Biextensions of 1-motives in Voevodsky's category of motives

0.15 MB·English

by  Cristiana Bertolin

BIEXTENSIONS OF 1-MOTIVES IN VOEVODSKY’S CATEGORY
OF MOTIVES
9
CRISTIANABERTOLINANDCARLOMAZZA
0
0
2 Abstract. Letk beaperfectfield. Inthispaper weprovethatbiextensions
of 1-motives define multilinear morphisms between 1-motives in Voevodsky’s
n
triangulatedcategoryDMeff(k,Q)ofeffectivegeometricalmotivesoverkwith
a gm
rationalcoefficients.
J

3

]
T
Contents
K
. Introduction 1
h
Acknowledgment 2
t
a Notation 2
m
1. 1-motives in Voevodsky’s category 4
[ 2. Bilinear morphisms between 1-motives 6

3. Multilinear morphisms between 1-motives 7
1
v References 8
1
3
3
0
Introduction
.
1
0 Let k be a perfect field. In [O] Orgogozoconstructs a fully faithful functor
9
(0.1) O :Db(1−Isomot(k))−→DMeff(k,Q)
0 gm
:
v fromthe bounded derivedcategoryofthe category1−Isomot(k) of1-motivesover
i k defined modulo isogenies to Voevodsky’s triangulated category DMeff(k,Q) of
X gm
effectivegeometricalmotivesoverk withrationalcoefficients. IfM (fori=1,2,3)
i
r
a is a 1-motive defined over k modulo isogenies, in this paper we prove that the
group of isomorphism classes of biextensions of (M ,M ) by M is isomorphic
1 2 3
to the group of morphisms of the category DMeff(k,Q) from the tensor product
gm
O(M )⊗ O(M ) to O(M ):
1 tr 2 3
Theorem 0.1. Let M (for i=1,2,3) be a 1-motive defined over a perfect field k.
i
Then
Biext1(M1,M2;M3)⊗Q∼=HomDMeff(k,Q)(O(M1)⊗trO(M2),O(M3)).
gm
ThisisomorphismanswersaquestionraisedbyBarbieri-VialeandKahnin[BK1]
Remark 7.1.3 2). In loc. cit. Proposition 7.1.2 e) they prove the above theorem in
the casewhereM is asemi-abelianvariety. Ourproofis ageneralizationoftheirs.
3
1991 Mathematics Subject Classification. 14F,14K.
Key words and phrases. multilinearmorphisms,biextensions,1-motives.
1
2 CRISTIANABERTOLINANDCARLOMAZZA
If k is a field ofcharacteristic0 embeddable in C,by [D](10.1.3)we havea fully
faithful functor
(0.2) T:1−Mot(k)−→MR(k)
fromthecategory1−Mot(k)of1-motivesoverktotheTannakiancategoryMR(k)
ofmixedrealizationsoverk(see[J]I2.1),whichattachestoeach1-motiveitsHodge
realizationforanyembeddingk֒→C,itsdeRhamrealization,itsℓ-adicrealizations
for any prime number ℓ, and its comparison isomorphisms. According to [B1]
Theorem4.5.1,if M (for i=1,2,3)is a 1-motive defined over k modulo isogenies,
i
thegroupofisomorphismclassesofbiextensionsof(M ,M )byM isisomorphicto
1 2 3
the groupof morphisms of the category MR(k) from the tensor product T(M )⊗
1
T(M ) of the realizations of M and M to the realization T(M ) of M . Putting
2 1 2 3 3
together this result with Theorem 0.1, we get the following isomorphisms
Biext1(M1,M2;M3)⊗Q ∼= HomDMeff(k;Q)(O(M1)⊗trO(M2),O(M3))
gm
(0.3) ∼= HomMR(k)(cid:0)T(M1)⊗T(M2),T(M3)(cid:1).
Theseisomorphismsfitintothefollowingcontext: in[H]Huberconstructsafunctor
H:DMeff(k,Q)−→D(MR(k))
gm
from Voevodsky’s category DMeff(k,Q) to the triangulated category D(MR(k))
gm
of mixed realizations over k, which respects the tensor structures. Extending the
functor T (0.2) to the derivedcategoryDb(1−Isomot(k)), we obtain the following
diagram
(0.4) Db(1−Isomot(k)) T // D(MR(k))
O (cid:15)(cid:15) mmmmmmHmmmmmmm66
DMeff(k,Q)
gm
Theisomorphisms(0.3)meanthatbiextensionsof1-motivesdefineinacompatible
way bilinear morphisms between 1-motives in each category involved in the above
diagram. Barbieri-Viale and Kahn informed the authors that in [BK2] they have
proved the commutativity of the diagram (0.4) in an axiomatic setting. If k = C,
they can prove its commutativity without assuming axioms.
