelian groups on is an Étale sheaf if it restricts to an Étale sheaf on each in that is if:
i. The sequence
is exact for every surjective Étale morphism of smooth schemes,
We demonstrate Lemma 3.2.
Proof. We want the tensor product , which induces to tensor triangulated structure on the derived category of Étale sheaves of R-modules with transfers10 defined in other expositions . Considering Proposition 3.1, we have:
Then, it is sufficient to demonstrate that , preserve quasi-isomorphisms. The details can be found in .
Then the tensor product as pre-sheaf to Étale sheaves can have a homology space of zero dimension that vanishes in certain component right exact functor from the category of pre-sheaves of R-modules with transfers to the category of the Étale sheaves of R-modules and transfers. Then every derived functor vanishes on to certain complex of Étale.
Then, all right exact functors are acyclic. This is the machinery to demonstrate the functor exactness and resolution in modules through of induce from a tensor-triangulated structure to a derived category more general that
Also we have:
Lemma 3.3. Fix Y, and set If F, is a pre-sheaf of R-modules with transfers such that then
4. Some considerations to mathematical physics
Remember that in the derived geometry we work with structures that must support R-modules with characterizations that should be most general to the case of singularities, where it is necessary to use irregular connections, if it is the case, for example in field theory in mathematical physics when studying the quantum field equations on a complex Riemann manifold with singularities.
Through the characterization of connections for derived tensor products, we search precisely generalize the connections through pre-sheaves with certain special properties, as can be the Étale sheaves.
Remember we want to generalize the field theory on spaces that admit decomposing into components that can be manageable in the complex manifolds whose complex varieties can be part of those components called motives, creating a decomposition in the derived category of its spectrum considering the functor Spec, and where solutions of the field equations are defined in a hypercohomology.11 Likewise, this goes focused to obtain a good integrals theory (solutions) in the hypercohomology context considering the knowledge of spectral theory of the cycle sequences in motivic theory that searches the solution of the field equations even with singularities of the complex Riemann manifold.
We can demonstrate that induces a tensor-triangulated structure to a derived category more general than as for example, which is our objective. In this case, we want geometrical motives, where this last category can be identified for the derived category .
We consider and fix Y, and the right exact functor from the category of pre-sheaves of R-modules with transfers to the category of the Étale sheaves of R-modules and transfers. Likewise, their left functors are the homology sheaves of the total left derived functor Considering a chain complex C, the hypercohomology spectral sequence is:
Then the corresponding infinite sequence is exact.
We consider A, and where is a category as has been defined before.
We have the following proposition.
Proposition 4.1. There is equivalence between categories
Then a hypercohomology as given to can be obtained through double functor work through an inclusion of a category in Then is had the result.
Theorem 4.1. The left adjoint to the inclusion functor is defined by In particular, the image of under this functor is
The derived tensor product is a regular tensor product.
Theorem 4.2. The character for an adjoint lifts for a homotopically meaningful adjunction complies:
Meaning that, it is an adjunction of categories, which induces an adjunction to level of homotopy categories.
We define the cotangent complex required in derived geometry and QFT.
Definition 4.1. The cotangent complex is the image of functor under the left functor of the Kahler differentials module . Likewise, if be a free resolution then
The cotangent complex as defined in (69) lives in the derived category We observe that choosing the particular resolution of then is a co-fibrant object in the derived category which no exist distinction between the derived tensor product and the usual tensor product. Then to any representation automorphic of the can be decomposed as the tensor product This last fall in the geometrical Langlands ramifications.
Example 4.1. (66) in the context of solution of field equations as has solution in the hypercohomology of a spectral sequence of (established on the infinite sequence ) when its functors whose image have as its cotangent complex the image under of the functor which is the functor image under the left derived functor of Kahler differentials.
To demonstrate this, it is necessary to define an equivalence between derived categories in the level of derived categories and where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles. The integrals are those whose functors image will be in which is the variety of opers on the formal disk or neighborhood of all points in a surface in a complex Riemannian manifold .
As was shown, the geometrical motives required in our research are a result of embedding the derived category (geometrical motives category) in the considering the category of smooth schemes on the field k.
