+ All Categories
Home > Documents > Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety...

Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety...

Date post: 07-Jan-2020
Category:
Upload: others
View: 0 times
Download: 0 times
Share this document with a friend
27
Tensor products on free abelian categories and Nori motives Mike Prest School of Mathematics Alan Turing Building University of Manchester [email protected] April 28, 2018 April 28, 2018 1 / 15
Transcript
Page 1: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Tensor products on free abelian categories and Norimotives

Mike PrestSchool of MathematicsAlan Turing Building

University of [email protected]

April 28, 2018

April 28, 2018 1 / 15

Page 2: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Free abelian categories

Freyd showed that, given a skeletally small preadditive category R, for instance aring, or the category mod-S of finitely presented modules over a ring, there is anembedding R → Ab(R) of R into an abelian category which has the followinguniversal property.

for every additive functor M : R → A, where A is an abelian category, there is aunique-to-natural-equivalence extension of M to an exact functor M making thefollowing diagram commute.

R //

M""EEEEEEEEE Ab(R)

M��A

The category Ab(R) is realised as the category of finitely presented functors onfinitely presented left R-modules, or as the category of pp-pairs for left R-modules.

Theorem

For any ring or small preadditive category R, there are natural equivalencesAb(R) ' (R-mod,Ab)fp ' RLeq+. Furthermore, with reference to the diagram

above, M = Meq+, the enrichment of the R-module M by pp-imaginaries.

April 28, 2018 2 / 15

Page 3: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Free abelian categories

Freyd showed that, given a skeletally small preadditive category R, for instance aring, or the category mod-S of finitely presented modules over a ring, there is anembedding R → Ab(R) of R into an abelian category which has the followinguniversal property.

for every additive functor M : R → A, where A is an abelian category, there is aunique-to-natural-equivalence extension of M to an exact functor M making thefollowing diagram commute.

R //

M""EEEEEEEEE Ab(R)

M��A

The category Ab(R) is realised as the category of finitely presented functors onfinitely presented left R-modules, or as the category of pp-pairs for left R-modules.

Theorem

For any ring or small preadditive category R, there are natural equivalencesAb(R) ' (R-mod,Ab)fp ' RLeq+. Furthermore, with reference to the diagram

above, M = Meq+, the enrichment of the R-module M by pp-imaginaries.April 28, 2018 2 / 15

Page 4: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

For Example:

The free abelian category on the quiver A3 •→ •→ •(rather, on its path algebra, equivalently on the preadditive category freelygenerated by A3):

00

00

01

00

00

11

00

10

11

00

00

10

00

10

10

00

01

10

00

11

10 0

111

10

00

01

00

00

10

00

01

11

00

01

10

00

01

01

00

01

00

00

11

10

00

11

00

00

10

00

00

April 28, 2018 3 / 15

Page 5: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori motives

(Grothendieck) The motive of a variety should be its abelian avatar: given asuitable category V of varieties (or schemes), there should be a functor from V toits category of motives. That category should be abelian and such that everyhomology or cohomology theory on V factors through the functor from V to itscategory of motives. So that functor itself should be a kind of universal(co)homology theory for V.In the case that V is the category of nonsingular projective varieties over C, thereis such a category of motives. But the question of existence for possibly singular,not-necessarily projective varieties - the conjectural category of mixed motives - isopen.

In the 90s Nori described the construction of an abelian category which is acandidate for the category of mixed motives. His idea is to construct from acategory of varieties V a (very large) quiver D such every (co)homology theory onV gives a representation of D (or Dop). A particular representation - singularhomology - is then used to construct this category of motives.

(There is more involved than this, in particular a product structure on D is neededto give a tensor product operation on the category of motives.)