WefinishgeneralizingTheorem0.1tomultilinearmorphismsbetween1-motives.
Acknowledgment
The authors are very grateful to Barbieri-Viale and Kahn for several useful re-
marks improving the first draft of this paper.
Notation
IfC isanadditivecategory,wedenotebyC⊗QtheassociatedQ-linearcategory
which is universal for functors from C to a Q-linear category. Explicitly, the cate-
goryC⊗QhasthesameobjectsasthecategoryC,butthesetsofarrowsofC⊗Qare
the sets of arrows of C tensored with Q, i.e. HomC⊗Q(−,−)=HomC(−,−)⊗ZQ.
We give a quick review of Voevodsky’s category of motives (see [V]). Denote
by Sm(k) the category of smooth varieties over a field k. Let A = Z or Q be
the coefficient ring. Let SmCor(k,A) be the category whose objects are smooth
BIEXTENSIONS AND 1-MOTIVES 3
varieties over k and whose morphisms are finite correspondences with coefficients
in A. It is an additive category.
The triangulated category DMeff(k,A) of effective geometrical motives
gm
over k is the pseudo-abelianenvelopeofthe localizationofthe homotopycategory
Hb(SmCor(k,A))ofboundedcomplexesoverSmCor(k,A)withrespecttothethick
subcategory generated by the complexes X× A1 →X and U ∩V →U ⊕V →X
k k
for any smooth variety X and any Zariski-coveringX =U ∪V.
The category of Nisnevich sheaves on Sm(k), Sh (Sm(k)), is the category
Nis
of abelian sheaves on Sm(k) for the Nisnevich topology.
A presheaf with transfers on Sm(k) is an additive contravariant functor from
SmCor(k,A) to the category of abelian groups. It is called a Nisnevich sheaf with
transfersifthecorrespondingpresheafofabeliangroupsonSm(k)isasheafforthe
Nisnevich topology. Denote by Sh (SmCor(k,A)) the category of Nisnevich
Nis
sheaves with transfers. By [V] Theorem 3.1.4 it is an abelian category.
ApresheafwithtransfersF iscalledhomotopyinvariantifforanysmoothvariety
X the natural map F(X)→F(X× A1) induced by the projectionX× A1 →X
k k k k
is an isomorphism. A Nisnevich sheaf with transfers is called homotopy invariant
if it is homotopy invariant as a presheaf with transferts.
Thecategory DMeff(k,A) of effective motivic complexesisthefullsubcat-
−
egory of the derived category D−(Sh (SmCor(k,A))) of complexes of Nisnevich
Nis
sheaves with transfers bounded from the above, which consists of complexes with
homotopy invariant cohomology sheaves.
There exists a functor L :SmCor(k,A) →Sh (SmCor(k,A)) which associates
Nis
to each smooth variety X a Nisnevich sheaf with transfers given by L(X)(U) =
c(U,X) ,wherec(U,X) isthefreeA-modulegeneratedbyprimecorrespondences
A A
from U to X. This functor extends to complexes furnishing a functor
L:Hb(SmCor(k,A))−→D−(Sh (SmCor(k,A))).
Nis
There exists also a functor C∗ :ShNis(SmCor(k,A)) → DMe−ff(k,A) which asso-
ciatestoeachNisnevichsheafwithtransfersF theeffectivemotiviccomplexC∗(F)
given by C (F)(U) = F(U ×∆n) where ∆∗ is the standard cosimplicial object.
n
This functor extends to a functor
(0.5) RC∗ :D−(ShNis(SmCor(k,A)))−→DMe−ff(k,A)
whichisleftadjointtothenaturalembedding. Moreover,thisfunctoridentifiesthe
categoryDMeff(k,A)withthelocalizationofD−(Sh (SmCor(k,A)))withrespect
− Nis
to the localizing subcategory generated by complexes of the form L(X × A1) →
k k
L(X) for any smooth variety X (see [V] Proposition 3.2.3).
If X and Y are two smooth varieties over k, the equality
(0.6) L(X)⊗L(Y)=L(X × Y)
k
defines a tensor structure on the category Sh (SmCor(k,A)), which extends to
Nis
thederivedcategoryD−(Sh (SmCor(k,A))).ThetensorstructureonDMeff(k,A),
Nis −
that we denote by ⊗tr, is the descent with respect to the projection RC∗ (0.5) of
the tensor structure on D−(Sh (SmCor(k,A))) .
