Derived Tensor Products and Their Applications

In this research we studied t he tensor product on derived categories of Étale sheaves with transfers considering as fundamental, the tensor product of categories X ⊗ Y ¼ X (cid:2) Y , on the category Cor k , (finite correspondences category) by under-standing 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.


Introduction
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 ⊗ L , 1 on the category PST(k) , considering extensions of the tensor products ⊗ R A ð Þ , 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 DMett k, =m ð Þ, of Étale motives to be equivalent to the derived category of discrete =mmodules over the Galois group G ¼ Gal k sep =k À Á , which says on the equivalence of functors of tensor triangulated categories 2 .
Then the mean result of derived tensor products will be in tensor triangulated category DM eff,À Nis k, R ð Þ, of effective motives and their subcategory of effective geometric motives DM eff,À gm k, R ð Þ: Likewise, the motive M(X), of a scheme X, is an object of DM eff,À Nis k, R ð Þ, and belongs to DM eff,À gm k, R ð Þ, 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.

Fundaments of derived tensor products
We consider the Abelian category Ab, which is conformed by all functor images that are contravariant additive functors F : A ! Ab, on small category of  A ð Þ: Likewise,  A ð Þ, is the category of all additive pre-sheaves on A. Likewise, we can define this category as of points space: Likewise, we have the Yoneda embedding as the mapping 3 : which has correspondence rule or We need a generalization of the before categories and functors, therefore we give a ring R, originating the ring structure A R ð Þ, to be the Abelian category of the additive functors being R-mod, the category of the modules that originate the ring structure. Then h X , is the functor 2 Theorem. If 1=m ∈ k, the space L, ⊗ L ð Þ , is a tensor triangulated category and the functors 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 ⊗ R A ð Þ , that can be extended to presheaves ⊗ L , first using the projective objects of R A ð Þ , and define the projective resolution to infinite complexes sequence. Lemma 1.1. Every representable pre-sheaf h X , is a projective object of R A ð Þ , every projective object of R A ð Þ is a direct summand of a direct sum of representable functors, and every F, in R A ð Þ, has a projective resolution. Proof. We consider an analogue to (6) in the functor context: Then each object h X , is a projective object in R A ð Þ: Likewise, each F ∈ R A ð Þ, is a quotient then there exist a surjection x, such that Then from the additive category until functional additive category modulus A ⊕ ⊂  A ð Þ, we have: which proves the lemma.

■
Now suppose that A, with an additive symmetric monoidal structure ⊗ , is such that This means that ⊗ , commutes with direct sum. Let N α , α ∈ A, and M, be R-modules; then is clear that: We extend ⊗ , on A ⊕ , in the same way, and this extends to tensor product of corresponding projectives. Then ⊗ , can be extended to a tensor product on all of R A ð Þ. Likewise, if F, G ∈ R A ð Þ, then we have a pre-sheaf tensor product in the following way: However, this does not correspond to R A ð Þ, since F ⊗ R G, is not additive. However, this could be additive when one component F, or G, is element of A ⊕ . But if we want to get a tensor product on R A ð Þ, we need a more complicated or specialized construction. For this, we consider X, Y ∈ A, then h X ⊗ h Y , of their representable pre-sheaves should be represented by X ⊗ Y. As a first step, we can extend ⊗ , to a tensor product commuting with ⊕ . Thus if L 1 , L 2 ∈ Ch À A ⊕ À Á , of the above co-chain complexes as follows: the chain complex L 1 ⊗ L 2 , is defined as the total complex of the double complex L * 1 ⊗ L *

.
Then we can define a legitimate tensor product between two categories F, G ∈ R A ð Þ, as follows: we define F ⊗ L G, 4 to be P ⊗ Q, which means that the tensor product is total having that Tot P * ⊗ Q * ð Þ : Then the tensor product to these pre-sheaves and the Hom, pre-sheaves is defined as: and The relation (17) means the chain homotopy equivalent of the F ⊗ L G, is well defined up to chain homotopy equivalence, and analogous for Hom F, G ð Þ: In particular, given that h X , and h Y , are projective, we have Likewise, the ring R A ð Þ, is an additive symmetric monoidal category. We consider the following lemma. Lemma 1.2. The functor Hom F, • ð Þ, is right adjoint to F ⊗ •: In particular Hom F, • ð Þ, is left exact and F ⊗ •, is right exact. Proof. Let be We consider 4 ⊗ L , is a total tensor product. 5 Hom Then in (20) we have: But also, where the lemma is proved.

