# Functorial splitting of $\displaystyle l$-adic cohomology
of an extension of group varieties

Victor de Vries

###### Abstract.

In this document we consider an exact sequence of group varieties
$\displaystyle e\to N\to G\to Q\to$ $\displaystyle e$ over an algebraically
closed field. We show that for $\displaystyle l\neq\mathrm{char}(k)$ a prime
there exists an isomorphism of graded $\displaystyle\mathbb{Q}_{l}$-algebras
$\displaystyle\mathrm{H}_{\mathrm{\acute{e}t}}^{*}(G,\mathbb{Q}_{l})\cong\mathrm{H}_{\mathrm{\acute{e}t}}^{*}(N,\mathbb{Q}_{l})\otimes_{\mathbb{Q}_{l}}\mathrm{H}_{\mathrm{\acute{e}t}}^{*}(Q,\mathbb{Q}_{l})$
that is compatible with pullback homomorphisms $\displaystyle f^{*}$ of
endomorphisms $\displaystyle f:G\to G$ that stabilize $\displaystyle N$.

##  1\. Introduction

Let $\displaystyle G$ be an algebraic group over an algebraically closed field
$\displaystyle k$. We will call it a group variety if it is smooth and
connected. For $\displaystyle l\neq\mathrm{char}(k)$ a prime the
$\displaystyle l$-adic cohomology ring of a variety $\displaystyle X$ is
denoted
$\displaystyle\mathrm{H}^{*}(X):=\mathrm{H}_{\mathrm{\acute{e}t}}^{*}(X,\mathbb{Q}_{l})=\bigoplus_{r\geq
0}\mathrm{H}_{\mathrm{\acute{e}t}}^{r}(X,\mathbb{Q}_{l})$ and we denote the
tensor product over $\displaystyle\mathbb{Q}_{l}$ by $\displaystyle\otimes$.
For an exact sequence of group varieties, we study the property that we call
$\displaystyle(*)$ related to the $\displaystyle l$-adic cohomology of the
involved group varieties:

######  Definition.

An exact sequence of group varieties $\displaystyle e\to
N\overset{\iota}{\to}G\overset{\pi}{\to}Q\to e$ has $\displaystyle(*)$ if
there is an isomorphism of graded $\displaystyle\mathbb{Q}_{l}$-algebras
$\displaystyle
s\otimes\pi^{*}:\mathrm{H}^{*}(N)\otimes\mathrm{H}^{*}(Q)\to\mathrm{H}^{*}(G)$
where $\displaystyle s$ is a section to $\displaystyle\iota^{*}$.

Our main theorem asserts that any exact sequence of group varieties has this
property.

######  Theorem.

An exact sequence of group varieties $\displaystyle e\to N\to G\to Q\to e$ has
$\displaystyle(*)$.

The way that we prove the theorem is as follows: By first showing that the
property $\displaystyle(*)$ holds for certain specific exact sequence. For
example a group variety $\displaystyle G$ fits into an exact sequence
$\displaystyle e\to G_{\mathrm{lin}}\to G\to G_{\mathrm{ab}}\to e$, where
$\displaystyle G_{\mathrm{lin}}$ is linear and $\displaystyle G_{\mathrm{ab}}$
is an abelian variety and we show that this sequence has $\displaystyle(*)$.

Then we introduce the following generalization of an exact sequence.

######  Definition.

A sequence of group varieties $\displaystyle e\to
N\overset{\iota}{\to}G\overset{\pi}{\to}Q\to e$ is called almost exact if
either:

  * •

There exists an exact sequence $\displaystyle e\to
M\overset{j}{\to}G\overset{\pi}{\to}Q\to e$ is exact and an isogeny
$\displaystyle q:N\to M$ such that $\displaystyle\iota=j\circ q$.

  * •

There exists an exact sequence $\displaystyle e\to
N\overset{\iota}{\to}G\overset{p}{\to}P\to e$ and an isogeny $\displaystyle
q:P\to Q$ is such that $\displaystyle\pi=q\circ p$.

We also say that an almost exact sequence has $\displaystyle(*)$ if it
satisfies the condition in Definition 1.

The use of this definition lies in the fact that whenever $\displaystyle e\to
N\to G\to Q\to e$ is exact, the induced sequences $\displaystyle e\to
N_{\mathrm{lin}}\to G_{\mathrm{lin}}\to Q_{\mathrm{lin}}\to e$ and
$\displaystyle e\to N_{\mathrm{ab}}\to G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to
e$ will generally not be exact but will be almost exact.
After showing that exact sequences $\displaystyle e\to N\to G\to Q\to e$,
where either $\displaystyle N,G,Q$ are all linear group varieties or are all
abelian varieties, have $\displaystyle(*)$, we show that the almost exact
sequences $\displaystyle e\to N_{\mathrm{lin}}\to G_{\mathrm{lin}}\to
Q_{\mathrm{lin}}\to e$ and $\displaystyle e\to N_{\mathrm{ab}}\to
G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to e$ both have $\displaystyle(*)$. This
will allow us to conclude that the exact sequence $\displaystyle e\to N\to
G\to Q\to e$ has $\displaystyle(*)$.

One use of this theorem lies in the following: Let $\displaystyle\sigma:G\to
G$ be an endomorphism that stabilizes the subgroup variety $\displaystyle N$
of $\displaystyle G$. Writing $\displaystyle\sigma_{N}:N\to N$ and
$\displaystyle\sigma_{Q}:Q\to Q$ for the induced endomorphisms, one sees that
through the isomorphism in Theorem Theorem, $\displaystyle\sigma^{*}$
corresponds to $\displaystyle\sigma_{N}^{*}\otimes\sigma_{Q}^{*}$. This
implies that we can calculate the graded trace for $\displaystyle\sigma$,
$\displaystyle\mathrm{tr}(\sigma^{*}):=\sum_{r\geq
0}(-1)^{r}\mathrm{tr}(\sigma^{*}\,|\,\mathrm{H}^{r}(G))$ by
$\displaystyle\mathrm{tr}(\sigma)=\mathrm{tr}(\sigma_{N})\cdot\mathrm{tr}(\sigma_{Q})$.
An application of our result is the following.