April 28, 2018 4 / 15

Page 6: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori motives

(Grothendieck) The motive of a variety should be its abelian avatar: given asuitable category V of varieties (or schemes), there should be a functor from V toits category of motives. That category should be abelian and such that everyhomology or cohomology theory on V factors through the functor from V to itscategory of motives. So that functor itself should be a kind of universal(co)homology theory for V.In the case that V is the category of nonsingular projective varieties over C, thereis such a category of motives. But the question of existence for possibly singular,not-necessarily projective varieties - the conjectural category of mixed motives - isopen.

In the 90s Nori described the construction of an abelian category which is acandidate for the category of mixed motives. His idea is to construct from acategory of varieties V a (very large) quiver D such every (co)homology theory onV gives a representation of D (or Dop). A particular representation - singularhomology - is then used to construct this category of motives.

(There is more involved than this, in particular a product structure on D is neededto give a tensor product operation on the category of motives.)

April 28, 2018 4 / 15

Page 7: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori motives

(Grothendieck) The motive of a variety should be its abelian avatar: given asuitable category V of varieties (or schemes), there should be a functor from V toits category of motives. That category should be abelian and such that everyhomology or cohomology theory on V factors through the functor from V to itscategory of motives. So that functor itself should be a kind of universal(co)homology theory for V.In the case that V is the category of nonsingular projective varieties over C, thereis such a category of motives. But the question of existence for possibly singular,not-necessarily projective varieties - the conjectural category of mixed motives - isopen.

In the 90s Nori described the construction of an abelian category which is acandidate for the category of mixed motives. His idea is to construct from acategory of varieties V a (very large) quiver D such every (co)homology theory onV gives a representation of D (or Dop). A particular representation - singularhomology - is then used to construct this category of motives.

(There is more involved than this, in particular a product structure on D is neededto give a tensor product operation on the category of motives.)

April 28, 2018 4 / 15

Page 8: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

It turns out that Nori’s category of motives is a Serre quotient of the free abeliancategory on D, the quotient being determined by the representation given bysingular homology.

In essence this first appeared in a paper of Barbieri-Viale, Caramello and Lafforgue(arXiv:1506:06113), though it is not said this way. In that paper Caramello usedthe methods of categorical model theory, in particular classifying toposes forregular logic, and showed that Nori’s category is the effectivisation of the regularsyntactic category for a regular theory associated to Nori’s diagram D. This is amuch simpler construction than Nori’s original one, in particular there is no needto approximate the final result through finite subdiagrams of D or to go viacoalgebra representations.

In that paper additivity appears at a relatively late stage of the construction. If webuild that in from the beginning then (Barbieri-Viale and Prest,arXiv:1604:00153), we are able to apply the existing model theory of additivestructures and, in particular, to realise Nori’s category of motives as a localisation

of the free abelian category on the preadditive category Z−→D generated by Nori’s

diagram D.

(−→D is the category freely generated by D - so Z

−→D is essentially the path algebra

of D).

April 28, 2018 5 / 15

Page 9: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

It turns out that Nori’s category of motives is a Serre quotient of the free abeliancategory on D, the quotient being determined by the representation given bysingular homology.

In essence this first appeared in a paper of Barbieri-Viale, Caramello and Lafforgue(arXiv:1506:06113), though it is not said this way. In that paper Caramello usedthe methods of categorical model theory, in particular classifying toposes forregular logic, and showed that Nori’s category is the effectivisation of the regularsyntactic category for a regular theory associated to Nori’s diagram D. This is amuch simpler construction than Nori’s original one, in particular there is no needto approximate the final result through finite subdiagrams of D or to go viacoalgebra representations.

In that paper additivity appears at a relatively late stage of the construction. If webuild that in from the beginning then (Barbieri-Viale and Prest,arXiv:1604:00153), we are able to apply the existing model theory of additivestructures and, in particular, to realise Nori’s category of motives as a localisation

of the free abelian category on the preadditive category Z−→D generated by Nori’s

diagram D.

(−→D is the category freely generated by D - so Z

−→D is essentially the path algebra

of D).