■
We consider the following examples. Example 1.1. We consider the category A, of free R-modules over a commutative ring R A ð Þ. This category is equivalent to the category of all R-modules where pre-sheaf associated to M, is M ⊗ R , and Hom, and ⊗ , are the familiar Hom R , and ⊗ R : Here, for any two modules A, B ∈ R A ð Þ, we have: Example 2.1. Let A, be the category of R-modules M, such that: where K, is a fraction field 6 and M tor , is the torsion submodule of M: Then associated to M, is 1 ⊗ R M, which is pre-sheaf. Here Hom: and ⊗ , are Hom R , and ⊗ tor : Example 3.1. Let R, be a simplicial commutative ring and R A ð Þ ! A, be a category cofibrant replacement. Here, the pre-sheaf associated to M, which is the Kähler 1-differentials module, is M ⊗ L R A ð Þ , and here Hom, and ⊗ , are Hom R , and ⊗ L R A ð Þ :. Here the category is of the cotangent complexes of R. Proposition 1.1. If F i , and G i , are in R A ð Þ, then there is a natural mapping compatible with the monoidal pairing Proof. We have Hom, as defined in (18): and We consider the Universal mapping which is commutative: Then (31) is compatible with the monoidal pairing.

■
If the (projective) objects h X , are flat, that is to say, h X ⊗ •, is an exact functor then ⊗ , is called a balanced functor [2]. Here F ⊗ L G, agrees with the usual left derived functor L F ⊗ • ð ÞG: But here we do not know when the h X , are flat. This is true in Example 1.1. But it is not true in PST k ð Þ ¼  Cor k ð Þ: Then we need to extend ⊗ L , to a total tensor product on the category Ch À R A ð Þ, of bounded above cochain complexes (15). This would be the usual derived functor if ⊗ , were balanced [2], and our construction is parallel. Likewise, if C, is a complex in Ch À R A ð Þ, there is a quasi-isomorphism Pffi! C, 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 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 C ⊗ L D, is bounded above. Then, since P, and Q , are defined up to chain homotopy, the complex C ⊗ L D, 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 ⊗ L , 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 ⊗ L , and ⊗ tr

L,ét
We enounce the following proposition. Proposition 3.1. The derived category D À R A ð Þ, equipped with ⊗ Lstructure is a tensor-triangulated category.
Proof. We consider a projective object X ∈ P, where P, is a projective category defined as the points set We consider the application Λ, defined by the mapping: where the objects Λ X ð Þ, are those that are determined by Then we have via the chain homotopy. For other side which is risked from ⊗ L À structure when ⊗ ffi ⊗ L , in P, 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 C, and D, be bounded complexes of pre-sheaves. There is a canonical mapping: which was foresee in the Definition 1.1. By right exactness of ⊗ R , and ⊗ , given in Lemma 1.1, it suffices to construct a natural mapping of pre-sheaves For U, in A, η U , is the monoidal product in A, followed by the diagonal mapping of triangle: that is to say, satisfies the triangle 7 : where With all these dispositions and generalities, now we can specialize to the case when 8 and ⊗ , is the tensor product Then we have the Yoneda embedding: We denote as ⊗ tr , for the tensor product on PST k and ⊗ tr L , for ⊗ L . Then there is a natural mapping Here ⊗ tr L , is the tensor product induced to  Cor k ð Þ: But, before we will keep using the product ⊗ tr , which we can define as: being h X ¼ R tr X ð Þ, ∀h X ∈ Hom, ∀X ∈ A ⊕ : Def. If X, Y ∈ Cor k , their tensor product X ⊗ Y, is defined to be the product underlying schemes over k, The above can be generalized through the following lemma. Lemma 3.1. The pre-sheaf  tr X 1 , x 1 ð Þ∧⋯∧ X n , x n ð Þ ð Þ , is a direct summand of  tr X 1 Â ⋯ Â X n ð Þ . In particular, it is projective object of PST. Likewise, for the following sequence of pre-sheaves with transfers, the exactness is explicit 9 : Then, it is sufficient to demonstrate that ⊗ tr L,ét , preserve quasi-isomorphisms. Definition 3.2. A pre-sheaf with transfers is a contravariant additive functor: we write 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 ⊗ tr , their extension to ⊗ tŕe t : 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 F ⊗ tr G: If C, and D, are bounded above complexes of pre-sheaves with transfers, we shall write C ⊗ tŕ et D, for C ⊗ tr D ð Þ et , and where P, and Q , are complexes of representable sheaves with transfers, P ffi C, and Q ffi D: Then there is a natural mapping induced by Lemma 3.2. If F, and F 0 , 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 F, of Abelian groups on Sm=k, is an Étale sheaf if it restricts to an Étale sheaf on each X, in Sm=k, , that is if: is exact for every surjective Étale morphism of smooth schemes, ii We demonstrate Lemma 3.2.
Proof. We want the tensor product ⊗ tr L,ét , which induces to tensor triangulated structure on the derived category of Étale sheaves of R-modules with transfers 10 defined in other expositions [4]. Considering Proposition 3.1, we have: Then, it is sufficient to demonstrate that ⊗ tr L,ét , preserve quasi-isomorphisms. The details can be found in [5].
Then the tensor product ⊗ tŕ et , as pre-sheaf to Étale sheaves can have a homology space of zero dimension that vanishes in certain component right exact functor