The authors of [2] studied the fixed point count of the $\displaystyle n$’th
iterate of $\displaystyle\sigma$ and came up with the following formula ([2]
Theorem 7.2.1), which holds in the case that $\displaystyle
k=\overline{\mathbb{F}}_{p}$:

$\\#\mathrm{Fix}(\sigma^{n})=|d_{n}|c^{n}r_{n}|n|_{p}^{s_{n}}p^{-t_{n}|n|_{p}^{-1}}$

Here we have $\displaystyle c\in\mathbb{Q}_{>0}$ and we have that
$\displaystyle s_{n},t_{n}\in\mathbb{Z}_{>0}$ and $\displaystyle
r_{n}\in\mathbb{Q}_{>0}$ are periodic. The term $\displaystyle d_{n}$ may be
defined even when $\displaystyle k\neq\overline{\mathbb{F}}_{p}$ for any
$\displaystyle p$ (see [2] Definition 12.3.2) and can be given a cohomological
interpretation as follows: The group variety $\displaystyle G$ fits into an
exact sequence $\displaystyle e\to G_{\mathrm{lin}}\to G\to G_{\mathrm{ab}}\to
e$ where $\displaystyle G_{\mathrm{lin}}$ is a linear and fully characteristic
subgroup variety and $\displaystyle G_{\mathrm{ab}}$ is an abelian variety.
The term $\displaystyle d_{n}$ equals
$\displaystyle\mathrm{tr}(\sigma_{\mathrm{lin}}^{n})\cdot\mathrm{tr}(\sigma_{\mathrm{ab}}^{n})$,
where $\displaystyle\sigma_{\mathrm{lin}}$ is the induced endomorphism on
$\displaystyle G_{\mathrm{lin}}$ and $\displaystyle\sigma_{\mathrm{ab}}$ the
one on $\displaystyle G_{\mathrm{ab}}$.

One may ask the natural question, whether $\displaystyle d_{n}$ is equal to
$\displaystyle\mathrm{tr}(\sigma^{n})$? The discussion below Theorem Theorem
shows that this is indeed the case. In the case that $\displaystyle
k=\overline{\mathbb{F}}_{p}$ this isomorphism was shown to exist by the
authors of [2] by using the result of Arima ([1] Theorem 1), which says that
there exists an isogeny $\displaystyle
G_{\mathrm{lin}}\times_{\overline{\mathbb{F}}_{p}}G_{\mathrm{ab}}\to G$.
However over a general algebraically closed field $\displaystyle k$, such an
isogeny need not exist, showing the need to use a different approach to tackle
this problem in general.

##  2\. Generalities and strategy

First we define the property of group varieties in which we are interested.

######  Definition 1.

An exact sequence of group varieties $\displaystyle e\to
N\overset{\iota}{\to}G\overset{\pi}{\to}Q\to e$ has $\displaystyle(*)$ if
there is an isomorphism of graded $\displaystyle\mathbb{Q}_{l}$-algebras
$\displaystyle
s\otimes\pi^{*}:\mathrm{H}^{*}(N)\otimes\mathrm{H}^{*}(Q)\to\mathrm{H}^{*}(G)$
where $\displaystyle s$ is a section to $\displaystyle\iota^{*}$.

The aim of this document is to show that any group variety has this property
and hence to prove the following theorem.

######  Theorem 2.

An exact sequence of group varieties $\displaystyle e\to N\to G\to Q\to e$ has
$\displaystyle(*)$.

We now state two known lemmas that are key for proving Theorem Theorem. The
first one is a structure theorem on algebraic groups (see [8]).

######  Lemma 3.

The following statements about group varieties are true:

  * •

A group variety $\displaystyle G$ is an extension of an abelian variety
$\displaystyle G_{\mathrm{ab}}$ by a linear group variety $\displaystyle
G_{\mathrm{lin}}$.

  * •

A linear group variety is the extension of a reductive group variety
$\displaystyle G_{\mathrm{red}}$ by a unipotent group variety
$\displaystyle\mathrm{R}_{\text{u}}(G)$.

  * •

A reductive group variety $\displaystyle G$ is the extension of a semisimple
group variety $\displaystyle G_{\mathrm{ss}}$ by a torus
$\displaystyle\mathrm{R}(G)$

  * •

A group variety $\displaystyle G$ has a largest normal subgroup variety
$\displaystyle G_{\mathrm{ant}}$, called the anti-affine group variety of
$\displaystyle G$, such that $\displaystyle G/G_{\mathrm{ant}}$ is affine.

The second lemma says that an isogeny induces an isomorphism on cohomology,
which is a well known fact.

######  Lemma 4.

Let $\displaystyle\varphi:G\to H$ be an isogeny of group varieties. Then
$\displaystyle\varphi^{*}:\mathrm{H}^{*}(H)\to\mathrm{H}^{*}(G)$ is an
isomorphism.

###### Proof.

It suffices to prove the statement for separable and purely inseparable
isogenies. In the purely inseparable case, $\displaystyle\ker(\varphi)$ is
connected, so
$\displaystyle\mathbb{Z}/l^{n}\mathbb{Z}\to\varphi_{*}\varphi^{*}\mathbb{Z}/l^{n}\mathbb{Z}$
is an isomorphism of sheaves. Since $\displaystyle\ker(\varphi)$ is zero
dimensional, the Leray spectral sequence map
$\displaystyle\mathrm{H}^{n}(H,\varphi_{*}\varphi^{*}\mathbb{Z}/l^{n}\mathbb{Z})\to\mathrm{H}^{n}(G,\varphi^{*}\mathbb{Z}/l^{n}\mathbb{Z})$
is an isomorphism, which gives the result.
In the separable case, $\displaystyle\ker(\varphi)$ may be thought of as a
finite group $\displaystyle M$ and $\displaystyle H$ is identified with
$\displaystyle G/M$. In this case, the map $\displaystyle\varphi^{*}$ is
injective and identifies $\displaystyle\mathrm{H}^{*}(H)$ with
$\displaystyle\mathrm{H}^{*}(G)^{M}$ by ([13] (5.10)). Since the action of
$\displaystyle M$ on $\displaystyle\mathrm{H}^{*}(G)$ is induced by the one of
$\displaystyle G(k)$ and $\displaystyle G$ is a connected algebraic group, the
action is trivial by ([5] Proposition 6.4). ∎

In order to share the strategy for the proof, we need to make one more
definition.

######  Definition 5.

A sequence of group varieties $\displaystyle e\to
N\overset{\iota}{\to}G\overset{\pi}{\to}Q\to e$ is called almost exact if
either:

  * •

There exists an exact sequence $\displaystyle e\to
M\overset{j}{\to}G\overset{\pi}{\to}Q\to e$ is exact and an isogeny
$\displaystyle q:N\to M$ such that $\displaystyle\iota=j\circ q$.

  * •

There exists an exact sequence $\displaystyle e\to
N\overset{\iota}{\to}G\overset{p}{\to}P\to e$ and an isogeny $\displaystyle
q:P\to Q$ is such that $\displaystyle\pi=q\circ p$.

