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38.8. Properties of algebraic group schemes

Recall that a scheme over a field $k$ is (locally) algebraic if it is (locally) of finite type over $\mathop{\mathrm{Spec}}(k)$, see Varieties, Definition 32.20.1. This is the sense of algebraic we are using in the title of this section.

Lemma 38.8.1. Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then $G$ is equidimensional and $\dim(G) = \dim_g(G)$ for all $g \in G$. For any closed point $g \in G$ we have $\dim(G) = \dim(\mathcal{O}_{G, g})$.

Proof. Let us first prove that $\dim_g(G) = \dim_{g'}(G)$ for any pair of points $g, g' \in G$. By Morphisms, Lemma 28.27.3 we may extend the ground field at will. Hence we may assume that both $g$ and $g'$ are defined over $k$. Hence there exists an automorphism of $G$ mapping $g$ to $g'$, whence the equality. By Morphisms, Lemma 28.27.1 we have $\dim_g(G) = \dim(\mathcal{O}_{G, g}) + \text{trdeg}_k(\kappa(g))$. On the other hand, the dimension of $G$ (or any open subset of $G$) is the supremum of the dimensions of the local rings of $G$, see Properties, Lemma 27.10.3. Clearly this is maximal for closed points $g$ in which case $\text{trdeg}_k(\kappa(g)) = 0$ (by the Hilbert Nullstellensatz, see Morphisms, Section 28.15). Hence the lemma follows. $\square$

The following result is sometimes referred to as Cartier's theorem.

Lemma 38.8.2. Let $k$ be a field of characteristic $0$. Let $G$ be a locally algebraic group scheme over $k$. Then the structure morphism $G \to \mathop{\mathrm{Spec}}(k)$ is smooth, i.e., $G$ is a smooth group scheme.

Proof. By Lemma 38.6.3 the module of differentials of $G$ over $k$ is free. Hence smoothness follows from Varieties, Lemma 32.25.1. $\square$

Remark 38.8.3. Any group scheme over a field of characteristic $0$ is reduced, see [Perrin-thesis, I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4] and also [Perrin, Proposition 4.2.8]. This was a question raised in [Oort, page 80]. We have seen in Lemma 38.8.2 that this holds when the group scheme is locally of finite type.

Lemma 38.8.4. Let $k$ be a perfect field of characteristic $p > 0$ (see Lemma 38.8.2 for the characteristic zero case). Let $G$ be a locally algebraic group scheme over $k$. If $G$ is reduced then the structure morphism $G \to \mathop{\mathrm{Spec}}(k)$ is smooth, i.e., $G$ is a smooth group scheme.

Proof. By Lemma 38.6.3 the sheaf $\Omega_{G/k}$ is free. Hence the lemma follows from Varieties, Lemma 32.25.2. $\square$

Remark 38.8.5. Let $k$ be a field of characteristic $p > 0$. Let $\alpha \in k$ be an element which is not a $p$th power. The closed subgroup scheme $$G = V(x^p + \alpha y^p) \subset \mathbf{G}_{a, k}^2$$ is reduced and irreducible but not smooth (not even normal).

The following lemma is a special case of Lemma 38.7.13 with a somewhat easier proof.

Lemma 38.8.6. Let $k$ be an algebraically closed field. Let $G$ be a locally algebraic group scheme over $k$. Let $g_1, \ldots, g_n \in G(k)$ be $k$-rational points. Then there exists an affine open $U \subset G$ containing $g_1, \ldots, g_n$.

Proof. We first argue by induction on $n$ that we may assume all $g_i$ are on the same connected component of $G$. Namely, if not, then we can find a decomposition $G = W_1 \amalg W_2$ with $W_i$ open in $G$ and (after possibly renumbering) $g_1, \ldots, g_r \in W_1$ and $g_{r + 1}, \ldots, g_n \in W_2$ for some $0 < r < n$. By induction we can find affine opens $U_1$ and $U_2$ of $G$ with $g_1, \ldots, g_r \in U_1$ and $g_{r + 1}, \ldots, g_n \in U_2$. Then $$g_1, \ldots, g_n \in (U_1 \cap W_1) \cup (U_2 \cap W_2)$$ is a solution to the problem. Thus we may assume $g_1, \ldots, g_n$ are all on the same connected component of $G$. Translating by $g_1^{-1}$ we may assume $g_1, \ldots, g_n \in G^0$ where $G^0 \subset G$ is as in Proposition 38.7.11. Choose an affine open neighbourhood $U$ of $e$, in particular $U \cap G^0$ is nonempty. Since $G^0$ is irreducible we see that $$G^0 \cap (Ug_1^{-1} \cap \ldots \cap Ug_n^{-1})$$ is nonempty. Since $G \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type, also $G^0 \to \mathop{\mathrm{Spec}}(k)$ is locally of finite type, hence any nonempty open has a $k$-rational point. Thus we can pick $g \in G^0(k)$ with $g \in Ug_i^{-1}$ for all $i$. Then $g_i \in g^{-1}U$ for all $i$ and $g^{-1}U$ is the affine open we were looking for. $\square$

Lemma 38.8.7. Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$. Then $G$ is quasi-projective over $k$.