April 28, 2018 5 / 15

Page 10: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

The Serre quotient associated to a representation

Theorem

Suppose that M is a representation of the small preadditive category R and let Mbe its exact extension to the free abelian category on R. The kernel of M,SM = {F ∈ Ab(R) : MF = 0}, is a Serre subcategory of Ab(R) and there is a

factorisation of M as a composition of exact functors through the quotientcategory A(M) = Ab(R)/SM .

R

M

��

j // Ab(R)

M

��

$$HHHHHHHHH

A(M)

Muujjjjjjjjjjjjjjjjjj

Ab

April 28, 2018 6 / 15

Page 11: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori’s diagram

For the vertices, we take triples (X ,Y , i) where X ,Y ∈ V, Y is a closedsubvariety of X and i ∈ Z.

The arrows of D are of two kinds:- for each morphism f : X → X ′ of V we have, for each i , a corresponding arrow(X ,Y , i) → (X ′,Y ′, i) provided fY ⊆ Y ′;- for each X ,Y ,Z ∈ V with Y ⊇ Z closed subvarieties of X , we add an arrow(Y ,Z , i) → (X ,Y , i − 1).

A homology theory H on V gives a representation of this quiver by sending(X ,Y , i) to the relative homology Hi (X ,Y ). Arrows of the first kind are sent tothe obvious maps between relative homology objects; those of the second kind aresent to the connecting maps in the long exact sequence for homology.Taking H to be singular homology, we obtain a representation of D and then the

corresponding Serre quotient A(H) = Ab(Z−→D )/SH of the free abelian category

turns out to be Nori’s category of motives.

In fact, more is needed. In particular there should be a tensor product structureon motives. This is needed, for example, to express the Kunneth formula.In Barbieri-Viale, Huber and Prest, arXiv:1803.00809, we show how to induce thisstructure. In particular we show how a tensor product on the category ofR-modules induces a tensor product on the free abelian category Ab(R).

April 28, 2018 7 / 15

Page 12: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori’s diagram

For the vertices, we take triples (X ,Y , i) where X ,Y ∈ V, Y is a closedsubvariety of X and i ∈ Z.

The arrows of D are of two kinds:- for each morphism f : X → X ′ of V we have, for each i , a corresponding arrow(X ,Y , i) → (X ′,Y ′, i) provided fY ⊆ Y ′;- for each X ,Y ,Z ∈ V with Y ⊇ Z closed subvarieties of X , we add an arrow(Y ,Z , i) → (X ,Y , i − 1).

A homology theory H on V gives a representation of this quiver by sending(X ,Y , i) to the relative homology Hi (X ,Y ). Arrows of the first kind are sent tothe obvious maps between relative homology objects; those of the second kind aresent to the connecting maps in the long exact sequence for homology.

Taking H to be singular homology, we obtain a representation of D and then the

corresponding Serre quotient A(H) = Ab(Z−→D )/SH of the free abelian category

turns out to be Nori’s category of motives.

In fact, more is needed. In particular there should be a tensor product structureon motives. This is needed, for example, to express the Kunneth formula.In Barbieri-Viale, Huber and Prest, arXiv:1803.00809, we show how to induce thisstructure. In particular we show how a tensor product on the category ofR-modules induces a tensor product on the free abelian category Ab(R).

April 28, 2018 7 / 15

Page 13: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori’s diagram

For the vertices, we take triples (X ,Y , i) where X ,Y ∈ V, Y is a closedsubvariety of X and i ∈ Z.

The arrows of D are of two kinds:- for each morphism f : X → X ′ of V we have, for each i , a corresponding arrow(X ,Y , i) → (X ′,Y ′, i) provided fY ⊆ Y ′;- for each X ,Y ,Z ∈ V with Y ⊇ Z closed subvarieties of X , we add an arrow(Y ,Z , i) → (X ,Y , i − 1).