Nis
Ifweassumektobeaperfectfield,by[V]Proposition3.2.6thereexistsafunctor
(0.7) i:DMeff(k,A)→DMeff(k,A)
gm −
4 CRISTIANABERTOLINANDCARLOMAZZA
whichisafullembeddingwithdenseimageandwhichmakesthefollowingdiagram
commutative
Hb(SmCor(k,A)) →L D−(Sh (SmCor(k,A)))
Nis
↓ ↓RC∗
DMeff(k,A) 99iK DMeff(k,A).
gm −
Remark 0.2. ForVoevodsky’stheoryofmotiveswithrationalcoefficients,the´etale
topology gives the same motivic answer as the Nisnevich topology: if we construct
the category of effective motivic complexes using the ´etale topology instead of the
Nisnevichtopology,wegetatriangulatedcategoryDMeff (k,A)whichisequivalent
−,´et
as triangulated category to the category DMeff(k,A) if we assume A = Q (see [V]
−
Proposition 3.3.2).
1. 1-motives in Voevodsky’s category
A 1-motive M =(X,A,T,G,u) over a field k (see [D] §10) consists of
• agroupschemeX overk,whichislocallyforthe´etaletopology,aconstant
group scheme defined by a finitely generated free Z-module,
• an extention G of an abelian k-variety A by a k-torus T,
• a morphism u:X −→G of commutative k-group schemes.
u
A 1-motive M =(X,A,T,G,u) can be viewed also as a length 1 complex [X →
G]ofcommutative k-groupschemes. In this paper, asa complexwe shallput X in
degree 0 and G in degree 1. A morphism of 1-motives is a morphism of complexes
of commutative k-groupschemes. Denote by 1−Mot(k) the category of 1-motives
over k. It is an additive category but it isn’t an abelian category.
Denote by 1−Isomot(k)the Q-linearcategory1−Mot(k)⊗Qassociatedto the
category of 1-motives over k. The objects of 1−Isomot(k) are called 1-isomotifs
and the morphisms of 1−Mot(k) which become isomorphisms in 1−Isomot(k)
are the isogenies between 1-motives, i.e. the morphisms of complexes [X → G] →
[X′ → G′] such that X → X′ is injective with finite cokernel, and G → G′ is
surjective with finite kernel. The category 1−Isomot(k) is an abelian category
(see [O] Lemma 3.2.2).
Assume now k to be a perfect field. The two mainingredients whichfurnish the
link between 1-motives and Voevodsky’s motives are:
(1) any commutative k-group scheme represents a Nisnevich sheaf with trans-
fers, i.e. an object of Sh (SmCor(k,A)) ([O] Lemma 3.1.2),
Nis
(2) if A (resp. T, resp. X) is an abelian k-variety (resp. a k-torus, resp. a
groupschemeoverk,whichislocallyforthe´etaletopology,aconstantgroup
scheme defined by a finitely generated free Z-module), then the Nisnevich
sheaf with transfers that it represents is homotopy invariant ([O] Lemma
3.3.1).
Since we can view 1-motives as complexes of smooth varieties over k, we have a
functor fromthe categoryof1-motivesto the categoryC(Sm(k)) of complexesover
Sm(k). Accordingto (1), this functor factorizesthroughthe categoryofcomplexes
over Sh (SmCor(k,A)):
Nis
1−Mot(k)−→C(Sh (SmCor(k,A)))
Nis
BIEXTENSIONS AND 1-MOTIVES 5
If we tensor with Q, we get an additive exact functor between abelian categories
1−Isomot(k)−→C(Sh (SmCor(k,A))⊗Q).
Nis
Takingtheassociatedboundedderivedcategories,weobtainatriangulatedfunctor
Db(1−Isomot(k))−→Db(Sh (SmCor(k,A))⊗Q).
Nis
Finally,accordingto(2)thislastfunctorfactorizesthroughthetriangulatedfunctor
O:Db(1−Isomot(k))−→DMeff(k,A)⊗Q.
−
By [O] Proposition 3.3.3 this triangulated functor is fully faithful, and by loc.
cit. Theorem 3.4.1 it factorizes through the thick subcategory d DMeff(k,Q) of
1 gm
DMeff(k,Q) generated by smooth varieties of dimension ≤1 over k and it induces
gm
an equivalence of triangulated categories, that we denote again by O,
O :Db(1−Isomot(k))−→d DMeff(k,Q).
1 gm
In order to simplify notation, if M is a 1-motive, we denote again by M its image
in d DMeff(k,Q)through the aboveequivalence of categoriesand also its image in
1 gm
DMeff(k,A) through the full embedding (0.7).