Proof. By Varieties, Lemma 32.15.1 we may assume that $k$ is algebraically closed. Let $G^0 \subset G$ be the connected component of $G$ as in Proposition 38.7.11. Then every other connected component of $G$ has a $k$-rational point and hence is isomorphic to $G^0$ as a scheme. Since $G$ is quasi-compact and Noetherian, there are finitely many of these connected components. Thus we reduce to the case discussed in the next paragraph.

Let $G$ be a connected algebraic group scheme over an algebraically closed field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over $k$ by Lemma 38.8.2. If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$ be the reduction of $G$. By Divisors, Proposition 30.17.9 it suffices to show that $H$ has an ample invertible sheaf. (For an algebraic scheme over $k$ having an ample invertible sheaf is equivalent to being quasi-projective over $k$, see for example the very general More on Morphisms, Lemma 36.42.1.) By Lemma 38.7.6 we see that $H$ is a group scheme over $k$. By Lemma 38.8.4 we see that $H$ is smooth over $k$. This reduces us to the situation discussed in the next paragraph.

Let $G$ be a quasi-compact irreducible smooth group scheme over an algebraically closed field $k$. Observe that the local rings of $G$ are regular and hence UFDs (Varieties, Lemma 32.25.3 and More on Algebra, Lemma 15.97.7). The complement of a nonempty affine open of $G$ is the support of an effective Cartier divisor $D$. This follows from Divisors, Lemma 30.16.6. (Observe that $G$ is separated by Lemma 38.7.3.) We conclude there exists an effective Cartier divisor $D \subset G$ such that $G \setminus D$ is affine. We will use below that for any $n \geq 1$ and $g_1, \ldots, g_n \in G(k)$ the complement $G \setminus \bigcup D g_i$ is affine. Namely, it is the intersection of the affine opens $G \setminus Dg_i \cong G \setminus D$ in the separated scheme $G$.

We may choose the top row of the diagram $$\xymatrix{ G & U \ar[l]_j \ar[r]^\pi & \mathbf{A}^d_k \\ & W \ar[r]^{\pi'} \ar[u] & V \ar[u] }$$ such that $U \not = \emptyset$, $j : U \to G$ is an open immersion, and $\pi$ is étale, see Morphisms, Lemma 28.34.20. There is a nonempty affine open $V \subset \mathbf{A}^d_k$ such that with $W = \pi^{-1}(V)$ the morphism $\pi' = \pi|_W : W \to V$ is finite étale. In particular $\pi'$ is finite locally free, say of degree $n$. Consider the effective Cartier divisor $$\mathcal{D} = \{(g, w) \mid m(g, j(w)) \in D\} \subset G \times W$$ (This is the restriction to $G \times W$ of the pullback of $D \subset G$ under the flat morphism $m : G \times G \to G$.) Consider the closed subset1 $T = (1 \times \pi')(\mathcal{D}) \subset G \times V$. Since $\pi'$ is finite locally free, every irreducible component of $T$ has codimension $1$ in $G \times V$. Since $G \times V$ is smooth over $k$ we conclude these components are effective Cartier divisors (Divisors, Lemma 30.15.7 and lemmas cited above) and hence $T$ is the support of an effective Cartier divisor $E$ in $G \times V$. If $v \in V(k)$, then $(\pi')^{-1}(v) = \{w_1, \ldots, w_n\} \subset W(k)$ and we see that $$E_v = \bigcup\nolimits_{i = 1, \ldots, n} D j(w_i)^{-1}$$ in $G$ set theoretically. In particular we see that $G \setminus E_v$ is affine open (see above). Moreover, if $g \in G(k)$, then there exists a $v \in V$ such that $g \not \in E_v$. Namely, the set $W'$ of $w \in W$ such that $g \not \in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$ such that the fibre of $W' \to V$ over $v$ has $n$ elements.

Consider the invertible sheaf $\mathcal{M} = \mathcal{O}_{G \times V}(E)$ on $G \times V$. By Varieties, Lemma 32.30.5 the isomorphism class $\mathcal{L}$ of the restriction $\mathcal{M}_v = \mathcal{O}_G(E_v)$ is independent of $v \in V(k)$. On the other hand, for every $g \in G(k)$ we can find a $v$ such that $g \not \in E_v$ and such that $G \setminus E_v$ is affine. Thus the canonical section (Divisors, Definition 30.14.1) of $\mathcal{O}_G(E_v)$ corresponds to a section $s_v$ of $\mathcal{L}$ which does not vanish at $g$ and such that $G_{s_v}$ is affine. This means that $\mathcal{L}$ is ample by definition (Properties, Definition 27.26.1). $\square$

Lemma 38.8.8. Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$. Then the center of $G$ is a closed subgroup scheme of $G$.