A homology theory H on V gives a representation of this quiver by sending(X ,Y , i) to the relative homology Hi (X ,Y ). Arrows of the first kind are sent tothe obvious maps between relative homology objects; those of the second kind aresent to the connecting maps in the long exact sequence for homology.Taking H to be singular homology, we obtain a representation of D and then the

corresponding Serre quotient A(H) = Ab(Z−→D )/SH of the free abelian category

turns out to be Nori’s category of motives.

In fact, more is needed. In particular there should be a tensor product structureon motives. This is needed, for example, to express the Kunneth formula.In Barbieri-Viale, Huber and Prest, arXiv:1803.00809, we show how to induce thisstructure. In particular we show how a tensor product on the category ofR-modules induces a tensor product on the free abelian category Ab(R).

April 28, 2018 7 / 15

Page 14: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Nori’s diagram

For the vertices, we take triples (X ,Y , i) where X ,Y ∈ V, Y is a closedsubvariety of X and i ∈ Z.

The arrows of D are of two kinds:- for each morphism f : X → X ′ of V we have, for each i , a corresponding arrow(X ,Y , i) → (X ′,Y ′, i) provided fY ⊆ Y ′;- for each X ,Y ,Z ∈ V with Y ⊇ Z closed subvarieties of X , we add an arrow(Y ,Z , i) → (X ,Y , i − 1).

A homology theory H on V gives a representation of this quiver by sending(X ,Y , i) to the relative homology Hi (X ,Y ). Arrows of the first kind are sent tothe obvious maps between relative homology objects; those of the second kind aresent to the connecting maps in the long exact sequence for homology.Taking H to be singular homology, we obtain a representation of D and then the

corresponding Serre quotient A(H) = Ab(Z−→D )/SH of the free abelian category

turns out to be Nori’s category of motives.

In fact, more is needed. In particular there should be a tensor product structureon motives. This is needed, for example, to express the Kunneth formula.In Barbieri-Viale, Huber and Prest, arXiv:1803.00809, we show how to induce thisstructure. In particular we show how a tensor product on the category ofR-modules induces a tensor product on the free abelian category Ab(R).

April 28, 2018 7 / 15

Page 15: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Lifting tensor product from modules to functors on modules

Suppose that R-mod has a tensor product. Then there is an induced tensorproduct on the free abelian category Ab(R) = (R-mod,Ab)fp, defined as follows.

Given A,B ∈ R-mod, define ⊗ on the corresponding representable functors by(A,−)⊗ (B,−) = (A⊗ B,−).Given morphisms f : A → A ′ and g : B → B ′ between finitely presented modules,define (f ,−)⊗ (g ,−) = (f ⊗ g ,−) : (A ′ ⊗ B ′,−) → (A⊗ B,−).The tensor product constructed on Ab(R) will be required to be right exact, sothat forces the rest of the construction.A typical object of Ab(R) is the cokernel of a morphism between representables:

(B,−)(f ,−)−−−→ (A,−)

π−→ Ff → 0

for some morphism f : A → B.Therefore if C ∈ R-mod then the value of (C ,−)⊗ Ff is forced by requiring thesequence

(C ,−)⊗ (B,−) → (C ,−)⊗ (A,−)π−→ (C ,−)⊗ Ff → 0

to be exact.That can then be repeated to compute the general case Fg ⊗ Ff .

April 28, 2018 8 / 15

Page 16: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Lifting tensor product from modules to functors on modules

Suppose that R-mod has a tensor product. Then there is an induced tensorproduct on the free abelian category Ab(R) = (R-mod,Ab)fp, defined as follows.

Given A,B ∈ R-mod, define ⊗ on the corresponding representable functors by(A,−)⊗ (B,−) = (A⊗ B,−).Given morphisms f : A → A ′ and g : B → B ′ between finitely presented modules,define (f ,−)⊗ (g ,−) = (f ⊗ g ,−) : (A ′ ⊗ B ′,−) → (A⊗ B,−).The tensor product constructed on Ab(R) will be required to be right exact, sothat forces the rest of the construction.