−
For the proof of Theorem 0.1, we will need the following
Proposition 1.1. Let M (for i = 1,2,3) be a 1-motive defined over k. The
i
forgetful triangulated functors
DMeff(k,A)֒→a D−(Sh (SmCor(k,A)))֒→b D−(Sh (Sm(k)))
− Nis Nis
induce an isomorphism
HomDMe−ff(k,A)(M1⊗trM2,M3)∼=HomD−(ShNis(Sm(k)))(M1⊗LM2,M3).
Proof. The forgetful functor a admits as left adjoint the functor RC∗ (0.5). The
forgetful functor b from the category of Nisnevich sheaves with transfers to the
category of Nisnevich sheaves admits as left adjoint the free sheaf with transfers
functor
(1.1) Φ:D−(Sh (Sm(k)))−→D−(Sh (SmCor(k,A)))
Nis Nis
([V]Remark1page202). IfX isasmoothvarietyoverk,letZ(X)bethesheafifica-
tionwithrespecttotheNisnevischtopologyofthepresheafU 7→Z[Hom (U,X)].
Sm(k)
ClearlyΦ(Z(X))istheNisnevichsheafwithtransfersL(X). IfY isanothersmooth
varietyoverk,wehavethatZ(X)⊗Z(Y)=Z(X× Y)(see[MVW]Lemma12.14)
k
and so by formula (0.6) we get
Φ(Z(X)⊗Z(Y))=Φ(Z(X))⊗ Φ(Z(Y)).
tr
The tensor structure on DMeff(k,A) is the descent of the tensor structure on
−
D−(ShNis(SmCor(k,A))) with respect to RC∗ and therefore
RC∗◦Φ(Z(X)⊗Z(Y))=RC∗◦Φ(Z(X))⊗trRC∗◦Φ(Z(Y)).
Using this equality and the fact that the composite RC∗◦Φ is the left adjoint of
b◦a, we have
HomD−(ShNis(Sm(k)))(M1⊗LM2,M3) ∼= HomDMe−ff(k,A)(RC∗◦Φ(M1⊗LM2),M3)
∼= HomDMeff(k,A)(RC∗◦Φ(M1)⊗trRC∗◦Φ(M2),M3).
−
6 CRISTIANABERTOLINANDCARLOMAZZA
Since1-motivesarecomplexesofhomotopyinvariantNisnevichsheaveswithtrans-
ferts, the counit arrows RC∗◦Φ(Mi)→Mi (for i=1,2) are isomorphisms and so
we can conclude. (cid:3)
2. Bilinear morphisms between 1-motives
LetK =[A →ui B ](fori=1,2,3)bealength1complexofabeliansheaves(over
i i i
anytoposT)withA indegree1andB indegree0. Abiextension(B,Ψ ,Ψ ,λ)
i i 1 2
of (K ,K ) by K consists of
1 2 3
(1) a biextension of B of (B ,B ) by B ;
1 2 3
(2) a trivialization Ψ (resp. Ψ ) of the biextension (u ,id )∗B of (A ,B )
1 2 1 B2 1 2
by B (resp. of the biextension (id ,u )∗B of (B ,A ) by B ) obtained
3 B1 2 1 2 3
as pull-back of B via (u ,id ):A ×B →B ×B (resp. via (id ,u ):
1 B2 1 2 1 2 B1 2
B × A → B × B ). These two trivializations have to coincide over
1 2 1 2
A ×A ;
1 2
(3) amorphismλ:A ⊗A →A suchthatthecompositeA ⊗A −λ→A −u→3
1 2 3 1 2 3
B is compatible with the restrictionoverA ×A ofthe trivializationsΨ
3 1 2 1
and Ψ .
2
We denote by Biext(K ,K ;K ) the category of biextensions of (K ,K ) by K .
1 2 3 1 2 3
The Baer sum of extensions defines a group law for the objects of the category
Biext(K ,K ;K ), which is therefore a Picard category (see [SGA7] Expos´e VII
1 2 3
2.4, 2.5 and 2.6). Let Biext0(K ,K ;K ) be the group of automorphisms of any
1 2 3
biextensionof (K ,K ) by K , and let Biext1(K ,K ;K ) be the groupof isomor-
1 2 3 1 2 3
phism classes of biextensions of (K ,K ) by K .
1 2 3
According to the main result of [B2], we have the following homological inter-
pretation of the groups Biexti(K ,K ;K ):
1 2 3
(2.1) Biexti(K ,K ;K )∼=Exti(K ⊗LK ,K ) (i=0,1)
1 2 3 1 2 3
Since we can view 1-motives as complexes of commutative S-group schemes of
length 1, all the above definitions apply to 1-motives.
