THE TATE CONJECTURES FOR PRODUCT AND QUOTIENT VARIETIES by RACHID EJOUAMAI A thesis submitted to the Department of Mathematics and Statistics in conformity with the requirements for the degree of Doctor of Philosophy Queen’s University Kingston, Ontario, Canada September 2013 Copyright c Rachid Ejouamai, 2013
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THE TATE CONJECTURES FOR PRODUCT AND QUOTIENT VARIETIES
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THE TATE CONJECTURES FOR PRODUCTAND QUOTIENT VARIETIES
∪i,j : H i(Xet, Z/NZ)×Hj(Xet, Z/NZ)→ H i+j(Xet, Z/NZ),
7
for all i, j. It follows immediately from Milne [19], Proposition V.1.16, that these are
GK-equivariant and are functorial with respect to K-morphisms f : X → Y .
It is a fundamental (but difficult) theorem that the Hk(Xet, Z/NZ) are finite
groups if N is invertible in K; cf. Katz [16] p.22. In addition one has that
Hk(Xet, Z/NZ) = 0 for k > 2 dim(X). (1)
We now fix a prime ` which is invertible in K, and put
Hk(Xet, Z`) = lim←−n
Hk(Xet, Z/`nZ).
It is immediate that Hk(Xet, Z`) is a Z`-module with a continuous GK action. More-
over, it follows from Milne [19], Lemma V.1.11 and the previous finiteness assertion
that Hk(Xet, Z`) is a finitely generated Z`-module. In particular,
Hk` (X) := Hk(Xet, Z`)⊗Q`
is a finite dimensional Q`-vector space. It thus follows from the previous discussion
that Hk` (X) satisfies the following properties:
1. Finiteness and Vanishing: If ` is different from the characteristic of K
then: Hk` (X) is a finite dimensional vector space over Q` and it is zero for
k > 2 dim(X).
2. Galois Structure: The absolute Galois group GK = AutK(K) acts continu-
ously on Hk` (X).
3. Functoriality: If f : X → Y is a K-morphism of schemes then we have a GK-
equivariant morphism on cohomologies: f ∗ : Hk` (Y )→ Hk
` (X).
2.1.2 Cohomology with compact support
As before, let X be a separated scheme of finite type over K. By Nagata [24], there
exists an embedding j : X ↪→ X ′, where X ′/K is proper. Put
Hkc (Xet, Z/NZ) := Hk(X ′
et, j!(Z/NZ)).
8
where j!(Z/NZ) is defined as on p.76 of Milne [19]. It turns out (cf. Milne [19], p.277)
that this group does not depend on the choice of the pair (X ′, j), so in particular
Hkc (Xet, Z/NZ) = Hk(Xet, Z/NZ)
if X/K is proper.
If ` is a prime, then we have the `-adic cohomology group with compact support:
Hk`,c(X) = lim←−
n
Hkc (Xet, Z/`nZ)⊗Q`
These are again finite dimensional Q`-vector spaces with a continuous GK-action and
vanish for k > 2 dim(X); cf. Katz [16], p.22.
If f : X → Y is a morphism of K-schemes, then in general we do not have an
induced map: f ∗ : Hk`,c(Y )→ Hk
`,c(X) unless f is proper; cf. Milne [19], p.229.
We conclude this subsection with the following proposition:
Proposition 2.1.1. (a) Hk`,c(X) coincides with the ordinary `- adic cohomology when
X/K is proper.
(b) If X is irreducible of dimension d and U is an open dense subset then one has
an isomorphism: fU : H2d`,c(U)→ H2d
`,c(X).
Proof. The first statement follows from the previous discussion. For the second state-
ment, we observe that the proof of Lemma VI.11.3. in Milne [19] shows that we have
isomorphisms:
jU,N : H2dc (U, Z/NZ)→ H2d
c (X, Z/NZ)
for each N . Using the compatibility of these isomorphisms as N vary and taking
inverse limit over N = `n we get an isomorphism on the projective limits. Finally
tensoring with Q` gives the second statement.
2.1.3 Tate Twists
Since we will consider Tate twists of cohomology groups later, it is convenient to
introduce them first more generally.
9
For n ≥ 1, let µ(`n) denote the group of `n-th roots of the unity in K. The Galois
group acts on each µ(`n) via the mod `n cyclotomic character: χ`n : GK → (Z/`nZ)×.
Moreover, the groups µ(`n) form a projective system, and one defines the Tate module
T`(µ) to be the inverse limit T`(µ) = lim←−(µ(`n)); this is a free Z` module of rank one,
on which the Galois group GK acts via the `-adic cyclotomic character χ` : GK → Z×` .
Now let V`(µ) = T`(µ)⊗Q`; this is a one dimensional vector space over Q`, on which
the Galois group acts by acting on the first factor. For m ∈ Z we define the m-th
Tate twist of V`(µ) by:
V`(µ)(m) =
{(V`(µ))⊗m, m ≥ 0;
HomQ`((V`(µ))⊗−m, Q`), m < 0.
