Open access peer-reviewed chapter

Derived Tensor Products and Their Applications

By Francisco Bulnes

Submitted: December 9th 2019Reviewed: May 18th 2020Published: June 25th 2020

DOI: 10.5772/intechopen.92869

Downloaded: 66

Abstract

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.

Keywords

  • algebraic variety
  • additive pre-sheaves
  • derived categories
  • derived tensor products
  • finite correspondences category
  • schemes
  • singularities
  • varieties
  • 2010 Mathematics Subject classification: 13D09
  • 18D20
  • 13D15

1. 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,1on the category PST(k), considering extensions of the tensor products RA,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 DMéttkZ/m,of Étale motives to be equivalent to the derived category of discrete Z/mmodules over the Galois group G=Galksep/k,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 DMNiseff,kR,of effective motives and their subcategory of effective geometric motives DMgmeff,kR.Likewise, the motive M(X), of a scheme X, is an object of DMNiseff,kR,and belongs to DMgmeff,kR,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 F:AAb,on small category of ZA.Likewise, ZA,is the category of all additive pre-sheaves on A. Likewise, we can define this category as of points space:

ZA=FF:AAb,E1

Likewise, we have the Yoneda embedding as the mapping3:

h:AZA,E2

which has correspondence rule

XXi,E3

or

hX=hXi,E4

We need a generalization of the before categories and functors, therefore we give a ring R, originating the ring structure AR,to be the Abelian category of the additive functors

F:ARmod,E5

being R-mod, the category of the modules that originate the ring structure. Then hX, is the functor

hX:ARZHomAAX,E6

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 RA,that can be extended to pre-sheaves L,first using the projective objects of RA, and define the projective resolution to infinite complexes sequence.

Lemma 1.1. Every representable pre-sheaf hX,is a projective object of RA, every projective object of RAis a direct summand of a direct sum of representable functors, and every F,in RA,has a projective resolution.

Proof. We consider an analogue to (6) in the functor context:

HomRAhXFFX,E7

Then each object hX,is a projective object in RA.Likewise, each FRA,is a quotient

F=hX/A,E8

then there exist a surjection x,such that

x:hXF,E9

Then from the additive category until functional additive category modulus AZA,we have:

F=XAxFXx0hX,E10

which proves the lemma.

Now suppose that A,with an additive symmetric monoidal structure ,is such that

A=Cork,E11

This means that ,commutes with direct sum. Let Nα,αA,and M, be R-modules; then is clear that:

MαANααAMNα,E12

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 RA.

Likewise, if F,GRA,then we have a pre-sheaf tensor product in the following way:

FRGX=FXRGX,E13

However, this does not correspond to RA,since FRG,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 RA,we need a more complicated or specialized construction. For this, we consider X,YA,then hXhY,of their representable pre-sheaves should be represented by XY. As a first step, we can extend ,to a tensor product

:A×AA,E14

commuting with . Thus if L1,L2ChA,of the above co-chain complexes as follows:

Fn0,E15

the chain complex L1L2,is defined as the total complex of the double complex L1L2.

Then we can define a legitimate tensor product between two categories F,GRA,as follows:

Definition 1.1. Let be F,GRA,choosing projective resolutions

PF,QG,E16

we define FLG,4to be PQ,which means that the tensor product is total having that TotPQ.Then the tensor product to these pre-sheaves and the Hom,pre-sheaves is defined as:

FG=H0FLG,E17

and

HomFG:XHomRAFhXG,E18

The relation (17) means the chain homotopy equivalent of the FLG,is well defined up to chain homotopy equivalence, and analogous for HomFG.

In particular, given that hX,and hY,are projective, we have

hXLhY=hXhY=hXY,X,YA.E19

Likewise, the ring RA,is an additive symmetric monoidal category.

We consider the following lemma.

Lemma 1.2. The functor HomF,is right adjoint to F.In particular HomF,is left exact and F,is right exact.

Proof. Let be

F=HomhYG,E20

Then5

HomRAhXF=HomRAhXHomhYG,E21

We consider

HomhYG=HomRAhXhYG,E22

Then in (20) we have:

HomRAhXHomRAhXhYG=GXY,E23

But also,

GXY=HomRAhXhYG,E24

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 RA. This category is equivalent to the category of all R-modules where pre-sheaf associated to M, is MR,and Hom,and ,are the familiar HomR,and R.