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 ⊗ tr L,ét , induces a tensor-triangulated structure to a derived category more general than D À R A ð Þ, as for example, DM eff et ,À k, =m ð Þ, which is our objective. In this case, we want geometrical motives, where this last category DM eff et ,À k, =m ð Þ, can be identified for the derived category DM À gm k, R ð Þ. We consider and fix Y, and the right exact functor Φ F ð Þ ¼ R tr Y ð Þ⊗ tŕ et F, from the category PST k, R ð Þ, of pre-sheaves of R-modules with transfers to the category of the Étale sheaves of R-modules and transfers. Likewise, their left functors L P Φ F ð Þ, are the homology sheaves of the total left derived functor Φ F ð Þ ¼ R tr Y ð Þ ⊗ tr L,ét F: Considering a chain complex C, the hypercohomology spectral sequence is: then Then the corresponding infinite sequence is exact. We consider A, and B ∈ A, where A, is a category as has been defined before. We have the following proposition. 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.
The cotangent complex as defined in (69) lives in the derived category Mod B : We observe that choosing the particular resolution of B, then Ω P • =A , is a co-fibrant object in the derived category Mod P • , which no exist distinction between the derived tensor product and the usual tensor product. Then to any representation automorphic of G A ð Þ, the G F ð Þ=G A ð Þ, can be decomposed as the tensor product ⊗ n i¼1 π I : 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 D L Bun, D À Á , and D L Loc, O À Á , where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles. The integrals are those whose functors image will be in Spec H SymT OP L G D ð Þ ð Þ , which is the variety of opers on the formal disk D, or neighborhood of all points in a surface Σ, in a complex Riemannian manifold [6].

Applications
As was shown, the geometrical motives required in our research are a result of embedding the derived category DM À gm k, R ð Þ, (geometrical motives category) in the DM eff et ,À k, =m ð Þ, considering the category of smooth schemes on the field k.
We consider the following functors. For each F ∈ D À Sh Nis Cor k ð Þ ð Þ , there is L A 1 F ∈ D eff À k ð Þ, 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 D eff À k ð Þ, into a tensor category as follows. We consider the Nisnevich sheaf  tr k ð Þ, with transfer tr : Y ! c Y, X ð Þ: We define 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 M, is a complex Riemannian manifold with singularities. Let X, and Y, be smooth projective varieties in M 12 . We know that solutions of the field equations dda ¼ 0, are given in a category Spec Sm k ð Þ, (see Example 4). Context Solutions of the quantum field equations for dda ¼ 0, are defined in hyper-cohomology on coefficients from the category Sm k , defined on a numerical field k, considering the derived tensor product ⊗ tŕ et , of pre-sheaves. Then the following triangulated tensor category scheme is true and commutative: The category DM eff gm k, R ð Þ, has a tensor structure and the tensor product of its motives is as defined in (75)  The inverting of the objects À ⊗ : 1 ð Þ: Remember that a scheme is a mathematical structure that enlarges the concept of algebraic variety in several forms, such as taking account of multiplicities. The schemes can to be of a same algebraic variety different and allowing "varieties" defined over any commutative ring. In many cases, the family of all varieties of a type can be viewed as a variety or scheme, known as a moduli space.