Now if H is a Q`- vector space with a GK action, we define its m-th Tate twist by:
H(m) = H ⊗ V`(µ)(m). It is clear from the definition that Tate twists satisfy the
identity: H(m + n) = H(m)(n) for all m, n ∈ Z.
2.1.4 Cup product and Cohomology ring
It follows from the discussion of sections 2.1.1 and 2.1.3 that for any integers k, j, m, n
one has Galois equivariant pairings:
∪k,j : Hk` (X)(n)×Hj
` (X)(m) −→ Hk+j` (X)(n + m)
with the property that if f : X → Y is a K-morphism of schemes then we have a
commutative diagram
Hk` (Y )(n)×Hj
` (Y )(m)f∗×f∗ //
∪k,j
��
Hk` (X)(n)×Hj
` (X)(m)
∪k,j
��
Hk+j` (Y )(n + m)
f∗// Hk+j
` (X)(n + m)
As is mentioned in Milne [19] p.268, the cup product makes the direct sum:
H∗` (X) =
⊕k
Hk` (X)([k/2])
10
into a graded ring, called the cohomology ring. It is clear from the property of the
cup product above that a morphism of schemes f : X → Y induces a homomorphism
of graded rings:
f ∗ : H∗` (Y )→ H∗
` (X).
2.1.5 The fundamental class and the trace map
In this subsection X will be a smooth irreducible variety over an algebraically closed
field K except at the end.
Relative Cohomology
Let X be an algebraic variety and Z a closed subvariety of X, one can define coho-
mology groups (functors) HkZ(X,−) called cohomologies with support in Z; cf. Milne
[19], p.91. It can be shown that for Z a complete subvariety there is a canonical
morphism (natural transformation):
fZ : HkZ(X,−)→ Hk
c (X,−),
where, as before, Hkc (X,−) is the cohomology with compact support; cf. Milne
[19], Proposition I.1.29. In particular one has morphisms fZ,N : HkZ(X, Z/NZ) →
Hkc (X, Z/NZ) for each integer N which are compatible as N varies. Thus, letting
N = `n for a prime ` and taking inverse limits, we get Z`-linear homomorphism:
fZ,` : HkZ(X, Z`)→ Hk
c (X, Z`).
Now tensoring with Q` gives a morphism:
fZ,` : HkZ(X, Q`)→ Hk
c (X, Q`).
Smooth pairs
As is defined in Milne [19] Chapter VI, 5. A smooth K-pair (Z,X) of codimension
n is a closed immersion of smooth K-schemes i : Z ↪→ X with Z has codimension n.
A morphism of smooth K-pairs f : (Z ′, X ′) → (Z,X) is a morphism of K-schemes
f : X ′ → X such that: Z ′ = Z ×X X ′.
11
Fundamental class
As is explained in Milne [19], to any smooth K-pair (Z,X) of pure codimension
n one can associate a cohomology class: SZ/X,N ∈ H2nZ (X, Z/NZ)(n), N prime to
the characteristic of K, which is uniquely determined by the four properties listed
in Theorem VI.6.1. of Milne [19]. By varying N = `r, we thus obtain an element
SZ/X,` ∈ H2nZ (X, Q`)(n).
Proposition 2.1.2. If f : (Z ′, X ′)→ (Z,X) is a morphism of smooth K-pairs, then
under the map : f ∗ : H2nZ (X, Z/NZ)(n)→ H2n
Z′ (X ′, Z/NZ)(n) we have f ∗(SZ/X,N) =
SZ′/X′,N for all integers N prime to the characteristic of K, and hence f ∗(SZ/X,`) =
SZ′/X′,`, if ` 6= char(K).
Proof. For the first statement see Milne [19] part c) of Theorem VI.6.1. The second
statement follows from the first by taking inverse limits.
Let X be smooth of dimension d, and let P ∈ X be a closed point, then (P, X) is a
smooth pair and so we have an associated cohomology class SP/X ∈ H2dP (X, Q`)(d).
Recall that we have a canonical morphism:
fP,` : H2dP (X, Q`)(d)→ H2d
c (X, Q`)(d)
We let clX(P ) = fP,`(SP/X). It turns out that clX(P ) does not depend on the partic-
ular closed point P ∈ X; cf. Milne [19] Theorem VI.11.1.(a)
Now if X is an arbitrary variety over K, let U = Xns be the smooth part of X.
Since U is a dense open subset of X, we can use the isomorphism fU of Proposition
2.1.1 and we define clX(P ) = fU(clU(P )).
Trace Map
Theorem 2.1.3. If X/K is a variety, then there exists a unique isomorphism
tX : H2dc (X, Q`)(d)→ Q`
such that tX(clX(P )) = 1, for all P ∈ Xns.
12
Proof. For X smooth this follows from Milne [19] Theorem VI.11.1.(a) by taking
inverse limits.