Here, for any two modules A,BRA,we have:

ARB:AB=BA,E25

Example 2.1. Let A,be the category of R-modules M, such that:

KRMKRM/Mtor,E26

where K,is a fraction field6 and Mtor,is the torsion submodule of M.Then associated to M,is 1RM,which is pre-sheaf. Here Hom.and ,are HomR,and tor.

Example 3.1. Let R, be a simplicial commutative ring and RAA,be a category cofibrant replacement. Here, the pre-sheaf associated to M, which is the Kähler 1-differentials module, is MRAL,and here Hom,and ,are HomR,and RAL. Here the category is of the cotangent complexes of R.

Proposition 1.1. If Fi,and Gi,are in RA,then there is a natural mapping

HomF1G1HomF2G2HomF1G1F2G2,E27

compatible with the monoidal pairing

HomAU×A1X1HomAU×A2X2HomAU×U×A1×A2X1×X2HomAU×A1×A2X1×X2,E28

Proof. We have Hom,as defined in (18):

HomF1G1:X1HomRAF1hX1G1,E29

and

HomF2G2:X2HomRAF2hX2G2,E30

If F1,G1,F2,G2RA., then

HomF1G1HomF2G2=HomRAF1hX1F2hX2G1G2=HomRAF1F2hXYG1G2=HomF1F2G1G2,E31

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, hX,is an exact functor then ,is called a balanced functor [2]. Here FLG,agrees with the usual left derived functor LFG.But here we do not know when the hX, are flat. This is true in Example 1.1. But it is not true in PSTk=ZCork.Then we need to extend L,to a total tensor product on the category ChRA,of bounded above co-chain 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 ChRA,there is a quasi-isomorphism PC,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 [3].

Likewise, if D, is any complex in ChRA,and

QD,E32

is a projective resolution, we define

CLD=PQ,E33

Now, how do we understand the extensions of these tensor products in chain homotopy equivalence?

Since P, and Q, are bounded above, each

PQn=i+j=nPiQj,E34

is a finite sum, and CLD,is bounded above. Then, since P, and Q, are defined up to chain homotopy, the complex CLD,is independent (up to chain homotopy equivalence) of the choice of P, and Q. Then there exists a mapping

CLDCD,E35

which extends the mapping

FLGFG,E36

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 L,éttr

We enounce the following proposition.

Proposition 3.1. The derived category DRA,equipped with Lstructure is a tensor-triangulated category.

Proof. We consider a projective object XP, where P, is a projective category defined as the points set

P=XRAAis addative withRstructure,E37

We consider the application Λ,defined by the mapping:

Λ:PKP,E38

where the objects ΛX,are those that are determined by

H0ΛLQ=HomhXHomΛhYG,E39

or

ΛX=HomRAhXΛDRA,E40

Then we have

DRAKP,E41

via the chain homotopy. For other side

XHomhXHomΛhYG,E42

which is risked from Lstructure when 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:

CRDCD,E43

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

η:hXRhYhXY,E44
X,YA.

For U,in A,ηU,is the monoidal product in A,followed by the diagonal mapping of triangle:

η:UUU,E45

that is to say,

hXURhYUHomAUXRHomAUY,E46

satisfies the triangle7:

HomAUXRHomAUYHomAUUXYΔΔHomAUXYE47

where

HomAUXY=hXYU,E48

With all these dispositions and generalities, now we can specialize to the case when8

A=Cork,E49

and ,is the tensor product

XY=X×Y',E50

Then we have the Yoneda embedding:

PSTkCorkCork,E51

We denote as tr,for the tensor product on PSTk=ZCork,or

PSTkR=RCork,E52

and trL,for L. Then there is a natural mapping

CLtrDCtrD,E53

Here trL,is the tensor product induced to ZCork.But, before we will keep using the product tr,which we can define as:

XY=X×Y.
RtrXtrRtrY=RtrXY,E54

being hX=RtrX,hXHom,XA.

The above can be generalized through the following lemma.

Lemma 3.1. The pre-sheaf ZtrX1x1Xnxn,is a direct summand of ZtrX1××Xn. In particular, it is projective object of PST. Likewise, for the following sequence of pre-sheaves with transfers, the exactness is explicit9:

0ZtrX1××XniZtrXii,jZtrXi×Xji,jZtrX1××Xj××Xj××XniZtrX1××X̂j××XnZtrX1××XnZtrX1Xn0,E55

Then, it is sufficient to demonstrate that L,éttr, preserve quasi-isomorphisms.

