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
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
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
which is representable of the
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.
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
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
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:
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.
Then in (20) we have:
where the lemma is proved. ■
We consider the following examples.
Here, for any two modules we have:
where is a fraction field6 and is the torsion submodule of Then associated to is which is pre-sheaf. Here and are and
compatible with the monoidal pairing
If , then
We consider the Universal mapping which is commutative:
Then (31) is compatible with the monoidal pairing. ■
If the (projective) objects
is a projective resolution, we define
Now, how do we understand the extensions of these tensor products in chain homotopy equivalence?
is a finite sum, and is bounded above. Then, since
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.
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.
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.
Then, it is sufficient to demonstrate that , preserve quasi-isomorphisms.
to describe the functor category on the field
Likewise, analogously we can define to the tensor product their extension to
Likewise, we have the definition.
the Étale sheaf associated to
induces an isomorphism
Remember that a pre-sheaf is defined as:
i. The sequence
is exact for every surjective Étale morphism of smooth schemes,
We demonstrate Lemma 3.2.
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
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:
4. Some considerations to mathematical physics
Remember that in the derived geometry we work with structures that must support
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
Then the corresponding infinite sequence is exact.
We have the following proposition.
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.
The derived tensor product is a regular tensor product.
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.
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.
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
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.
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 .
We demonstrate the Theorem 5.1.
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
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 Mod
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.
Adams JF. Stable homotopy and generalised homology. In: Chicago Lectures in Mathematics, MR 53 #6534. Chicago, IL: University of Chicago Press; 1974
Mazza C, Voevodsky V, Weibel C, editors. Lecture Notes on Motivic Cohomology. Vol. 2. Cambridge, MA, USA: AMS Clay Mathematics Institute; 2006
Aubry M. Homotopy Theory and Models. Boston, MA: Birkhäuser; 1995
Milne JS. Étale Cohomology. Princeton, NJ: Princeton University Press; 1980. ISBN: 0-691-08238-3
Bosch S. Algebraic Geometry and Commutative Algebra. NY, USA: Springer; 2013
Bulnes F. Extended d-cohomology and integral transforms in derived geometry to QFT-equations solutions using Langlands correspondences. Theoretical Mathematics and Applications. 2017; 7(2):51-62
Voevodsky V. Triangulated categories of motives over a field. In: Cycles, Transfers, and Motivic Homology Theories, Vol. 143 of Annals of Mathematics Studies. USA: Princeton University Press; 2000. pp. 188-238
- 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, 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,X⊗Y=X×Y.
- 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: