A going down theorem for Grothendieck Chow motives

Abstract

Let X be a geometrically split, geometrically irreducible variety over a
field F satisfying Rost nilpotence principle. Consider a field extension E/F
and a finite field K. We provide in this note a motivic tool giving sufficient
conditions for so-called outer motives of direct summands of the Chow motive of
X_E with coefficients in K to be lifted to the base field. This going down
result has been used S. Garibaldi, V. Petrov and N. Semenov to give a complete
classification of the motivic decompositions of projective homogeneous
varieties of inner type E_6 and to answer a conjecture of Rost and Springer.

Full text

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arXiv:1001.0645v1 [math.AG] 5 Jan 2010
A going-down theorem for Chow-Grothendieck motives
Charles De Clercq
January 6, 2010
Abstract
Let (M(X),p) be a direct summand of the motive associated with a geometrically split, ge-
ometrically irreducible variety over a field F satisfying the nilpotence principle. We show that
under some conditions, if (M(XE),pE) is a direct summand of another motive ME over a field
extension E, then (M(X),p) is a direct summand of M over F.
I Introduction
Throughout this note, Λ will always be a finite field. Given a field F, an F-variety will be understood as
a separated scheme of finite type over F. Given such Λ and an F-variety X, we can consider CHi(X;Λ),
the Chow group of i-dimensional cycles on X with coefficients in Λ, defined by CHi(X) ⊗ZΛ. These
groups are the first step in the construction of the category CM(F;Λ) of Chow-Grothendieck motives
with coefficients in Λ. This category is constructed as the pseudo-abelian enveloppe of the category
CR(F;Λ) of correspondences with coefficients in Λ. Our main reference for the construction and the
main properties of these categories is [2].
Definition I.1. Let X be an F-variety. A field extension E/F is a splitting field of X if the E-motive
XEis isomorphic to a finite direct sum of twisted Tate motives.
Following [2], we will write CH(X;Λ) for the colimit of all CH(XE;Λ), where E runs through
all field extensions E/F. We will say that an F-variety X is geometrically split if X splits over the
algebraic closure of F.
For any field extension E/F, we will say that any element of CH(X;Λ) lying in the image of the
restriction morphism CH(XE;Λ) −→ CH(X;Λ) is E-rational.
The purpose of this note is to generalize the following result, proved by N. Karpenko ([3, Proposition
4.6]).
Theorem I.2. Let X be a geometrically irreducible, geometrically split variety satisfying the nilpotence
principle and M be a motive. Assume that there exists a field extension E/F such that
1. the field extension E(X)/F(X) is purely transcendental;
2. the upper indecomposable summand of M(X)E is also lower and is a summand of ME.
Then the upper indecomposable summand of M(X) is a summand of M.
The class of geometrically split, geometrically irreducible F-varieties satisfying the nilpotence prin-
ciple is quite large (see [2], [1], [4]). N. Karpenko uses this result for instance to prove several motivic
decompositions in CM(F;Λ) ([3, lemma 7.1]). Here we generalize theorem I.2 by replacing the motive
M(X) by the direct summand (M(X),p) associated with a projector p ∈ CHdim(X)(X × X;Λ).
I would like to thank Nikita Karpenko, my Ph.D. thesis adviser, for raising this question and
guiding me during this work.
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II The Chow group of a geometrically split F-variety
Let X be a geometrically split F-variety. Let us consider the motivic decomposition
M(XE) ≃
m
?
i=1
Λ(ai)
where m and aiare integers, over a splitting field E of X.
By definition of morphisms in CM(E;Λ), HomCM(E;Λ)(Λ(i),Λ(j)) ≃ δijΛ, where δij stands for
the Kronecker symbol. Therefore there is an integer n called the rank of X such that
CH(XE;Λ) ≃
n
?
i=1
Λ.
Moreover, the isomorphism between M(XE) and the previous direct sum of twisted Tate motives
defines an homogeneous basis (xk)n
k=1of the Λ-module CH(XE;Λ).
Proposition II.1. Let X be a geometrically split F-variety. then the pairing
Ψ :
CH(X;Λ) × CH(X;Λ)
(α,β)
−→
?−→
Λ
deg(α · β)
is bilinear, symetric and non-degenerate.
Proof. c.f. [5, Remark 5.6]
Since the bilinear form Ψ is non-degenerate, it gives rise to an isomorphism of Λ-modules between
CH(X;Λ) and its dual space HomΛ(CH(X;Λ),Λ) given by
f :
CH(X;Λ)
u
−→
?−→
HomΛ(CH(X;Λ),Λ)
Ψ(u,·)
Now consider the dual basis (x′
i)n
is the Kronecker symbol.
i)n
i=0of the basis (xi)n
i). By definition of the antedual basis, we have Ψ(xi,x∗
i=0. We define the antedual basis of (xi)n
i=0by
(x∗
i=0, where x∗
i:= f−1(x′
j) = δij, where δij
Proposition II.2. Let M and N be two motives in CM(F;Λ), with M split.
isomorphism of Λ-modules
Then there is an
CH∗(M;Λ) ⊗ CH∗(N;Λ) −→ CH∗(M ⊗ N;Λ)
Proof. c.f. [2, Proposition 64.3].
Computations become much simpler on split varieties.
Lemma II.3. Let X be a split variety of rank n and two smooth complete varieties Y and Y′. Consider
the homogeneous basis (xk)n
y ∈ CH(Y ;Λ), y′∈ CH(Y′;Λ) and for all 1 ≤ i,j ≤ n,
k=1defined previously and its antedual basis (x∗
k)n
k=1. Then for all cycles
(xi× y) ◦ (y′× x∗
j) = δij(y′× y).
Proof. We set 1 for the identity class, either in CH(Y ;Λ) or in CH(Y′;Λ).
(xi× y) ◦ (y′× x∗
j) = (Y′pY
X)∗
?
?(y′× x∗
j· xi)(y′× y)
(Y′×XpY)∗(y′× x∗
j× 1) · (1 × xi× y)?
j) · (pX×Y
Y′
)∗(xi× y)
?
= (Y′pY
X)∗
= (Y′pY
= deg(x∗
= δij(y′× y)
X)∗
?y′× (x∗
j· xi) × y?
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III Direct summands of geometrically split F-varieties
Let X be an F-variety and p ∈ CHdim(X)(X × X;Λ) be the projector defining the direct summand
(X,p). A direct summand of the motive (X,p) is given by a projector in EndCM(F;Λ)((X,p),(X,p)),
by definition of the category CM(F;Λ).
Lemma III.1. Let X be an F-variety, and p ∈ CHdim(X)(X × X;Λ) the projector defining the direct
summand (X,p). Then every direct summand of (X,p) is a summand (X,q) associated with a projector
q ∈ CHdim(X)(X × X;Λ) satisfying p ◦ q ◦ p = q.
Proof. Indeed we have EndCM(F;Λ)((X,p),(X,p)) = p◦CHdim(X)(X×X;Λ)◦p, thus for any projector
q ∈ EndCM(F;Λ)((X,p),(X,p)), there is a cycle s ∈ CHdim(X)(X × X;Λ) such that q = p ◦ s ◦ p. But
p is itself a projector, therefore p ◦ q ◦ p = p2◦ s ◦ p2= p ◦ s ◦ p = q.
We now study the notion of upper and lower direct summands of a direct summand (M(X),p).
From now on, we consider a geometrically split F-variety X of rank n. We keep the notation (xk)n
for the homogeneous basis of CH(X;Λ) and (x∗
the pairing Ψ.
k=1
k)n
k=1for the antedual basis of (xk)n
k=1associated with
Definition III.2. Let X a geometrically split F-variety and p ∈ CHdim(X)(X × X;Λ) be a non-
zero projector. We denote by p the image of p under the restriction map to a splitting field E of X.
Considering the bases (xi)n
i=1of CH(XE;Λ), we have
i=1and (x∗
i)n
p =
n
?
i=1
n
?
j=1
pij(xi× x∗
j).
We define the lowest codimension of p by
cdmin(p) := min
pij?=0(codim(xi)))
where codim(xi) stands for the codimension in CH(X;Λ).
As X is geometrically split, the summand (X,p) associated with the projector p ∈ CHdim(X)(X ×
X;Λ) splits over a field extension E/F :
(XE,pE) ≃
m
?
i=1
(Λ(ai))mi
(III.1)
where ai, miand m are integers, and we set i < j ⇒ ai< aj.
Definition III.3. Let X be a geometrically split F-variety, and (X,p) be a summand of X. Let also
(X,q) be a direct summand of the motive (X,p). Consider the decomposition III.1 of (X,p) in a
splitting field E of X. The summand (X,q) is :
1. upper in (X,p) if CHa1((XE,qE)) ?= 0.
2. lower in (X,p) if CHam((XE,qE)) ?= 0.
By definition, a direct summand (X,q) is upper in (X,p) if and only if (X(−dim(X)),tq) is lower
in the dual motive (X(−dim(X)),tp). We can now study the link between the lowest codimension of
a summand and the position of its motivic decomposition.
Proposition III.4. Let X be a geometrically split F-variety, and p ∈ CHdim(X)(X×X;Λ) the projec-
tor defining the summand (X,p). Consider the motivic decomposition III.1 given above of the motive
(XE,pE), where E is a splitting field of X. Then
a1= cdmin(p)
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Proof. Let us show first that cdmin(p) ≥ a1. Indeed we consider the motivic decomposition III.1,
thus
CHa1((XE,pE);Λ) ≃
m
?
i=1
HomCM(E;Λ)(Λ(ai),Λ(a1))mi≃ Λm1?= 0.
Consequently CHa1(XE,Λ) ◦ pE?= 0 by definition of Chow groups in CM(E;Λ).
Let x ∈ CHa1(XE;Λ) such that x ◦ pE ?= 0. Since (xi)n
base (xk)k∈K of the Λ-module CHa1(XE;Λ) and write x =?
x ◦ pE?= 0 ⇒
i=1is homogeneous, we can consider the
k∈Kλkxk. Then

??
n
?
k∈K
λkxk
?
◦


n
?
i=1
n
?
j=1
pij(xi× x∗
j)
?= 0
⇒
i=1
?
k∈K
λkpikxi?= 0
⇒ ∃(i0,k0) ∈ {1,..,n}× K, pi0k0?= 0
Then since codim(xi0) = dim(XE)−codim(x∗
k0) = codim(xk0) = a1we have proven that cdmin(p) ≤
a1.
Suppose on the other hand that cdmin(p) = s < a1. Then there are two index i0, j0in {1,..,n} such
that pi0j0?= 0, and codim(xi0) = s. Then xj0◦pE=?n
CHs(XE;Λ) ◦ pE= HomCM(E;Λ)((XE,pE);Λ(s))
m
?
= 0
i=0pij0xi?= 0, therefore CHs(XE;Λ)◦pE?= 0,
as codim(xi0) = codim(xj0). But this contradicts the fact that
=
i=1
HomCM(E;Λ)(Λ(ai);Λ(s))mi
Therefore cdmin(p) = a1.
Proposition III.5. Let X be a geometrically split F-variety, and p the projector defining the summand
(M(X),p). Let also be (M(X),q) a direct summand of (M(X),p). Then
cdmin(p) = cdmin(q) ⇔ (X,q) is upper in (X,p).
Proof. We keep the same notations as in proposition III.4.
As q defines a direct summand of (X,p) we have q = p ◦ q ◦ p by lemma III.1, thus for any k
CHk((XE,qE);Λ) = CHk(XE;Λ) ◦ qE
= CHk(XE;Λ) ◦ (pE◦ qE◦ pE)
But qE◦ pE= (pE◦ qE◦ pE) ◦ pE= pE◦ qE◦ pE= qE, thus
CHk((XE,qE);Λ) =?CHk(XE;Λ) ◦ pE)?◦ qE= CHk((XE,pE);Λ) ◦ qE.(∗)
Therefore for any k such that CHk((XE,pE);Λ) = 0, we have shown that CHk((XE,qE);Λ) = 0.
Now write the decomposition of (XE,qE) :
(XE,qE) ≃
s
?
j=1
(Λ(bj))si
with i < j ⇒ bi< bj. Equality (∗) implies that for all j ∈ {1,..,s}, bj ≥ a1. But proposition III.4
implies that b1= cdmin(q), thus
cdmin(p) = cdmin(q) ⇔ b1= a1
⇔ CHa1((XE,qE);Λ) ?= 0
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IV Proof of the main theorem
Following [3], we recall some results on the category generated by geometrically split F-varieties
satisying the nilpotence principle in CM(F;Λ). We will say that a pseudo-abelian category satisfies
the Krull-Schmidt principle if every object of this category decomposes in a unique way as a direct
sum of indecomposable objects (up to permutation of these indecomposable summands).
Lemma IV.1. ([3, Corollary 3.2]) Let X be a geometrically split F-variety satisfying the nilpotence
principle. Then an appropriate power of any endomorphism of the motive X is a projector.
Proposition IV.2. ([3, Corollary 3.3])The Krull-Schmidt principle holds for the pseudo-abelian Tate
subcategory of CM(F,Λ) generated by the motives of geometrically split F-varieties satisfying the
nilpotence principle.
Remark IV.3. More generally, proposition IV.2 is true when Λ is any finite commutative ring (see
[3, Corollary 3.2]). The Krull-Schmidt principle also holds for the category generated by motives of
quadrics when the ring of coefficients Λ is Z. Nevertheless, the Krull-Schmidt principle does not hold
in the category generated by the motives of projective homogeneous varieties when Λ = Z (see [6]) and
this is one of the reasons why we suppose that Λ is a finite field.
Let X be a geometrically split F-variety. Let p ∈ CHdim(X)(X × X;Λ) be the projector defining
the direct summand (X,p) of X. As seen before, the homogeneous base (xk)n
(x∗
k=1relatively to the pairing Ψ allow us to decompose p in a splitting field E of the motive X, that
is to say pE=?n
Definition IV.4. Let X be a geometrically split F-variety and E a splitting field of the motive X. Let
p be the projector defining the direct summand (X,p). We say that couple (i,j) ∈ {1,..,n}2is contained
in pE if pij?= 0. We will also say that an element xi× x∗
contained in pE if the couple (i,j) is contained in pE.
k=1and its antedual base
k)n
i=1
?n
j=1pij(xi× x∗
j).
jof the base of CHdim(XE)(XE× XE;Λ) is
Notation IV.5. The set of all elements xi× x∗
the projector pE will be called the base of pE, and denoted B(pE).
jof CHdim(XE)(XE× XE;Λ) which are contained in
Definition IV.6. Let X be a geometrically split F-variety and two projectors p and q defining the
two direct summand (X,p) and (X,q). Let E be a splitting field of X. We say that p contains q if
B(qE) ⊂ B(pE). We will say that q intersects p if B(qE) ∩ B(pE) ?= ∅.
Remark IV.7. Given a geometrically split F-variety X and two direct summands (X,p) and (X,q) of
the motive X defined by the two projectors p and q, we will also say that the summand (X,p) contains
the summand (X,q) if the projector p contains the projector q.
Since Λ is a finite field, we have seen that the category generated by motives of geometrically split
F-varieties satisfying the nilpotence principle satisfies the Krull-Schmidt principle (proposition IV.2).
By definition of the category CM(F;Λ), direct summands of the motive of an F-variety correspond
to projectors in EndCM(F;Λ)(X), thus in a splitting field E of the motive X, every F-rational cy-
cle in CHdim(X)(X × X;Λ) is a linear combination of minimal F-rational cycles, corresponding to
indecomposable direct summands of X. We now prove the key-lemma.
Lemma IV.8. Let X be a geometrically split, geometrically irreducible F-variety satisfying the nilpo-
tence principle. Let p be the projector defining the direct summand N := (X,p) of X. Let also Y be
a smooth complete F-variety, and M := (Y,q) the direct summand of Y induced by the projector q.
Assume that there is a field extension E/F and two E-rational correspondences h : XE → ME and
k : ME→ XE such that
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