Definition 3.2. A pre-sheaf with transfers is a contravariant additive functor:

F=CorkAb,E56

we write

PreShCorkPSTk=PST,E57

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 trét.

Likewise, we have the definition.

Definition 3.3. If F, and G, are pre-sheaves of R-modules with transfers, we write:

FtrGétFéttrG,E58

the Étale sheaf associated to FtrG.

If C, and D, are bounded above complexes of pre-sheaves with transfers, we shall write CéttrD,for CtrDét,and

CLtrDPéttrQ,E59

where P, and Q, are complexes of representable sheaves with transfers, PC,and QD.Then there is a natural mapping

CL,éttrDCéttrD,E60

induced by

CL,trDCtrD,E61

Lemma 3.2. If F, and F',are Étale sheaves of R-modules with transfers, and F, is locally constant, the mapping:

hXURhYU=HomAUXHomAUYHomAUUXYΔ'HomAUXY=hXYU,E62

induces an isomorphism

FétF'FéttrF',E63

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:

  1. i. The sequence

0FXdiagFU+FU×XU,E64

  1. is exact for every surjective Étale morphism of smooth schemes,

UX,E65

  1. ii. FX¯Y=FXFY,X,Y,schemes.

We demonstrate Lemma 3.2.

Proof. We want the tensor product L,éttr, which induces to tensor triangulated structure on the derived category of Étale sheaves of R-modules with transfers10 defined in other expositions [4]. Considering Proposition 3.1, we have:

CL,éttrDDL,éttrC,E66

Then, it is sufficient to demonstrate that L,éttr, preserve quasi-isomorphisms. The details can be found in [5].

Then the tensor product éttr,as pre-sheaf to Étale sheaves can have a homology space of zero dimension that vanishes in certain component right exact functor ΦF=RtrYéttrF,from the category PSTkR,of pre-sheaves of R-modules with transfers to the category of the Étale sheaves of R-modules and transfers. Then every derived functor LnΦ,vanishes on H0C˜,to certain complex of Étale.

Then, all right exact functors RtrYéttrF,are acyclic. This is the machinery to demonstrate the functor exactness and resolution in modules through of induce from L,éttr,a tensor-triangulated structure to a derived category more general that DRA.

Also we have:

Lemma 3.3. Fix Y, and set Φ=RtrYéttr.If F, is a pre-sheaf of R-modules with transfers such that Fét=0,then LnΦF=0,n.

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 L,éttr,induces a tensor-triangulated structure to a derived category more general than DRA,as for example, DMéteff,kZ/m,which is our objective. In this case, we want geometrical motives, where this last category DMéteff,kZ/m,can be identified for the derived category DMgmkR.

We consider and fix Y, and the right exact functor ΦF=RtrYéttrF,from the category PSTkR,of pre-sheaves of R-modules with transfers to the category of the Étale sheaves of R-modules and transfers. Likewise, their left functors LPΦF,are the homology sheaves of the total left derived functor ΦF=RtrYL,éttrF.Considering a chain complex C, the hypercohomology spectral sequence is:

Ep,q2=LpΦHqC,E67

then

Lp+qΦC=0,E68

Then the corresponding infinite sequence is exact.

We consider A, and BA,where A,is a category as has been defined before.

We have the following proposition.

Proposition 4.1. There is equivalence between categories AbCRingA//BModB.

Then a hypercohomology as given to dda=0,can be obtained through double functor work ABB,through an inclusion of a category ModB,in CRingA//B.Then is had the result.

Theorem 4.1. The left adjoint to the inclusion functor ModB,CRingA//B.is defined by XΩX/AXB.In particular, the image of ABB,under this functor is BΩX/A.

The derived tensor product is a regular tensor product.

Theorem 4.2. The character for an adjoint lifts for a homotopically meaningful adjunction complies:

ChBBsCRingA//B,E69

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 LA/B,is the image of functor ABB,under the left functor of the Kahler differentials module MRAL,. Likewise, if PB,be a free resolution then

LA/B=ΩP/APB,E70

The cotangent complex as defined in (69) lives in the derived category ModB.We observe that choosing the particular resolution of B,then ΩP/A,is a co-fibrant object in the derived category ModP,which no exist distinction between the derived tensor product and the usual tensor product. Then to any representation automorphic of GA,the GF/GA,can be decomposed as the tensor product i=1nπI.This last fall in the geometrical Langlands ramifications.