A typical object of Ab(R) is the cokernel of a morphism between representables:

(B,−)(f ,−)−−−→ (A,−)

π−→ Ff → 0

for some morphism f : A → B.Therefore if C ∈ R-mod then the value of (C ,−)⊗ Ff is forced by requiring thesequence

(C ,−)⊗ (B,−) → (C ,−)⊗ (A,−)π−→ (C ,−)⊗ Ff → 0

to be exact.That can then be repeated to compute the general case Fg ⊗ Ff .

April 28, 2018 8 / 15

Page 17: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Lifting tensor product from modules to functors on modules

Suppose that R-mod has a tensor product. Then there is an induced tensorproduct on the free abelian category Ab(R) = (R-mod,Ab)fp, defined as follows.

Given A,B ∈ R-mod, define ⊗ on the corresponding representable functors by(A,−)⊗ (B,−) = (A⊗ B,−).Given morphisms f : A → A ′ and g : B → B ′ between finitely presented modules,define (f ,−)⊗ (g ,−) = (f ⊗ g ,−) : (A ′ ⊗ B ′,−) → (A⊗ B,−).The tensor product constructed on Ab(R) will be required to be right exact, sothat forces the rest of the construction.A typical object of Ab(R) is the cokernel of a morphism between representables:

(B,−)(f ,−)−−−→ (A,−)

π−→ Ff → 0

for some morphism f : A → B.Therefore if C ∈ R-mod then the value of (C ,−)⊗ Ff is forced by requiring thesequence

(C ,−)⊗ (B,−) → (C ,−)⊗ (A,−)π−→ (C ,−)⊗ Ff → 0

to be exact.That can then be repeated to compute the general case Fg ⊗ Ff .

April 28, 2018 8 / 15

Page 18: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Example:

R = K [ε : ε2 = 0] is commutative so we have the usual ⊗ on R-mod

First, we compute Ab(K [ε]):

We have the exact sequence

0 → Kj−→ R

p−→ K → 0,

where K is the unique simple R-module.

Using this, we get the projective presentations of the two simple functors onR-mod:

0 → (K ,−)(p,−)−−−→ (R,−)

πS−→ S = Fp → 0

0 → (K ,−)(p,−)−−−→ (R,−)

(j,−)−−−→ (K ,−)πT−−→ T = Fj → 0.

April 28, 2018 9 / 15

Page 19: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Example:

R = K [ε : ε2 = 0] is commutative so we have the usual ⊗ on R-mod

First, we compute Ab(K [ε]):

We have the exact sequence

0 → Kj−→ R

p−→ K → 0,

where K is the unique simple R-module.

Using this, we get the projective presentations of the two simple functors onR-mod:

0 → (K ,−)(p,−)−−−→ (R,−)

πS−→ S = Fp → 0

0 → (K ,−)(p,−)−−−→ (R,−)

(j,−)−−−→ (K ,−)πT−−→ T = Fj → 0.

April 28, 2018 9 / 15

Page 20: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

Example:

R = K [ε : ε2 = 0] is commutative so we have the usual ⊗ on R-mod

First, we compute Ab(K [ε]):

We have the exact sequence

0 → Kj−→ R

p−→ K → 0,

where K is the unique simple R-module.

Using this, we get the projective presentations of the two simple functors onR-mod:

0 → (K ,−)(p,−)−−−→ (R,−)

πS−→ S = Fp → 0

0 → (K ,−)(p,−)−−−→ (R,−)

(j,−)−−−→ (K ,−)πT−−→ T = Fj → 0.