We consider the following functors. For each there is the resulting functor is:
which is exact and left-adjoint to the inclusion
Also the functor (70) descends to an equivalence of triangulated categories. This is very useful to make into a tensor category as follows. We consider the Nisnevich sheaf with transfer We define
Then it can be demonstrated that the operation realized in (70) can be extended to give with the structure of a triangulated tensor category. Then the functor induces a tensor operation on making that also a triangulated tensor category. Likewise, explicitly in this gives us the functor
where we have the formula
If we consider the embedding theorem, then we can establish the following triangulated scheme
which has implications in the geometrical motives applied to bundle of geometrical stacks in mathematical physics.
Theorem 5.1 (F. Bulnes). Suppose that is a complex Riemannian manifold with singularities. Let X, and Y, be smooth projective varieties in 12. We know that solutions of the field equations are given in a category (see Example 4). Context Solutions of the quantum field equations for are defined in hyper-cohomology on coefficients from the category Smk, defined on a numerical field k, considering the derived tensor product of pre-sheaves. Then the following triangulated tensor category scheme is true and commutative:
The category has a tensor structure and the tensor product of its motives is as defined in (75) .
Triangulated category of geometrical motives or written simply as is defined formally inverting the functor of the Tate objects13 (are objects of a motivic category called Tannakian category) , to be image of the complex where the motive in degree will be or to any motive
Likewise, the important fact is that the canonical functor is full embedding . Therefore we work in the category .
Likewise, for X, and Y, smooth projective varieties and for any integer i, there exists an isomorphism:
We demonstrate the Theorem 5.1.
Proof. the category extends to a pseudo-tensor equivalence of cohomological categories over motives on k, that is to say, is the image of functors
which is an equivalence of the underlying triangulated tensor categories.
On the other hand, the category DQFT can be defined for the motives in a hypercohomology from the category defined as:
which comes from the hypercohomology
We observe that if a Zariski sheaf of modules with transfers F, is such that for all C, a complex defined on modules (being a special case when ), where the cohomology groups of the isomorphism , can be vanished for .
Then survives a hypercohomology If we consider we can to have the quantum version of this hyper-cohomology with an additional work on moduli stacks of the category ModB, in a study on equivalence between derived categories in the level of derived categories and , where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles14.
For other way, with other detailed work of quasi-coherent sheaves  we can to obtain the category The functors are constructed using the Mukai-Fourier transforms. ■
Posdoctoral research was supported by State of Mexico Council of Scientific Research COMECYT-077/111/21.
- L, is a Lefschetz motive Z1, .
- Theorem. If 1/m∈k, the space L⊗L, is a tensor triangulated category and the functorsD−GZ/m→π∗L→DWA‐1=DMÉteff,−kZ/m,are equivalences of tensor triangulated categories. Here π∗, is a triangulated functor from D−GZ/m, until the category D−ShétCorkZ/m.
- The obtained image by the Yoneda embedding has the pre-sheaf A⊕⊂ZA.
- ⊗L, is a total tensor product.
- The field of fractions of an integral domain is the smallest field in which this domain can be embedded.
- Def. If X, Y∈Cork, their tensor product X⊗Y, is defined to be the product underlying schemes over k,
- ZtrX≅Ztr⊕ZtrXx, ZtrX1×X2≅Ztr⊕ZtrX1x1⊕ZtrX2x2⊕ZtrX1∧X2.
- Definition. A pre-sheaf with transfers is a contravariant additive functor from the category Cork, to the category of abelian groups Ab.
- Definition. A hyperhomology or hypercohomology of a complex of objects of an abelian category is an extension of the usual homology of an object to complexes. The mechanism to give a hypercohomology is suppose that A, is an abelian category with enough injectives and Φ, a left exact functor to another abelian category B. If C, is a complex of objects of A, bounded on the left, the hypercohomology HiC, of C, (for an integer i) is calculated as follows: take a quasi-isomorphism ψ:C→I, where I, is a complex of injective elements of A. The hypercohomology HiC, of C, is then the cohomology HiΦI, of the complex ΦI.
- Singular projective varieties useful in quantization process of the complex Riemannian manifold. The quantization condition compact quantizable Käehler manifolds can be embedded into projective space.
- Let MTZ, denote the category of mixed Tate motives unramified over Z. It is a Tannakian category with Galois group GalMT.
- We consider the functor F, defined as: where KFr, the kernel space of the functor Fr, is the functor that induces the equivalence ModTDX×YX≅⊥KFr, and the operator T=Fr∘F, acting on category DX×YX.