Example 4.1. (66) in the context of solution of field equations as dda=0,has solution in the hypercohomology of a spectral sequence of DRA,(established on the infinite sequence Fn0.[6]) when its functors whose image ΩB/A,have as its cotangent complex the image under of the functor LA/B,which is the functor image ABB,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 DLBunD,and DLLocO,where geometrical motives can be risked with the corresponding moduli stack to holomorphic bundles. The integrals are those whose functors image will be in SpecHSymTOPLGD,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].

5. Applications

As was shown, the geometrical motives required in our research are a result of embedding the derived category DMgmkR,(geometrical motives category) in the DMéteff,kZ/m,considering the category of smooth schemes on the field k.

We consider the following functors. For each FDShNisCork,there is LA1FDeffk,the resulting functor is:

LA1:DShNisCorkDeffk,E71

which is exact and left-adjoint to the inclusion

DeffkDShNisCork,E72

Also the functor (70) descends to an equivalence of triangulated categories. This is very useful to make Deffk,into a tensor category as follows. We consider the Nisnevich sheaf Ztrk,with transfer tr:YcYX.We define

ZtrkZtrkZtrX×kY,E73

Then it can be demonstrated that the operation realized in (70) can be extended to give DShNisCork,with the structure of a triangulated tensor category. Then the functor LA1,induces a tensor operation on DA1ShNisCork,making that DA1ShNisCorkalso a triangulated tensor category. Likewise, explicitly in DMeffk,this gives us the functor

m:SmkDMeffk,E74

defined by

mXCSusZtrX,E75

where we have the formula

mX×kY=mXmY,E76

If we consider the embedding theorem, then we can establish the following triangulated scheme

SmkDMgmeffkmIdDMgmeffkE77

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 M12. We know that solutions of the field equations dda=0,are given in a category SpecSmk,(see Example 4). Context Solutions of the quantum field equations for dda=0,are defined in hyper-cohomology on Qcoefficients from the category Smk, defined on a numerical field k, considering the derived tensor product éttr,of pre-sheaves. Then the following triangulated tensor category scheme is true and commutative:

DQFTiFMDgmQMDOYE78

The category DMgmeffkR,has a tensor structure and the tensor product of its motives is as defined in (75) mXmY=mX×Y.

Triangulated category of geometrical motives DMgmkR,or written simply as DMgmk,is defined formally inverting the functor of the Tate objects13 (are objects of a motivic category called Tannakian category) Z1, to be image of the complex 1Speck,where the motive in degree p=2,3,will be mp=mZ1p,or to any motive mDMgmeffk,p.

Likewise, the important fact is that the canonical functor DMgmeffk,DMgmk,is full embedding [7]. Therefore we work in the category DMgmk.

Likewise, for X, and Y, smooth projective varieties and for any integer i, there exists an isomorphism:

HomDMgmeffkmXmYi2iAm+iX×Y,m=dimY,E79

We demonstrate the Theorem 5.1.

Proof. XSmk,the category Smk,extends to a pseudo-tensor equivalence of cohomological categories over motives on k, that is to say, MMk,is the image of functors

DMeffkDMgmk,E80

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 Smk,defined as:

HomDMgmeffkmXQqpHNisXQq=Hp,qXNisQq,E81

which comes from the hypercohomology

HLp,qXQ=Hp,qXQ,E82

We observe that if a Zariski sheaf of Qmodules with transfers F, is such that F=HqC,for all C, a complex defined on Qqmodules (being a special case when C=Qq), where the cohomology groups of the isomorphism HétpXFétHNispXFNis, can be vanished for p>dimY.

Then survives a hypercohomology HqXQ.If we consider SpecSmk,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 DLBunD,and DLLocO, 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 [6] we can to obtain the category MOOY.The functors are constructed using the Mukai-Fourier transforms. ■

Acknowledgments

Posdoctoral research was supported by State of Mexico Council of Scientific Research COMECYT-077/111/21.

Notes

  • L, is a Lefschetz motive Z1, [1].
  • 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.
  • HomhYG:Y→HomRAF⊗hYG.
  • The field of fractions of an integral domain is the smallest field in which this domain can be embedded.
  • ηU∘Δ=Δ'.
  • 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:

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Francisco Bulnes (June 25th 2020). Derived Tensor Products and Their Applications, Advances on Tensor Analysis and their Applications, Francisco Bulnes, IntechOpen, DOI: 10.5772/intechopen.92869. Available from:

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