We also say that an almost exact sequence has $\displaystyle(*)$ if it
satisfies the condition in Definition 1.

Now we give the strategy for proving the main thoerem.

######  Strategy 6.

The strategy for proving Theorem Theorem will be as follows:

  1. (1)

We show that the exact sequence $\displaystyle e\to G_{\mathrm{lin}}\to G\to
G_{\mathrm{ab}}\to e$ has $\displaystyle(*)$. For this we make use of the
existence of the anti-affine group variety. We also show that for
$\displaystyle G$ linear the exact sequence $\displaystyle
e\to\mathrm{R}_{\text{u}}(G)\to G\to G_{\mathrm{red}}\to e$ and for
$\displaystyle G$ reductive the exact sequence $\displaystyle
e\to\mathrm{R}(G)\to G\to G_{\mathrm{ss}}\to e$ both have $\displaystyle(*)$.

  2. (2)

We show that an exact sequence of abelian varieties (resp. tori, resp.
semisimple group varieties) has $\displaystyle(*)$.

  3. (3)

We show that certain almost exact sequences such as $\displaystyle e\to
N_{\mathrm{lin}}\to G_{\mathrm{lin}}\to Q_{\mathrm{lin}}\to e$ have
$\displaystyle(*)$ and in the end we conclude that any exact sequence
$\displaystyle e\to N\to G\to Q\to e$ has $\displaystyle(*)$.

All of the steps will use Lemma 4.
We know quite a bit about $\displaystyle\mathrm{H}^{*}(G)$ for $\displaystyle
G$ a general group variety. For $\displaystyle\mu:G\times_{k}G\to G$ the
multiplication, $\displaystyle\mathrm{inv}:G\to G$ inversion and
$\displaystyle e:\mathrm{Spec}(k)\to G$ the identity, the data
$\displaystyle(\mathrm{H}^{*}(G),\mu^{*},\mathrm{inv}^{*},e^{*})$ defines a
$\displaystyle\mathbb{Q}_{l}$-Hopf-algebra (see [10] for an introduction to
Hopf-algebras). An important subspace of a Hopf-algebra is the primitive
subspace.

######  Definition 7.

Let $\displaystyle(\mathrm{H},\Delta,\iota,\epsilon)$ be a Hopf algebra over a
field $\displaystyle l$. Then the primitive subspace of
$\displaystyle\mathrm{H}$ is defined by
$\displaystyle\mathrm{P}\mathrm{H}:=\\{x\in\mathrm{H}\,|\,\Delta(x)=x\otimes
1+1\otimes x\\}$.

Note that a homomorphism of Hopf-algebras preserves the primitive subspace.
Since $\displaystyle\mathrm{H}^{*}(G)$ is finite dimensional over
$\displaystyle\mathbb{Q}_{l}$ and graded-commutative we can use the following
theorem of Hopf.

######  Theorem 8 (Part 2.5D [3]).

Let $\displaystyle\mathrm{H}^{*}$ be a graded commutative Hopf-algebra over a
field $\displaystyle l$ with $\displaystyle\mathrm{char}(l)=0$ such that
$\displaystyle\mathrm{H}^{n}$ is finite dimensional for all $\displaystyle
n\geq 0$. Then there is a natural isomorphism of graded $\displaystyle
k$-algebras
$\displaystyle\mathrm{H}^{*}=\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}:=\bigoplus_{r\geq
0}\bigwedge^{r}\mathrm{P}\mathrm{H}^{*}$, where for
$\displaystyle\\{z_{i_{j}}\\}_{1\leq j\leq r}$ homogeneous elements of
$\displaystyle\mathrm{P}\mathrm{H}^{*}$ of degree $\displaystyle i_{j}$, the
element $\displaystyle z_{i_{1}}\wedge...\wedge z_{i_{r}}$ is homogeneous of
degree $\displaystyle\sum_{j=1}^{r}i_{j}$. In particular we have
$\displaystyle\mathrm{H}^{*}(G)=\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(G)$.

Using the primitive subspace allows us to formulate the property
$\displaystyle(*)$ in a simpler way.

######  Lemma 9.

Let $\displaystyle e\to N\overset{\iota}{\to}G\overset{\pi}{\to}Q\to e$ be an
exact sequence of group varieties. The sequence has $\displaystyle(*)$ if and
only if $\displaystyle
0\to\mathrm{P}\mathrm{H}^{*}(Q)\overset{\pi^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(G)\overset{\iota^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(N)\to
0$ is an exact sequence of graded $\displaystyle\mathbb{Q}_{l}$-vector spaces.

###### Proof.

If the sequence of primitive subspaces is exact, then any section
$\displaystyle s$ to $\displaystyle\iota^{*}$ induces an isomorphism:

$\mathrm{H}^{*}(N)\otimes\mathrm{H}^{*}(Q)=\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(N)\otimes\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(Q)\overset{\pi^{*}\otimes
s}{\to}\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(G)=\mathrm{H}^{*}(G)$

Conversely if the sequence has $\displaystyle(*)$, then note that pullback
homomorphisms give Hopf-algebra homomorphisms on cohomology, which preserve
primitive subspaces. It follows that
$\displaystyle\iota^{*}:\mathrm{P}\mathrm{H}^{*}(G)\overset{\iota^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(N)$
is surjective and that
$\displaystyle\pi^{*}:\mathrm{P}\mathrm{H}^{*}(Q)\overset{\pi^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(G)$
is injective. Moreover
$\displaystyle\dim\mathrm{P}\mathrm{H}^{*}(Q)+\dim\mathrm{P}\mathrm{H}^{*}(N)=\mathrm{P}\mathrm{H}^{*}(G)$
and $\displaystyle\ker(\pi^{*})\subset\mathrm{Im}(\pi^{*})$, so we obtain an
exact sequence of primitive subspaces. ∎

##  3\. The proof

Now we move towards the proof of Theorem Theorem as described in Strategy 6.

### Step 1

First we deal with the case that $\displaystyle G$ is commutative, so
$\displaystyle G$ will be a commutative group variety until we say that this
is no longer the case. Let $\displaystyle\mathcal{C}$ denote the category of
commutative algebraic groups.

######  Proposition 10.

For all $\displaystyle n\geq 1$ there is an isomorphism of functors
$\displaystyle\mathrm{Ext}(-,\mathbb{Z}/l^{n}\mathbb{Z})\to\mathrm{H}^{1}(-,\mathbb{Z}/l^{n}\mathbb{Z})$,
where both are seen as functors $\displaystyle\mathcal{C}\to\mathrm{Grp}$.

###### Proof.

That there is a natural transformation is well-known and may be found in [11]
Chapter 7. Miyanishi ([9] Lemma 4) shows that it is in fact an isomorphism. ∎

######  Lemma 11.

There is an exact sequence $\displaystyle
0\to\mathrm{H}^{1}(G_{\mathrm{ab}})\overset{\pi^{*}}{\to}\mathrm{H}^{1}(G)\overset{\iota^{*}}{\to}\mathrm{H}^{1}(G_{\mathrm{lin}})\to
0$ of $\displaystyle\mathbb{Q}_{l}$-vector spaces.

###### Proof.

Since $\displaystyle\mathbb{Q}_{l}$ is flat over
$\displaystyle\mathbb{Z}_{l}$, it suffices to prove it with
$\displaystyle\mathbb{Z}_{l}$ coefficients. As the groups
$\displaystyle\mathrm{H}^{1}(G,\mathbb{Z}/l^{n}\mathbb{Z})$ are finite, we
obtain that passing to the limit preserves exactness by the Mittag-Leffler
condition. So it suffices to show that the above statement holds with
$\displaystyle\mathbb{Z}/l^{n}\mathbb{Z}$ coefficients. By Proposition 10 this
translates into a statement about
$\displaystyle\mathrm{Ext}(-,\mathbb{Z}/l^{n}\mathbb{Z})$, which is true by
Theorem 7.13 in [11]. ∎

We can now prove the following proposition.

######  Proposition 12.

When $\displaystyle G$ is commutative the sequence $\displaystyle e\to
G_{\mathrm{lin}}\to G\to G_{\mathrm{ab}}\to e$ has $\displaystyle(*)$.

###### Proof.

It suffices to show that
$\displaystyle\mathrm{H}^{1}(G)=\mathrm{P}\mathrm{H}^{*}(G)$ by Lemma 9 and
Lemma 11. The Künneth formula gives an isomorphism of graded
$\displaystyle\mathbb{Q}_{l}$-vector spaces $\displaystyle p_{1}^{*}\cup
p_{2}^{*}:\mathrm{H}^{*}(G)\otimes\mathrm{H}^{*}(G)\to\mathrm{H}^{*}(G\times_{k}G)$.
On the degree $\displaystyle 1$ part a section is given by $\displaystyle
x\mapsto\iota_{1}^{*}(x)+\iota_{2}^{*}(x)$ for $\displaystyle\iota_{1}:G\to
G\times_{k}\\{e\\}$ and $\displaystyle\iota_{2}:G\to\\{e\\}\times_{k}G$. So
for $\displaystyle x\in\mathrm{H}^{1}(G)$ we have that
$\displaystyle\iota_{1}^{*}\mu^{*}(x)+\iota_{2}^{*}\mu^{*}(x)=x+x=\iota_{1}^{*}(p_{1}^{*}(x)+p_{2}^{*}(x))=\iota_{2}^{*}(p_{1}^{*}(x)+p_{2}^{*}(x))$
and hence
$\displaystyle\mathrm{H}^{1}(G)\subset\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(G)$.
Now it suffices to show that:

(1) $\mathrm{H}^{r}(G)=0\text{ for all }r>\dim\mathrm{H}^{1}(G)$

Since $\displaystyle G_{\mathrm{lin}}$ is commutative it is an extension of a
torus $\displaystyle T$ by a unipotent group
$\displaystyle\mathrm{R}_{\text{u}}(G_{\mathrm{lin}})$. Let $\displaystyle
H=G/\mathrm{R}_{\text{u}}(G_{\mathrm{lin}})$. As
$\displaystyle\mathrm{R}_{\text{u}}(G_{\mathrm{lin}})$ is isomorphic to an
affine space, it follows that
$\displaystyle\mathrm{H}^{*}(H)\to\mathrm{H}^{*}(G)$ is an isomorphism by
looking at the leray spectral sequence, so it suffices to prove (1) for
$\displaystyle H$ and so we may assume that $\displaystyle G$ is the extension
of an abelian variety by a torus. In this case we have
$\displaystyle\mathrm{H}^{1}(G_{\mathrm{lin}})=\mathbb{Q}_{l}^{n}$ and
$\displaystyle\mathrm{H}^{1}(G_{\mathrm{ab}})=\mathbb{Q}_{l}^{2g}$ where
$\displaystyle n=\dim(G_{\mathrm{lin}})$ and $\displaystyle
g=\dim(G_{\mathrm{ab}})$. Since the abelian variety and torus have
cohomological dimension $\displaystyle 2g$ and $\displaystyle n$ respectively,
the Leray spectal gives that $\displaystyle\mathrm{H}^{r}(G)=0$ for all
$\displaystyle r>2g+n$. By Lemma 11 we have
$\displaystyle\dim\mathrm{H}^{1}(G)=2g+n$, which gives that (1) holds and thus
the proposition holds. ∎

From now on we no longer require $\displaystyle G$ to be commutative.

######  Proposition 13.

Let $\displaystyle G$ be a group variety and consider $\displaystyle e\to
G_{\mathrm{lin}}\to G\to G_{\mathrm{ab}}\to e$. There exists an isomorphism of
graded $\displaystyle\mathbb{Q}_{l}$-vector spaces
$\displaystyle\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})\oplus\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}})\cong\mathrm{P}\mathrm{H}^{*}(G)$.

###### Proof.

Let $\displaystyle D:=G_{\mathrm{ant}}$, which is a commutative group variety.
We have an exact sequence of algebraic groups $\displaystyle
0\to\ker\overset{\iota}{\to}D\times G_{\mathrm{lin}}\overset{m}{\to}G\to 0$,
where $\displaystyle\ker\cong D\cap G_{\mathrm{lin}}$ and the map
$\displaystyle D\cap G_{\mathrm{lin}}\to D\times G_{\mathrm{lin}}$ is given by
$\displaystyle a\mapsto(-a,a)$. This gives rise to the following commuting
diagram:

(2)
$\displaystyle{e}$$\displaystyle{D_{\mathrm{lin}}}$$\displaystyle{D\times_{k}G_{\mathrm{lin}}}$$\displaystyle{Q}$$\displaystyle{e}$$\displaystyle{e}$$\displaystyle{D\cap
G_{\mathrm{lin}}}$$\displaystyle{D\times_{k}G_{\mathrm{lin}}}$$\displaystyle{G}$$\displaystyle{e}$$\displaystyle\scriptstyle{j^{\prime}}$$\displaystyle\scriptstyle{\iota_{\mathrm{lin}}}$$\displaystyle\scriptstyle{m^{\prime}}$$\displaystyle\scriptstyle{\mathrm{Id}}$$\displaystyle\scriptstyle{q}$$\displaystyle\scriptstyle{j}$$\displaystyle\scriptstyle{m}$

By Proposition 12 the embedding $\displaystyle-\iota_{D}:D_{\mathrm{lin}}\to
D$ induces a surjection
$\displaystyle\mathrm{H}^{*}(D)\to\mathrm{H}^{*}(D_{\mathrm{lin}})$. Moreover
$\displaystyle\mathrm{H}^{*}(D\times_{k}G_{\mathrm{lin}})\to\mathrm{H}^{*}(D)$
is surjective by Künneth, and hence the pullback
$\displaystyle(j^{\prime})^{*}:\mathrm{H}^{*}(D\times_{k}G_{\mathrm{lin}})\to\mathrm{H}^{*}(D_{\mathrm{lin}})$
is surjective. Since $\displaystyle D_{\text{lin}}$ is a commutative linear
group variety, the fiber bundle $\displaystyle D\times_{k}G_{\mathrm{lin}}\to
Q$ is Zariski locally trivial by Proposition 7.6 in [11].
These two statements imply that the Leray-Hirsch principle applies (see [6]
p.245 for a proof in topology, which carries over to our case since the
fibration is Zariski locally trivial implying that the Mayer-Vietoris argument
of the reference can be used). This gives an isomorphism
$\displaystyle\mathrm{H}^{*}(G_{\mathrm{lin}}\times_{k}D)\cong\mathrm{H}^{*}(Q)\otimes\mathrm{H}^{*}(D_{\text{lin}})$
as $\displaystyle\mathrm{H}^{*}(Q)$-modules. By the Künneth formula we have
$\displaystyle\mathrm{H}^{*}(G_{\mathrm{lin}}\times_{k}D)\cong\mathrm{H}^{*}(G_{\mathrm{lin}})\otimes\mathrm{H}^{*}(D)$
and by Proposition 12 we obtain that:

(3)
$\mathrm{H}^{*}(Q)\otimes\mathrm{H}^{*}(D_{\text{lin}})\cong\mathrm{H}^{*}(G_{\mathrm{lin}})\otimes\mathrm{H}^{*}(D_{\text{lin}})\otimes\mathrm{H}^{*}(D_{\mathrm{ab}})$

Since $\displaystyle D_{\mathrm{lin}}$ has finite index in $\displaystyle
D\cap G_{\mathrm{lin}}$, the map $\displaystyle q$ is an isogeny. This map
induces a homomorphism $\displaystyle q_{\mathrm{ab}}:D_{\mathrm{ab}}\to
G_{\mathrm{ab}}$. Since $\displaystyle G=D\cdot G_{\mathrm{lin}}$ it also
follows that $\displaystyle q_{\mathrm{ab}}$ is an isogeny. An isogeny induces
an isomorphism on the cohomology. This means that the isomorphism in (3)
reads:

$\mathrm{H}^{*}(G)\otimes\mathrm{H}^{*}(D_{\text{lin}})\cong\mathrm{H}^{*}(G_{\mathrm{lin}})\otimes\mathrm{H}^{*}(G_{\mathrm{ab}})\otimes\mathrm{H}^{*}(D_{\mathrm{lin}})$

Since $\displaystyle\mathrm{H}^{*}(D_{\text{lin}})$ is a nonzero graded
$\displaystyle\mathbb{Q}_{l}$-vector space we see that there is an isomorphism
of graded $\displaystyle\mathbb{Q}_{l}$-vector space isomorphism
$\displaystyle\mathrm{H}^{*}(G)\cong\mathrm{H}^{*}(G_{\mathrm{ab}})\otimes\mathrm{H}^{*}(G_{\mathrm{lin}})$.
Hence
$\displaystyle\bigwedge^{*}\mathrm{P}\mathrm{H}^{*}(G)\cong\bigwedge^{*}(\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})\oplus\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}}))$
and so we obtain that:

$\mathrm{P}\mathrm{H}^{*}(G)\cong\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})\oplus\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}})$

∎

Now we will use the above to show that the desired isomorphism indeed exists.

######  Theorem 14.

There is a section $\displaystyle s$ to
$\displaystyle\iota^{*}:\mathrm{H}^{*}(G)\to\mathrm{H}^{*}(G_{\mathrm{lin}})$
such that
$\displaystyle
s\otimes\pi^{*}:\mathrm{H}^{*}(G_{\mathrm{lin}})\otimes\mathrm{H}^{*}(G_{\mathrm{ab}})\to\mathrm{H}^{*}(G)$
is an isomorphism of graded $\displaystyle\mathbb{Q}_{l}$-algebras.

###### Proof.

We will show that $\displaystyle
0\to\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}})\overset{\pi^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(G)\overset{\iota^{*}}{\to}\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})\to
0$ is exact, which is sufficient by Lemma 9. Since
$\displaystyle\dim\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})+\dim\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}})=\dim\mathrm{P}\mathrm{H}^{*}(G)$
and since $\displaystyle\ker(\pi^{*})\subset\mathrm{Im}(\iota^{*})$ it will
suffice to show that $\displaystyle\iota^{*}$ is surjective and
$\displaystyle\pi^{*}$ is injective.
For this we look at (2) and we stick to the notation of Proposition 13. For
$\displaystyle p_{1}:D\times_{k}G_{\mathrm{lin}}\to D$ the first projection we
have $\displaystyle\pi\circ m=q_{\mathrm{ab}}\circ\pi_{D}\circ p_{1}$. This
implies that $\displaystyle
m^{*}\pi^{*}=p_{1}^{*}\pi_{D}^{*}q_{\mathrm{ab}}^{*}$. We have that
$\displaystyle m^{*}$ is injective by the Leray-Hirsch principle and
$\displaystyle\pi_{D}^{*}$ is injective by Proposition 12. Moreover,
$\displaystyle p_{1}^{*}$ and $\displaystyle q_{\mathrm{ab}}^{*}$ are also
injective and hence $\displaystyle\pi^{*}$ is injective.
To show that $\displaystyle\iota^{*}$ is surjective, pick $\displaystyle
x\in\mathrm{H}^{*}(G_{\mathrm{lin}})$ and write $\displaystyle
p_{2}:D\times_{k}G_{\mathrm{lin}}\to G_{\mathrm{lin}}$ for the second
projection. Then the Leray Hirsch principle applied to $\displaystyle
m^{\prime}:D\times_{k}G_{\mathrm{lin}}\to Q$ together with the fact that
$\displaystyle q\circ m^{\prime}=m$ implies that $\displaystyle
p_{2}^{*}x=\sum m^{*}\beta_{n}\cup\alpha_{r}^{\sigma_{r}}$ for
$\displaystyle\beta_{n}\in\mathrm{H}^{n}(G)$ and the
$\displaystyle\alpha_{r}^{\sigma_{r}}$ such that
$\displaystyle(j^{\prime})^{*}\alpha_{r}^{\sigma_{r}}$ generate
$\displaystyle\mathrm{H}^{r}(D_{\mathrm{lin}})$ freely. Notice that the
$\displaystyle\alpha_{r}^{\sigma_{r}}$ can be chosen of the form
$\displaystyle p_{1}^{*}\beta_{r}^{\sigma_{r}}$ for
$\displaystyle\beta_{r}^{\sigma_{r}}\in\mathrm{H}^{r}(D)$. Then note that by
using that $\displaystyle\iota_{2}^{*}p_{2}^{*}$ is the identity, we have
$\displaystyle p_{2}^{*}x=p_{2}^{*}\iota_{2}^{*}p_{2}^{*}x=\sum
p_{2}^{*}\iota_{2}^{*}m^{*}\beta_{n}\cup
p_{2}^{*}\iota_{2}^{*}p_{1}^{*}\beta_{r}^{\sigma_{r}}=p_{2}^{*}\iota_{2}^{*}m^{*}y$
for $\displaystyle y\in\mathrm{H}^{*}(G)$. So this gives $\displaystyle
m\circ\iota_{2}=\iota$ and since $\displaystyle p_{2}^{*}$ is injective, this
gives $\displaystyle x=\iota^{*}y$ and hence $\displaystyle\iota^{*}$ is
surjective. We conclude that the sequence of primitive subspaces is exact and
that hence the theorem is true. ∎

We still have to show that for linear $\displaystyle G$ the sequence
$\displaystyle e\to\mathrm{R}_{\text{u}}(G)\to G\to G_{\mathrm{red}}\to e$ and
for reductive $\displaystyle G$ the sequences $\displaystyle e\to R(G)\to G\to
G_{\mathrm{ss}}\to e$ give a tensor product decomposition of $\displaystyle
l$-adic cohomology. The first one follows trivially as
$\displaystyle\mathrm{R}_{\text{u}}(G)$ is isomorphic to an affine space and
hence the Leray spectral sequence gives an isomorphism
$\displaystyle\mathrm{H}^{*}(G_{\mathrm{red}})\to\mathrm{H}^{*}(G)$. For the
second one we have the following lemma:

######  Lemma 15.

Consider the commuting diagram of group varieties with exact rows and all
columns isogenies:

$\displaystyle{e}$$\displaystyle{N_{1}}$$\displaystyle{G_{1}}$$\displaystyle{Q_{1}}$$\displaystyle{e}$$\displaystyle{e}$$\displaystyle{N_{2}}$$\displaystyle{G_{2}}$$\displaystyle{Q_{2}}$$\displaystyle{e}$$\displaystyle\scriptstyle{\iota_{1}}$$\displaystyle\scriptstyle{\pi_{1}}$$\displaystyle\scriptstyle{\iota_{2}}$$\displaystyle\scriptstyle{q_{N}}$$\displaystyle\scriptstyle{\pi_{2}}$$\displaystyle\scriptstyle{q_{G}}$$\displaystyle\scriptstyle{q_{Q}}$

Then the top row has $\displaystyle(*)$ if and only if the bottom row has
$\displaystyle(*)$.

###### Proof.

Upon taking primitive subspaces of $\displaystyle l$-adic cohomology, we get
the following commuting diagram:

$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{1})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{1})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{1})}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{2})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{2})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{2})}$$\displaystyle{0}$$\displaystyle\scriptstyle{q_{N}^{*}}$$\displaystyle\scriptstyle{q_{G}^{*}}$$\displaystyle\scriptstyle{q_{Q}^{*}}$

Since $\displaystyle q_{N}^{*},q_{G}^{*}$ and $\displaystyle q_{Q}^{*}$ are
isomorphisms, the top sequence is exact if and only if the bottom sequence is
exact. ∎

This gives the following immediate corollary:

######  Corollary 16.

For reductive $\displaystyle G$, the sequence $\displaystyle e\to R(G)\to G\to
G_{\mathrm{ss}}\to e$ has $\displaystyle(*)$.

###### Proof.

Since $\displaystyle G$ is reductive, $\displaystyle
m:R(G)\times_{k}G_{\mathrm{der}}\to G$ is an isogeny. Hence there is a
commuting diagram:

$\displaystyle{e}$$\displaystyle{R(G)}$$\displaystyle{G}$$\displaystyle{G_{\mathrm{ss}}}$$\displaystyle{e}$$\displaystyle{e}$$\displaystyle{R(G)}$$\displaystyle{R(G)\times_{k}G_{\mathrm{der}}}$$\displaystyle{G_{\mathrm{der}}}$$\displaystyle{e}$$\displaystyle\scriptstyle{\iota_{1}}$$\displaystyle\scriptstyle{\mathrm{Id}}$$\displaystyle\scriptstyle{p_{2}}$$\displaystyle\scriptstyle{\mu}$

Now the previous lemma gives the result as the bottom sequence is split. ∎

This concludes step 1 of Strategy 6.

### Step 2

In this step we show that an exact sequence of abelian varieties, unipotent
groups, tori or semisimple groups give a tensor product decomposition of
$\displaystyle l$-adic cohomology. Note that the case of unipotent groups is
trivial as they have no higher cohomology. For the other cases we have the
following lemma.

######  Lemma 17.

An exact sequence $\displaystyle e\to N\to G\to Q\to e$ of abelian varieties
(resp. semisimple group varieties, resp. tori) has $\displaystyle(*)$.

###### Proof.

For any abelian variety $\displaystyle A$, the simple abelian subvarieties
$\displaystyle A_{1},...,A_{n}$ of $\displaystyle A$ satisfy that
multiplication $\displaystyle A_{1}\times_{k}...\times_{k}A_{n}\to A$ is an
isogeny by p.42 of [7]. So there exists an abelian subvarieties $\displaystyle
M\subset G$ and $\displaystyle H\subset G$ such that multiplication
$\displaystyle M\times_{k}H\to G$ is an isogeny. The result for abelian
varieties now follows from Lemma 15.
For any semisimple group variety $\displaystyle G$ there also exist almost
simple subgroup varieties $\displaystyle G_{1},...,G_{n}$ such that
multiplication $\displaystyle G_{1}\times_{k}...\times_{k}G_{n}\to G$ is an
isogeny by Theorem 21.51 of [8], so the proof is the same as above.
For tori it will follow from the fact that any exact sequence of them is
split. Since $\displaystyle N$ is a commutative algebraic group, any principle
$\displaystyle N$-bundle over $\displaystyle Q$ is Zariski locally trivial,
hence these principal bundles are parametrized by
$\displaystyle\mathrm{H}_{\mathrm{zar}}^{1}(Q,N)\cong\mathrm{H}_{\mathrm{zar}}^{1}(Q,\mathbb{G}_{m})^{n}$
for some $\displaystyle n$. Since we have
$\displaystyle\mathrm{H}_{\mathrm{zar}}^{1}(Q,\mathbb{G}_{m})=\mathrm{Pic}(\mathcal{O}(Q))=0$,
because $\displaystyle\mathcal{O}(Q)$ is a UFD, there is a regular section
$\displaystyle s:Q\to G$. We may assume that $\displaystyle s(e)=e$ which
means that $\displaystyle s$ is a homomorphism by Proposition 12.49 of [8]. So
the sequence is split and hence it has $\displaystyle(*)$ by the Künneth
formula. ∎

This finishes step 2 of Strategy 6.

### Step 3

In this step we finish the proof of Theorem Theorem. Recall the notion of an
almost exact sequence in Definition 5.

######  Example 18.

Notable examples of almost exact sequences are the following.

  * •

When $\displaystyle e\to N\to G\to Q\to e$ is an exact sequence of group
varieties, the induced sequences $\displaystyle e\to N_{\mathrm{lin}}\to
G_{\mathrm{lin}}\to Q_{\mathrm{lin}}\to e$ and $\displaystyle e\to
N_{\mathrm{ab}}\to G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to e$ are almost exact.
Checking this is routine and relies on the fact that $\displaystyle
N_{\mathrm{lin}}$ has finite index in $\displaystyle N\cap
G_{\mathrm{lin}}=\ker(G_{\mathrm{lin}}\to Q_{\mathrm{lin}})$.

  * •

When $\displaystyle e\to N\to G\to Q\to e$ is exact and all group varieties
are linear, the induced sequences $\displaystyle
e\to\mathrm{R}_{\text{u}}(N)\to\mathrm{R}_{\text{u}}(G)\to\mathrm{R}_{\text{u}}(Q)\to
e$ and $\displaystyle e\to N_{\mathrm{red}}\to G_{\mathrm{red}}\to
Q_{\mathrm{red}}\to e$ are almost exact.

  * •

When all group varieties in question are reductive, the sequences
$\displaystyle e\to\mathrm{R}(N)\to\mathrm{R}(G)\to\mathrm{R}(Q)\to e$ and
$\displaystyle e\to N_{\mathrm{ss}}\to G_{\mathrm{ss}}\to Q_{\mathrm{ss}}\to
e$ are almost exact.

We now show that certain almost exact sequences have $\displaystyle(*)$.

######  Lemma 19.

For $\displaystyle e\to N\to G\to Q\to e$ an exact sequence of group
varieties, the almost exact sequence $\displaystyle e\to N_{\mathrm{ab}}\to
G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to e$ has $\displaystyle(*)$. When all
group varieties are linear, the almost exact sequence
$\displaystyle\mathrm{R}_{\text{u}}(N)\to\mathrm{R}_{\text{u}}(G)\to\mathrm{R}_{\text{u}}(Q)\to
e$ has $\displaystyle(*)$ and when all group varieties are reductive, the
almost exact sequences $\displaystyle
e\to\mathrm{R}(N)\to\mathrm{R}(G)\to\mathrm{R}(Q)\to e$ and $\displaystyle
e\to N_{\mathrm{ss}}\to G_{\mathrm{ss}}\to Q_{\mathrm{ss}}\to e$ have
$\displaystyle(*)$.

###### Proof.

The case of unipotent groups is trivial as they have no higher cohomology.
Consider the exact sequence $\displaystyle e\to\frac{N}{N\cap
G_{\mathrm{lin}}}\to G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to e$. It has
$\displaystyle(*)$ by Lemma 17. Note that $\displaystyle
N_{\mathrm{ab}}\to\frac{N}{N\cap G_{\mathrm{lin}}}$ is an isogeny and so it
induces an isomorphism on cohomology. Hence it follows that $\displaystyle
e\to N_{\mathrm{ab}}\to G_{\mathrm{ab}}\to Q_{\mathrm{ab}}\to e$ has
$\displaystyle(*)$.
When all group varieties are reductive consider $\displaystyle
e\to\mathrm{R}(N)\to\mathrm{R}(G)\to\frac{\mathrm{R}(G)}{\mathrm{R}(N)}\to e$.
Since all radicals are tori, the sequence has $\displaystyle(*)$ by Lemma 17.
Since $\displaystyle\frac{R(G)}{R(N)}\to R(Q)$ is an isogeny, whose pullback
is an isomorphism, it follows that $\displaystyle
e\to\mathrm{R}(N)\to\mathrm{R}(G)\to\mathrm{R}(Q)\to e$ has
$\displaystyle(*)$.
For the sequence $\displaystyle e\to N_{\mathrm{ss}}\to G_{\mathrm{ss}}\to
Q_{\mathrm{ss}}\to e$, note that $\displaystyle
e\to\frac{N}{N\cap\mathrm{R}(G)}\to G_{\mathrm{ss}}\to Q_{\mathrm{ss}}\to e$
is exact and that $\displaystyle
N_{\mathrm{ss}}\to\frac{N}{N\cap\mathrm{R}(G)}$ is an isogeny, so this follows
as in the case of the abelian varieties. ∎

Now we can combine this lemma with several parts of Step 1 to obtain the
following.

######  Lemma 20.

An exact sequence $\displaystyle e\to N\to G\to Q\to e$ of reductive group
varieties has $\displaystyle(*)$. An induced sequence $\displaystyle e\to
N_{\mathrm{red}}\to G_{\mathrm{red}}\to Q_{\mathrm{red}}$ has
$\displaystyle(*)$.

###### Proof.

Let $\displaystyle e\to N\to G\to Q\to e$ be an exact sequence of reductive
group varieties. By considering the induced sequences of radicals and
semisimple groups and then passing to $\displaystyle l$-adic cohomology, one
gets the following commuting diagram:

$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(\mathrm{R}(N))}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(R(G))}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(R(Q))}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q)}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{\mathrm{ss}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ss}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{\mathrm{ss}})}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$

We have to show that the middle row is exact. The top and bottom row are exact
by Lemma 19. All columns are exact by Corollary 16. By applying the Snake
lemma twice, we see that
$\displaystyle\mathrm{P}\mathrm{H}^{*}(Q)\to\mathrm{P}\mathrm{H}^{*}(G)$ is
injective and that
$\displaystyle\mathrm{P}\mathrm{H}^{*}(G)\to\mathrm{P}\mathrm{H}^{*}(N)$ is
surjective. Since $\displaystyle\mathrm{P}\mathrm{H}^{*}(Q)$ is in the kernel
of the latter map, comparing dimensions gives the result.
For the second statement, note that $\displaystyle
e\to\frac{N}{N\cap\mathrm{R}_{\text{u}}(G)}\to G_{\mathrm{red}}\to
Q_{\mathrm{red}}\to e$ is an exact sequence of reductive group varieties and
that $\displaystyle
N_{\mathrm{red}}\to\frac{N}{N\cap\mathrm{R}_{\text{u}}(G)}$ is an isogeny. ∎

We now use this lemma to show that Theorem Theorem holds in the case that all
group varieties are linear.

######  Lemma 21.

An exact sequence $\displaystyle e\to N\to G\to Q\to e$ of linear group
varieties has $\displaystyle(*)$. Moreover for $\displaystyle e\to N\to G\to
Q\to e$ exact, the almost exact sequence $\displaystyle e\to
N_{\mathrm{lin}}\to G_{\mathrm{lin}}\to Q_{\mathrm{lin}}\to e$ has
$\displaystyle(*)$.

###### Proof.

The proof follows similarly to the above proof by considering the following
commuting diagram, which has exact columns and an exact bottom row by Lemma
20:

$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q)}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{\mathrm{red}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{red}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{\mathrm{red}})}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$

For the second statement, note that $\displaystyle e\to N_{\mathrm{lin}}\to
G_{\mathrm{lin}}\to
G_{\mathrm{lin}}\to\frac{G_{\mathrm{lin}}}{N_{\mathrm{lin}}}\to e$ has
$\displaystyle(*)$ since all group varieties are linear and that
$\displaystyle\frac{G_{\mathrm{lin}}}{N_{\mathrm{lin}}}\to Q_{\mathrm{lin}}$
is an isogeny. ∎

Now we can prove the main result of the paper, which is Theorem Theorem.

######  Theorem.

An exact sequence of group varieties $\displaystyle e\to N\to G\to Q\to e$ has
$\displaystyle(*)$.

###### Proof.

The proof is almost identical to the proof of Lemma 20. Consider the large
commuting diagram:

$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{\mathrm{lin}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{lin}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{\mathrm{lin}})}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G)}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q)}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(N_{\mathrm{ab}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(G_{\mathrm{ab}})}$$\displaystyle{\mathrm{P}\mathrm{H}^{*}(Q_{\mathrm{ab}})}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$$\displaystyle{0}$

Again it suffices to show that the middle row is exact. The columns are exact
by Theorem 14 of Step 1. The top and bottom rows are exact by Lemma 21 and
Lemma 19 respectively. Thus we can apply the snake lemma twice and then note
that
$\displaystyle\mathrm{P}\mathrm{H}^{*}(Q)\subset\ker(\mathrm{P}\mathrm{H}^{*}(G)\to\mathrm{P}\mathrm{H}^{*}(N))$
to obtain that the middle row is exact, which proves the theorem. ∎

We remark that the following generalization of the result can not be made.

######  Remark 22.

One might wonder if the condition that $\displaystyle N\subset G$ is normal
can be relaxed to $\displaystyle N=H$ being any subgroup variety of
$\displaystyle G$. In this case the quotient $\displaystyle G/H$ exists as a
variety over $\displaystyle k$ and one can ask whether one has a similar
isomorphism
$\displaystyle\mathrm{H}^{*}(G)\cong\mathrm{H}^{*}(H)\otimes\mathrm{H}^{*}(G/H)$.
This fails already in some very simple cases. For example, we let
$\displaystyle n\geq 2$ and take $\displaystyle G=\mathrm{GL}_{n}$ and the
torus $\displaystyle
H=\\{\mathrm{diag}(t_{1},...,t_{n})\,|\,t_{1},...,t_{n}\in
k^{*}\\}\cong\mathbb{G}_{m}^{n}$. We have by ([4] p.230) an isomorphism of
graded $\displaystyle\mathbb{Q}_{l}$-algebras
$\displaystyle\mathrm{H}^{*}(\mathrm{GL}_{n})\cong\bigwedge^{*}\mathrm{J}[\times
2-1]$ , where $\displaystyle\mathrm{J}$ is the
$\displaystyle\mathbb{Q}_{l}$-vector space spanned freely by the primitive
Weyl-group invariants of $\displaystyle\mathrm{GL}_{n}$. I.e.
$\displaystyle\mathrm{J}=S^{W}/S^{W}_{+}$ for $\displaystyle
S=\mathrm{Sym}(X^{*}(T)\otimes_{\mathbb{Z}}\mathbb{Q}_{l}))$, where
$\displaystyle S^{W}_{+}$ is the ideal of invariants of positive degree (see
[12] Section 7.1 for an introduction on the Weyl group). In our case the
primitive invariants are $\displaystyle e_{1},...,e_{n}$, the elementary
symmetric polynomials and $\displaystyle[\times 2-1]$ indicates that the
weight of $\displaystyle e_{i}$ is $\displaystyle 2i-1$. Hence
$\displaystyle\dim\mathrm{H}^{1}(\mathrm{GL}_{n})=1$, while
$\displaystyle\dim\mathrm{H}^{1}(\mathbb{G}_{m}^{n})=n$ by the Künneth
formula, so indeed the above generalization does not hold.

We end with the following question for possible further research.

######  Question.

The proof of Theorem Theorem relies on various structure theorems for
algebraic groups and being able to compare certain exact sequences to split
exact sequences. One may ask if there is a more intrinsic proof than ours. For
example, can one show directly that the $\displaystyle l$-adic Leray spectral
sequence of $\displaystyle G\to Q$ has $\displaystyle
E_{2}^{p,q}=\mathrm{H}^{p}(Q,\mathbb{Q}_{l})\otimes_{\mathbb{Q}_{l}}\mathrm{H}^{q}(N,\mathbb{Q}_{l})$
and that it degenerates at the $\displaystyle E_{2}$-page?

### Acknowledgements

The author wants to thank prof. Gunther Cornelissen for his supervision,
enthusiasm and helpful discussions during the research period.

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