April 28, 2018 9 / 15

Page 21: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

The category Ab(K [ε]):

(R,−)

''OOOOOOOOOOO

���

�������

(R,−)

%%LLLLLLLLLL

���

�������

(K ,−)

::uuuuuuuuu

$$JJJJJJJJJJ(R,−)/soc(R,−)

&&MMMMMMMMMMMM(K ,−)

$$JJJJJJJJJJ

::uuuuuuuuu(R,−)/soc

T

77nnnnnnnnnnnnn

��� S

<<yyyyyyyyyT

88rrrrrrrrrrr

���

April 28, 2018 10 / 15

Page 22: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

STS

��999999999

��

��������

STS

��999999999

��

��������

T

S

BB���������

��<<<<<<<<S

T

��<<<<<<<<T

S

��<<<<<<<<

BB��������� S

T

T

@@��������

��� S

AA��������T

@@��������

���

April 28, 2018 11 / 15

Page 23: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

We compute the values of ⊗ on Ab(R) using the projective presentations

(K ,−)(p,−)−−−→ (R,−)

πS−→ S → 0

and

(R,−)(j,−)−−−→ (K ,−)

πT−−→ T → 0

of the simple functors S and T .

To compute S ⊗ S :

(K ⊗ K ,−)(p⊗1K ,−) //

(1K⊗p,−)

��

(R ⊗ K ,−)πS⊗(1K ,−) //

(1R⊗p,−)

��

S ⊗ (K ,−)

1S⊗(p,−)

��

// 0

(K ⊗ R,−)(p⊗1R ,−) //

(1K ,−)⊗πS

��

(R ⊗ R,−)πS⊗(1R ,−) //

(1R ,−)⊗πS

��

S ⊗ (R,−)

1S⊗πS

��

// 0

(K ,−)⊗ S(p,−)⊗1S //

��

(R,−)⊗ SπS⊗1S //

��

S ⊗ S //

��

// 0

0 0 0

April 28, 2018 12 / 15

Page 24: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

We compute the values of ⊗ on Ab(R) using the projective presentations

(K ,−)(p,−)−−−→ (R,−)

πS−→ S → 0

and

(R,−)(j,−)−−−→ (K ,−)

πT−−→ T → 0

of the simple functors S and T .

To compute S ⊗ S :

(K ⊗ K ,−)(p⊗1K ,−) //

(1K⊗p,−)

��

(R ⊗ K ,−)πS⊗(1K ,−) //

(1R⊗p,−)

��

S ⊗ (K ,−)

1S⊗(p,−)

��

// 0

(K ⊗ R,−)(p⊗1R ,−) //

(1K ,−)⊗πS

��

(R ⊗ R,−)πS⊗(1R ,−) //

(1R ,−)⊗πS

��

S ⊗ (R,−)

1S⊗πS

��

// 0

(K ,−)⊗ S(p,−)⊗1S //

��

(R,−)⊗ SπS⊗1S //

��

S ⊗ S //

��

// 0

0 0 0

April 28, 2018 12 / 15

Page 25: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

which simplifies to:

(K ,−)1 //

1

��

(K ,−) //

(p,−)

��

0

��

// 0

(K ,−)(p,−) //

��

(R,−)πS //

πS

��

S

πS⊗1S��

// 0

0 //

��

SπS⊗1S //

��

S ⊗ S //

��

// 0

0 0 0

Hence S ⊗ S = S and πS ⊗ 1S = 1S .

April 28, 2018 13 / 15

Page 26: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

S T (K ,−) =T

SS

T (R,−) =

STS

S S 0 0 S S

T 0 (K ,−) (K ,−) T T

(K ,−) 0 (K ,−) (K ,−) (K ,−) (K ,−)

S

T S T (K ,−)S

TS

T

(R,−) S T (K ,−)S

T (R,−)

April 28, 2018 14 / 15

Page 27: Tensor products on free abelian categories and Nori motives(Grothendieck) The motive of a variety should be its abelian avatar: given a suitable category Vof varieties (or schemes),

A final remark: this shows how, when we have a tensor product on R-mod, toform the tensor product of pp formulas.

April 28, 2018 15 / 15


Recommended