Now for X non smooth, let U = Xns be the smooth part of X. By applying the
smooth case to U there is a unique map tU : H2d`,c(U)(d)→ Q` such that tU(clU(P )) =
1. Now recall from Proposition 2.1.1 that we have a canonical isomorphism: fU :
H2d`,c(U)(d)→ H2d
`,c(X)(d). We define the trace map for X by
tX = tU ◦ fU−1 : H2d
`,c(X)→ Q`.
which is an isomorphism, moreover, for P ∈ U = Xns we have tX(clX(P )) = (tU ◦fU
−1)(clX(P )) = (tU◦fU−1)(fU(clU(P ))) = tU(clU(P )) = 1. This proves the existence,
the uniqueness is trivial because tX is an isomorphism.
2.1.6 Poincare Duality Theorem
Let ϕX denote the composition of the cup product pairing
∪X : Hk` (X)(m)×H2d−k
` (X)(d−m) −→ H2d` (X)(d)
with the trace map:
tX : H2d` (X)(d)→ Q`.
The following is the Poincare duality theorem:
Theorem 2.1.4. Suppose that ` is different from the characteristic of K, and that
X/K is a smooth proper and geometrically irreducible variety of dimension d. Then
the pairing
ϕX = tX ◦ ∪X : Hk` (X)(m)×H2d−k
` (X)(d−m) −→ Q`
defines a perfect duality.
Proof. This follows from Corollary VI.11.2 of Milne [19], again by taking inverse limits
and noticing that cohomology with compact support coincides with the ordinary
cohomology because X is proper.
13
2.2 Algebraic Cycles and the Cycle map
2.2.1 Algebraic Cycles and Numerical Equivalence
Let X be a smooth projective variety of dimension d over an algebraically closed field
K. A prime cycle of codimension k on X is an irreducible subvariety V of X of
codimension k. A cycle C of codimension k on X is an element of the free abelian
group generated by the prime cycles that is, a formal linear combination of the form:
C =∑
nj · [Vj], where the nj’s are integers all zero except finitely many, and the Vj
’s are prime cycles of codimension k. We denote by Zk(X) the group of cycles of
codimension k.
For any i ≤ d, one has an intersection pairing
Zi(X)× Zd−i(X)→ Z
which associates to a pair (V, W ) an integer denoted by V.W ; cf Fulton [7] Ch. 8.3.
We say that a cycle V ∈ Zi(X) is numerically equivalent to zero if for any ir-
reducible element D ∈ Zd−i(X) we have V.D = 0. Cycles which are numerically
equivalent to zero form a subgroup of Zi(X) denoted by Zi0(X). One denotes the
quotient group Zi(X)/Zi0(X) by Zi(X); it is called the group of cycles up to numer-
ical equivalence.
2.2.2 Flat pullback of cycles
Let f : X → Y be a flat morphism of smooth varieties. For any irreducible subvariety
V of Y of codimension k set f ∗(V ) = [f−1(V )] =∑
nj · [Wj] where the Wj’s are the
irreducible components of f−1(V ) and the nj’s are their multiplicities (see Fulton [7],
page 15 for the definition of nj). We extend f ∗ by linearity to get a morphism of
groups of cycles
f ∗ : Zk(Y )→ Zk(X).
This map is called the flat pullback of cycles, see Fulton, [7] page 18.
14
2.2.3 The Cycle map
For any 0 ≤ k ≤ d, one has a cycle map
cyc : Zk(X)→ H2k` (X)(k)
This map is defined by means of the Poincare duality theorem as follows. If i : Z ↪→X is an irreducible closed subvariety of codimension k, then we have a restriction
morphism on cohomologies
i∗ : H2(d−k)` (X)→ H
2(d−k)` (Z).
Twisting by (d− k), this yields the following morphism
i∗(d−k) : H2(d−k)` (X)(d− k)→ H
2(d−k)` (Z)(d− k).
Composing i∗(d−k) with the trace map
tZ : H2(d−k)` (Z)(d− k)→ Q`,
we get the map
tZ ◦ i∗(d−k) : H2(d−k)` (X)(d− k)→ Q`.
By Poincare duality there exists a unique element ηZ ∈ H2k` (X)(k) such that:
ϕX(ηZ , ·) = tZ ◦ i∗(d−k)
where ϕX is the Poincare pairing. We define:
cyc(Z) = ηZ
Now that the image of a prime cycle is defined, one extends cyc by linearity to any
cycle.
15
2.3 L-functions
2.3.1 The finite field case
Here we assume that our variety X is defined over K = Fp. Let ` be a prime different
from p and let H i`(X) be the i-th `-adic cohomology group. The Galois group GFp acts
continuously on H i`(X). Let Fp ∈ GFp denote the Frobenius element, i.e, Fpx = xp
for all x ∈ Fp. We define the following polynomial:
FX,i,`(t) := det((1− tF−1p )|H i
`(X)).
The following result is due to Deligne; cf. [1] Theorem I.6.
Theorem 2.3.1. If X is smooth and projective, then we have the following:
1. The polynomial FX,i,`(t) has coefficients in Z and is independent of `.
2. If λ is a root of FX,i,`(t), then |λ| = pi/2.
Thus, one can attach to H i`(X) an L-series defined as follows
Li(X, s) = FX,i,`(t)(p−s).
2.3.2 Zeta function
Let X be any variety over Fp. The zeta function of X is defined by
Z(X, t) = exp
(∑n≥1
νn(X)
n· tn)
,
where νn(X) is the number of points with coordinates in X(Fpn). From its definition,
the object Z(X, t) belongs to Q((t)). Now the following theorem of Grothendieck
shows that it belongs to Q(t), cf. Milne [19], Theorem VI.13.1.
Theorem 2.3.2. Assume that X is proper of dimension d over Fp, and let ` 6= p.
With the same notations as in the previous subsection we have the identity
Z(X, t) =2d∏i=0
FX,i,`(t)(−1)(i+1)
.
16
2.3.3 The global case
Here we assume that our variety X is smooth and projective over Q. Let H i`(X) be
the i-th `-adic cohomology group. The Galois group GQ acts continuously on H i`(X).
Let p be a rational prime different from `, and fix a prime ideal P in the ring Z of all
algebraic integers, such that p ∈ P. Let DP and IP denote the decomposition and
inertia subgroups of GQ with respect to P. We say that H i`(X) is unramified at p if
IP acts trivially on H i`(X). For such a prime we define the following polynomial
Fp,i,`(X)(t) = det(1− tF−1p |H i
`(X))
This polynomial has coefficients in Q` and depend only on the conjugacy class of Fp.
Let Z(p) denotes the localization of Z at the prime ideal (p). We say that X has
good reduction at p if there exists a smooth and projective scheme X over Z(p) with
generic fiber X0 ' X. The special fiber Xp of X is a reduction of X modulo p; it
is smooth and projective over Fp. One considers as in the previous subsection the
polynomial
FXp,i,`(t) := det(1− tF−1p |H i
`(Xp)).
Using Theorem 2.3.1 this polynomial has coefficients in Z and is independent of `.
We have the following
Proposition 2.3.3. Assume that X is smooth and projective and that X has good
reduction Xp at p. If P is a prime of Z over p and ` 6= p, then there is an isomorphism
of DP-modules
H i`(X) ' H i
`(Xp),
where DP acts on H i`(Xp) via DP/IP ' GFp. In particular, H i
`(X) is unramified at p
and we have the equality
Fp,i,`(X)(t) = FXp,i,`(t),
so Fp,i,`(X)(t) has integral coefficients and is independent of `.
Proof. This is explained in Tate’s article [33], page 108.
17
Remark 2.3.4. It turns out that for X smooth and projective there are only fi-
nitely many primes p such that X has bad reduction at p; cf. Hindry-Silverman [9],
Proposition A.9.1.6.
2.3.4 Compatible systems of `-adic modules
Let M = (M`)` be a family of GQ-modules, where ` run over all rational primes,
and each M` is a Q`-vector space. We say that (M`)` is a strictly compatible system
of weight w, if there exists a finite set of primes S and an integer w such that the
following conditions are satisfied
1. For any ` and any p /∈ S ∪ {`} the module M` is unramified at p.
2. For any ` and any p /∈ S ∪ {`} the polynomial Fp,`(t) := det((1 − tF−1p )|M`)
has coefficients in Q and is independent of `. We can thus drop the ` from the
notation and denote it by Fp(t).
3. For any p /∈ S the roots of Fp(t) in C have absolute value pw2 .
It is immediate that if the conditions 1,2 and 3 are satisfied by two finite sets of
primes S1 and S2, then they are also satisfied by their intersection S1∩S2. We denote
by SM the smallest set of primes such that the three conditions above hold.
One can associate an L-series to any such system M as follows. For any p /∈ SM
define
L(M, s, p) = Fp(p−s).
Now define
L(M, s) =∏
p/∈SM
L(M, s, p)−1.
Proposition 2.3.5. If M is a strictly compatible system of weight w, then the L-
series L(M, s) converges absolutely for <(s) > 1 + w2.
Proof. For any p /∈ SM we can write
L(M, s, p)−1 =∏
1≤i≤m
1
1− λp,i · p−s,
18
where m = dimQ`(M`), ` is any prime different from p and λp,i are the roots of Fp,`(t)
in C. Taking absolute value we have
|L(M, s, p)−1| =∏
1≤i≤m
1
|1− λp,i · p−s|
Now using the facts that |1− |λp,i||p−s|| ≤ |1− λp,i · p−s| and |λp,i| = pw2 we get
|L(M, s, p)−1| ≤ 1
|1− pw2−<(s)|m
,
and so
|L(M, s)| ≤∏
p/∈SM
1
|1− pw2−<(s)|m
.
Now if <(s) > w2
+ 1 then |1− pw2−<(s)| = 1− p
w2−<(s) > 0, hence
|L(M, s)| ≤∏
p/∈SM
1
|1− pw2−<(s)|m
=∏
p/∈SM
1
(1− pw2−<(s))m
=∏
p∈SM
(1− pw2−<(s))m · ζ(<(s)− w
2)m,
which converges for <(s) > w2
+ 1.
For any finite set of primes S containing SM let LS(M, s) =∏
2. If w 6= −1, then L(M, s) has a pole of order n at s = w2
+ 1 if and only if
LS(M, s) has a pole of order n at s = w2
+ 1.
Proof. We have
L(M, s) = LS(M, s) ·∏
p/∈S\SM
L(M, s, p)−1
hence statement 1 follows. To prove statement 2), we need only to show that the
factor∏
p/∈S\SML(M, s, p)−1 has no poles or zeros at s = w
2+ 1. In fact we will prove
that for any p 6∈ SM the function s 7→ L(M, s, p) has no poles or zeros at s = w2
+ 1.
We have by definition L(M, s, p) = Fp(p−s), since the functions s 7→ p−s is analytic
19
on the whole complex plane then so is s 7→ L(M, s, p) (as composition of two analytic
functions) thus it has no poles at all. Now if s 7→ L(M, s, p) has a zero at s = w2
+ 1
then p−w2−1 will be a root for the polynomial Fp(t), hence by the weight condition we
would have p−w2−1 = p
w2 , which is false unless w = −1.
Let M and N be strictly compatible systems with exceptional sets SM and SN
and associated L-series L(M, s) and L(N, s). We say that L(M, s) and L(N, s) are
equal up to finitely many Euler factors and we write L(M, s) ∼ L(N, s), if there
exists a finite set of primes S containing SM and SN such that∏
p/∈S L(M, s, p)−1 =∏p/∈S L(N, s, p)−1.
In the following proposition we summarize some standard properties of strict com-
patible systems.
Proposition 2.3.7. 1. The system {Q`(r)}` with r ∈ Z is strictly compatible of
weight −2r. Moreover the L-series associated with this system is ζ(s+ r) where
ζ is the Riemann zeta function.
2. If M = {M`}` and N = {N`}` are two strictly compatible systems of the same
weight w, then the direct sum M ⊕N = {M` ⊕N`}` is also strictly compatible
of weight w. Moreover L(M ⊕N, s) ∼ L(M, s)L(N, s).
3. If M = {M`}` and N = {N`}` are two strictly compatible systems of weights
wM and wN then the tensor product M ⊗N = {M`⊗N`}` is strictly compatible
of weight wM +wN . In particular for any r ∈ Z the Tate twist M(r) = {M`(r)}`is a strictly compatible system.
4. If M = {M`}` is a strict compatible system of weight w then the dual M∨ =
{M`∨}` is strictly compatible of weight −w.
The main result of this subsection is the following result essentially due to Deligne.
Theorem 2.3.8. Let X be a smooth projective geometrically irreducible variety of
dimension d over Q. For any integer 0 ≤ k ≤ 2d the system of `-adic cohomology
groups {Hk` (X)}` is a strictly compatible system of weight k.
20
Proof. This follows from Theorem 2.3.1, Proposition 2.3.3 and Remark 2.3.4.
Using the previous theorem, one can associate an L-series to the system {Hk` (X)}`.
This L-series we denote by
Lk(X, s) := L(Hk` (X), s).
2.4 Tate’s conjectures in the smooth projective case
Let X be a smooth projective variety over a finitely generated field K and let
cyci : Zi(X)→ H2i` (X)(i)
be the cycle map defined earlier, and let Zi0,`(X) be the kernel of cyci. In his funda-
mental article, Tate [33] made the following preliminary conjecture
Conjecture 2.4.1. 1. The subgroup Zi0,`(X) does not depend on `,
2. Zi0,`(X) consists exactly of the cycles which are numerically equivalent to zero,
3. The quotient group Zi(X) = Zi(X)/Zi0,`(X) is finitely generated and
cyci⊗1 : Zi(X)⊗Z Q` → H2i` (X)(i)
is injective.
Remark 2.4.2. Statements 1 and 3 are known to be true in characteristic zero. For
i = 1, all of the statements 1, 2 and 3 are true in all characteristics. See Tate’s article
[33] pages 97, 98 for a sketch of the proof.
With the previous conjecture assumed to be true, Tate made some important
conjectures relating rational algebraic cycles to cohomology and L-series. We restrict
ourselves to K = Q and we start with the conjecture relating rational cycles to
cohomology. First, let Zi(X) be the subgroup of Zi(X) generated by the classes of
algebraic cycles which are defined over Q, that is,
Zi(X) = Zi(X)GQ = {ξ ∈ Zi(X) : σξ = ξ for all σ ∈ GQ}
21
Conjecture 2.4.3. The injective map in Conjecture 2.4.1 part 3) induces an isomor-
phism
cyci⊗1 : Zi(X)⊗Z Q` → H2i` (X)(i)GQ ,
or equivalently the following identity holds
dimQ`(Zi(X)⊗Z Q`) = dimQ`
H2i` (X)(i)GQ .
The other interesting conjecture relating Zi(X) to the L-series L2i(X, s) is the
following
Conjecture 2.4.4. The L-series L2i(X, s) has a pole at s = 1 + i of order equals to
− dimQ`(Zi(X)⊗Z Q`).
22
Chapter 3
Quotient Varieties
Let X be a smooth projective variety over a field K, G a finite group acting faithfully
on X by automorphisms. Then there exists a variety Y (cf. [8] Expose V, or [22]
Corollary 1.10) and a finite surjective map: π : X → Y of degree equals the order of
G, such that for any σ ∈ G we have: π ◦ σ = π. Moreover the variety Y is universal
for this property. In other words, if Z is an algebraic variety over K and if h : X → Z
is a map such that the diagram:
Xσ //
h @@@
@@@@
X
h~~~~~~
~~~
Z
is commutative for all σ ∈ G, then there is a unique map f : Y → Z such that
h = f ◦π. The pair (Y, π) is uniquely determined by this universal property up to an
isomorphism. It is called the quotient of X by G, and is denoted by X/G.
3.1 The `-adic cohomology of a quotient variety
It is a natural question to ask how the etale cohomology groups of a quotient variety
Y = X/G are related to those of X. Let Y = X/G be a quotient variety. Since the
group G acts on X, then it acts on Hk` (X) by functoriality. Now since π ◦ σ = π for
23
all σ in G, the image of the map: π∗ : Hk` (Y ) → Hk
` (X) is contained in Hk` (X)G. A
more precise result is the following:
Theorem 3.1.1 (Roth [29]). With the same notations as in the previous discussion
the map
π∗ : Hk` (Y )→ Hk
` (X)G
is an isomorphism.
3.2 Poincare duality for quotient varieties
The aim of this subsection is to formulate and prove a Poincare duality theorem for
quotient varieties over a field of characteristic zero. Let Y = X/G be a quotient
variety with X smooth and projective over a field of characteristic zero, and let
π : X → Y be the corresponding projection map. Moreover, let
ϕY : Hk` (Y )(m)×H2d−k
` (Y )(d−m) −→ Q`
be the pairing obtained by composing the cup product
∪Y : Hk` (Y )(m)×H2d−k
` (Y )(d−m) −→ H2d` (Y )(d)
and the trace map
tY : H2d` (Y )(d)→ Q`.
We have the following
Theorem 3.2.1. The Poincare pairing
ϕY : Hk` (Y )(m)×H2d−k
` (Y )(d−m)→ Q`
is a perfect duality. Moreover, we have a commutative diagram
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
ϕY
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
ϕX
��Q` ×|G|
// Q`
24
where ϕX is the Poincare pairing for X.
We need first to prove the following result.
Lemma 3.2.2. If π : X → Y is a finite morphism of degree n between two vari-
eties of dimension d over a field of characteristic zero, then the following diagram is
commutative
H2d`,c(Y )(d) π∗ //
tY
��
H2d`,c(X)(d)
tX
��Q` ×n
// Q`
Proof. First, assume that X and Y are smooth and pick any unramified point p in
Y , so |π−1(p)| = n. Set π−1(p) = {q1, q2, ..., qn}. Then we have a morphism of smooth
pairs; π : (X, {q1, ...qn})→ (Y, p). Hence by Proposition 2.1.2 we have:
π∗(Sp/Y ) = S{q1,...,qn}/X =∑
1≤i≤n
Sqi/X
Hence it follows that
π∗(clY (p)) =∑
1≤i≤n
clX(qi)
Now by Theorem 2.1.3 clX(qi) does not depend on the particular points qi and depends
only on X, hence we get the equation:
π∗(clY (p)) = n · π∗(clX(q))
where q is any point in the set π−1(p). Next, applying the trace map tX to this
equation yields
tX(π∗(clY (p))) = n · tX(clX(q)) = n · 1 = n.
and thus
tX ◦ π∗ = n · tY .
This proves the result when X and Y are smooth.
Now, for an arbitrary X and Y let U = Y ns denote the subset of Y obtained by
removing all of the singular points in Y , and let V = (π−1(U) ∩Xns) ⊂ X. Both U
25
and V are open dense subsets and we have a finite morphism of degree n, π : V → U .
Thus the first case applies to this situation and we have a commutative diagram:
H2d`,c(U)(d) π∗ //
tU
��
H2d`,c(V )(d)
tV
��Q` ×n
// Q`
Now consider the isomorphisms fU : H2d`,c(U)(d)→ H2d
`,c(Y )(d) and fV : H2d`,c(V )(d)→
H2d`,c(X)(d). Since we also have tY = tU ◦fU
−1 and tX = tV ◦fV−1, therefore composing
the previous diagram with the maps fU−1 and fV
−1 we get tX ◦ π∗ = n · tY which is
the desired result.
Now we give a proof of the previous theorem.
Proof. of Theorem 3.2.1
Recall the commutative diagram from Section 2.1.4
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
∪Y
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
∪X
��
H2d` (Y )(d)
π∗// H2d
` (X)(d)
Composing this diagram with that of Lemma 3.2.2 gives the following diagram:
Hk` (Y )(m)×H2d−k
` (Y )(d−m)π∗×π∗ //
ϕY
��
Hk` (X)(m)×H2d−k
` (X)(d−m)
ϕX
��Q` ×|G|
// Q`
This proves the second assertion of the theorem.
It remains to show that ϕY is non-degenerate. Since by Theorem 3.1.1 π∗ × π∗ is
injective with image Hk` (X)(m)
G × H2d−k` (X)(d−m)
G, it suffices to prove that the
restriction of ϕX (or equivalently the restriction of ∪X) to the subspace Hk` (X)(m)G×
H2d−k` (X)(d−m)G is non degenerate. Since ∪X is already non degenerate in the whole
space, it suffices to show that if an element α in Hk` (X)(m)G is such that α∪X β = 0
26
for all elements β in H2d−k` (X)(d − m)G, then α ∪X b = 0 for all elements b in
H2d−k` (X)(d−m) and that the same is true by interchanging the roles of α and β. To
do this, we first notice that from the diagram in Section 2.1.4 we have the following
identity:
g∗.a ∪X g∗.b = g∗.(a ∪X b)
which holds for all pairs (a, b) and all group elements g.
Now consider the element εG of Q`[G] given by :
εG =1
|G|∑g∈G
g
It is a general fact that for any Q`[G]-module V we have: V G = εGV .
Now let b in H2d−k` (X)(d − m). Then εG.b ∈ H2d−k
` (X)(d − m)G, hence by our
assumption about α we have
α ∪X εGb = 0 (2)
on the other hand, by using the linearity we have
α ∪X εGb =1
|G|∑g∈G
α ∪X g∗.b
Now using the facts that g∗.a ∪X g∗.b = g∗.(a ∪X b) and g∗α = α for all g we get
α ∪X εGb = εG(α ∪X b) (3)
hence from (2) and (3) we have
εG(α ∪X b) = 0 (4)
Now εG acts as 1 in the space H2d` (X)(d). Indeed we know that H2d
` (X)(d) is a one di-
mensional space, and we know also by Roth’s theorem that H2d` (X)(d)G ' H2d
` (Y )(d).
Thus, H2d` (X)(d)G is also a one dimensional space, and hence H2d
` (X)(d)G = H2d` (X)(d),
which means that G acts trivially on H2d` (X)(d) therefore εG acts as 1 in H2d
` (X)(d),
this together with equation (4) imply that α ∪X b = 0 as desired. Similarly one can
prove the same thing by interchanging the roles of Hk` (X)(m) and H2d−k
` (X)(d−m).
27
3.3 Cycles on a quotient variety and the cycle map
3.3.1 Cycles on a quotient variety
Let Y = X/G with X smooth and projective over a characteristic zero field K.
How does the group of i- cycles on Y = X/G relate to that on X. First take base
extensions X and Y over the algebraic closure of K. Then the group G acts on
X, and Y is the resulting quotient variety. Moreover the group G naturally acts on
the group of i-cycles Zi(X). Unfortunately one does not have a flat pullback π∗ in
this situation because the map π : X → Y is not flat in general. However there is
an alternative definition of π∗ given by Fulton [7] Example 1.7.6 page 20. Fulton’s
definition specialized to the case of char(K) = 0 is as follows:
For any cycle C in Zi(X) define the inertia group at C by
IC = {σ ∈ G : σ|C = idC}.
Now let [V ] be an irreducible cycle in Zi(Y ) and for any irreducible component C of
π−1(V ), let eC = |IC |. We define:
π∗([V ]) =∑
eC [C]
the sum is over all irreducible components C of π−1(V ). Note that since the field has
characteristic zero then the inseparability degree in Fulton’s definition is 1.
With this definition of π∗ one has an injective map
π∗ : Zi(Y )→ Zi(X)G
which becomes an isomorphism after tensoring with Q.
π∗ ⊗ 1 : Zi(Y )⊗Q→ Zi(X)G ⊗Q
Since the map π : X → Y is proper, one also have a push forward map
π∗ : Zi(X)→ Zi(Y ).
28
π∗ is defined as follows: π∗(V ) = deg(V/W )W where W = π(V ) and deg(V/W ) is
the degree of the finite map π|V : V → W . We have
π∗ ◦ π∗ = ×|G|; (5)
cf. Fulton [7], Example 1.7.6 page 20.
3.3.2 Intersection product on a quotient variety
In general there is no good intersection theory for singular varieties, but in the case of
a quotient variety Y = X/G one can define an intersection map in Y inherited from
that of X provided that X is smooth. If a ∈ Zi(Y ) and b ∈ Zd−i(Y ) one defines a · bby the following formula (cf. Fulton [7] Example 8.3.12 pages 142-143)
a · b =1
|G|(π∗(a) · π∗(b))
In contrast with the smooth case, a · b is not necessarily an integer but rather an
element of 1|G|Z. Thus the formula above defines an intersection mapping
Zi(Y )⊗Q× Zd−i(Y )⊗Q→ Q
Moreover, this mapping is compatible with that of X, namely, we have a projection
formula: for a ∈ Zi(Y ) and β ∈ Zi(X) we have
π∗(a) · β = a · π∗(β);
cf. Fulton [7] Example 8.3.12 pages 142-143. Despite the apparent dependence on X,
it is asserted in Fulton [7] Examples 16.1.3 and 17.4.10 that the intersection map on
Y is independent of X.
We say that a cycle a ∈ Zi(Y ) is numerically equivalent to zero, if for any cycle
b ∈ Zd−i(Y ) we have a·b = 0. We denote the subgroup of cycles numerically equivalent
to zero by Zi0(Y ). We have the following
Proposition 3.3.1. With the notations as in the previous discussion the following
holds
(π∗)−1(Zi0(X)) = Zi
0(Y )
29
Proof. Let a ∈ Zi0(Y ). We want to show that a ∈ (π∗)−1(Zi
0(X)) or equivalently
that π∗(a) ∈ Zi0(X). Let β ∈ Zd−i(X). Using the projection formula we have
π∗(a) · β = a · π∗(β). But a ∈ Zi0(Y ), hence a · π∗(β) = 0, thus π∗(a) · β = 0, which
means π∗(a) ∈ Zi0(X).
Conversely, if a ∈ (π∗)−1(Zi0(X)) then π∗(a) ∈ Zi
0(X). We want to show that
a ∈ Zi0(Y ). Let b ∈ Zd−i(Y ). Applying the projection formula again we have
π∗(a) · π∗(b) = a · π∗(π∗(b))
Now since π∗(a) ∈ Zi0(X) we have π∗(a) · π∗(b) = 0 thus,
a · π∗(π∗(b)) = 0.
Now using the fact that π∗(π∗(b)) = |G| · b the previous equation becomes
|G|(a · b) = 0
hence a · b = 0, which means a ∈ Zi0(Y ).
3.3.3 Cycle map on a quotient variety
Using Theorem 3.2.1(Poincare duality) one can define a cycle map for quotient va-
rieties in the same way as smooth varieties, see Subsection 2.2.3. The main result
about this cycle map is the following:
Theorem 3.3.2. If Y = X/G is a quotient variety of dimension d, then the following
diagram commutes
Zi(Y )cycY //
π∗
��
H2i` (Y )(i)
π∗
��
Zi(X) cycX
// H2i` (X)(i)
30
Proof. Let Z ⊂ Y be an irreducible i-cocycle and let W ⊂ π−1(Z) be an irreducible
component of π−1(Z), then we have the following diagram on varieties
W� � iW //
πW
��
X
π
��Z
� �
i// Y
where πW is the restriction of π to W and i and iW are the closed immersions. Hence
by functoriality we get the following commutative diagram on cohomology groups:
H2(d−i)` (Y )(d− i)
i∗ //
π∗
��
H2(d−i)` (Z)(d− i)
π∗W��
H2(d−i)` (X)(d− i)
i∗W
// H2(d−i)` (W )(d− i)
Composing this diagram with the trace maps:
tW : H2(d−i)` (W )(d− i)→ Q`
and
tZ : H2(d−i)` (Z)(d− i)→ Q`
and using Lemma 3.2.2 we get the following commutative diagram:
H2(d−i)` (Y )(d− i)
π∗ //
tZ◦i∗
��
H2(d−i)` (X)(d− i)
tW ◦i∗W��
Q` ×λW
// Q`
i.e, λW ·(tZ◦i∗) = (tW ◦i∗W )◦π∗, where λW is the degree of the finite map: πW : W → Z.
Now by definition of the cycle map we have
tZ ◦ i∗ = ϕY (cycY (Z), ·)
and
tW ◦ i∗W = ϕX(cycX(W ), ·)
31
where ϕX and ϕY are the Poincare pairings, and so we have
λW · ϕY (cycY (Z), ·) = ϕX(cycX(W ), ·) ◦ π∗ (6)
Multiplying (6) by the ramification index eW and summing over all irreducible com-
ponents W of π−1(Z) we get
(∑W |Z
eW · λW ) · ϕY (cycY (Z), ·) =∑W |Z
eW · ϕX(cycX(W ), ·) ◦ π∗ (7)
Using linearity, the right hand side of (7) becomes ϕX(cycX(∑
W |Z eW ·W ), π∗(·)),and using the identity
∑W |Z eW · λW = |G| which follows from equation (5) we get