Remark 2.1. Thehomologicalinterpretation(2.1)ofbiextensionscomputedin[B2]
is done for chain complexes K =[A −u→i B ] with A in degree 1 and B in degree
i i i i i
0. In this paper 1-motives are considered as cochain complexes M = [X →ui G ]
i i i
with X in degree 0 and G in degree 1. Therefore after switching from homological
notation to cohomological notation, the homological interpretation of the group
Biext1(M ,M ;M ) can be stated as follow:
1 2 3
Biext1(M ,M ;M )∼=Ext1(M [1]⊗LM [1],M [1])
1 2 3 1 2 3
where the shift functor [i] on a cochain complex C∗ acts as (C∗[i])j =Ci+j.
Proof of Theorem 0.1 By proposition 1.1, we have that
HomDMeff(k,Q)(M1⊗trM2,M3) ∼= HomDMeff(k,A)⊗Q(M1⊗trM2,M3)
gm −
∼= HomD−(ShNis(Sm(k)))(M1⊗LM2,M3)⊗Q.
On the other hand, according to the remark 2.1 we have the following homological
interpretation of the group Biext1(M ,M ;M ):
1 2 3
Biext1(M1,M2;M3)∼=Ext1(M1[1]⊗LM2[1],M3[1])∼=HomD−(ShNis(Sm(k)))(M1⊗LM2,M3)
BIEXTENSIONS AND 1-MOTIVES 7
and so we can conclude.
3. Multilinear morphisms between 1-motives
1-motives are endowed with an increasing filtration, called the weight filtration.
u
Explicitly, the weight filtration W∗ on a 1-motive M =[X →G] is
W (M) = M for each i≥0,
i
W−1(M) = [0−→G],
W−2(M) = [0−→Y(1)],
W (M) = 0 for each j ≤−3.
j
Defining GrW =W /W , we have GrW(M)=[X →0],GrW(M)=[0→A] and
i i i+1 0 −1
GrW(M) = [0 → Y(1)]. Hence locally constant group schemes, abelian varieties
−2
and tori are the pure 1-motives underlying M of weights 0,-1,-2 respectively.
The main property of morphisms of motives is that they have to respect the
weight filtration, i.e. any morphism f : A → B of motives satisfies the following
equality
f(A)∩W (B)=f(W (A)) ∀ i∈Z.
i i
Assume M and M ,...,M to be 1-motivesovera perfect field k andconsider a
1 l
morphism
F :⊗l M →M.
j=1 j
Because morphisms of motives have to respect the weight filtration, the only non
trivial components of the morphism F are the components of the morphism
⊗lj=1Mj.W−3(⊗lj=1Mj)−→M.
Using [B1] Lemma 3.1.3 with i=−3,we canwrite explicitly this last morphismin
the following way
X Xν1 ⊗···⊗Xνl−2 ⊗(Mι1 ⊗Mι2/W−3(Mι1 ⊗Mι2))−→M.
ι1<ι2 and ν1<···<νl−2
ι1,ι2∈/{ν1,...,νl−2}
To have the morphism
Xν1 ⊗···⊗Xνl−2 ⊗(Mι1 ⊗Mι2/W−3(Mι1 ⊗Mι2))−→M
is equivalent to have the morphism
Mι1 ⊗Mι2/W−3(Mι1 ⊗Mι2)−→Xν∨1 ⊗···⊗Xν∨l−2 ⊗M
whereX∨ isthek-groupschemeHom(X ,Z)forn=1,...,l−2. Butasobserved
νn νn
in[B1]§1.1“totensoramotivebyamotiveofweightzero”meanstotakeacertain
number of copies of this motive, and so applying Theorem 0.1 we get
Theorem 3.1. Let M and M ,...,M be 1-motives over a perfect field k. Then,
1 l
HomDMeff(k,Q)(M1⊗trM2⊗tr···⊗trMl,M)∼=
gm
Biext1(M ,M ;X∨ ⊗···⊗X∨ ⊗M)⊗Q
X ι1 ι2 ν1 νl−2
wherethesumistakenoverallthe(l−2)-uplets{ν1,...,νl−i+1}andallthe2-uplets
{ι1,ι2} of {1,··· ,l} such that {ν1,...,νl−2}∩{ι1,ι2} = ∅ and ν1 < ··· < νl−2,
ι <ι .
1 2
8 CRISTIANABERTOLINANDCARLOMAZZA
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NWF-I Mathematik,Universita¨tRegensburg,D-93040Regensburg
E-mail address: [email protected]
Dip. di Matematica,Universita` di Genova,Via Dodecaneso 35,I-16133Genova
E-mail address: [email protected]