Proof. Let $\text{Aut}(G)$ denote the contravariant functor on the category of schemes over $k$ which associates to $S/k$ the set of automorphisms of the base change $G_S$ as a group scheme over $S$. There is a natural transformation $$G \longrightarrow \text{Aut}(G),\quad g \longmapsto \text{inn}_g$$ sending an $S$-valued point $g$ of $G$ to the inner automorphism of $G$ determined by $g$. The center $C$ of $G$ is by definition the kernel of this transformation, i.e., the functor which to $S$ associates those $g \in G(S)$ whose associated inner automorphism is trivial. The statement of the lemma is that this functor is representable by a closed subgroup scheme of $G$.

Choose an integer $n \geq 1$. Let $G_n \subset G$ be the $n$th infinitesimal neighbourhood of the identity element $e$ of $G$. For every scheme $S/k$ the base change $G_{n, S}$ is the $n$th infinitesimal neighbourhood of $e_S : S \to G_S$. Thus we see that there is a natural transformation $\text{Aut}(G) \to \text{Aut}(G_n)$ where the right hand side is the functor of automorphisms of $G_n$ as a scheme ($G_n$ isn't in general a group scheme). Observe that $G_n$ is the spectrum of an artinian local ring $A_n$ with residue field $k$ which has finite dimension as a $k$-vector space (Varieties, Lemma 32.20.2). Since every automorphism of $G_n$ induces in particular an invertible linear map $A_n \to A_n$, we obtain transformations of functors $$G \to \text{Aut}(G) \to \text{Aut}(G_n) \to \text{GL}(A_n)$$ The final group valued functor is representable, see Example 38.5.4, and the last arrow is visibly injective. Thus for every $n$ we obtain a closed subgroup scheme $$H_n = \mathop{\mathrm{Ker}}(G \to \text{Aut}(G_n)) = \mathop{\mathrm{Ker}}(G \to \text{GL}(A_n)).$$ As a first approximation we set $H = \bigcap_{n \geq 1} H_n$ (scheme theoretic intersection). This is a closed subgroup scheme which contains the center $C$.

Let $h$ be an $S$-valued point of $H$ with $S$ locally Noetherian. Then the automorphism $\text{inn}_h$ induces the identity on all the closed subschemes $G_{n, S}$. Consider the kernel $K = \mathop{\mathrm{Ker}}(\text{inn}_h : G_S \to G_S)$. This is a closed subgroup scheme of $G_S$ over $S$ containing the closed subschemes $G_{n, S}$ for $n \geq 1$. This implies that $K$ contains an open neighbourhood of $e(S) \subset G_S$, see Algebra, Remark 10.50.6. Let $G^0 \subset G$ be as in Proposition 38.7.11. Since $G^0$ is geometrically irreducible, we conclude that $K$ contains $G^0_S$ (for any nonempty open $U \subset G^0_{k'}$ and any field extension $k'/k$ we have $U \cdot U^{-1} = G^0_{k'}$, see proof of Lemma 38.7.9). Applying this with $S = H$ we find that $G^0$ and $H$ are subgroup schemes of $G$ whose points commute: for any scheme $S$ and any $S$-valued points $g \in G^0(S)$, $h \in H(S)$ we have $gh = hg$ in $G(S)$.

Assume that $k$ is algebraically closed. Then we can pick a $k$-valued point $g_i$ in each irreducible component $G_i$ of $G$. Observe that in this case the connected components of $G$ are the irreducible components of $G$ are the translates of $G^0$ by our $g_i$. We claim that $$C = H \cap \bigcap\nolimits_i \mathop{\mathrm{Ker}}(\text{inn}_{g_i} : G \to G) \quad (\text{scheme theoretic intersection})$$ Namely, $C$ is contained in the right hand side. On the other hand, every $S$-valued point $h$ of the right hand side commutes with $G^0$ and with $g_i$ hence with everything in $G = \bigcup G^0g_i$.

The case of a general base field $k$ follows from the result for the algebraic closure $\overline{k}$ by descent. Namely, let $A \subset G_{\overline{k}}$ the closed subgroup scheme representing the center of $G_{\overline{k}}$. Then we have $$A \times_{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(\overline{k}) = \mathop{\mathrm{Spec}}(\overline{k}) \times_{\mathop{\mathrm{Spec}}(k)} A$$ as closed subschemes of $G_{\overline{k} \otimes_k \overline{k}}$ by the functorial nature of the center. Hence we see that $A$ descends to a closed subgroup scheme $Z \subset G$ by Descent, Lemma 34.34.2 (and Descent, Lemma 34.20.19). Then $Z$ represents $C$ (small argument omitted) and the proof is complete. $\square$

1. Using the material in Divisors, Section 30.17 we could take as effective Cartier divisor $E$ the norm of the effective Cartier divisor $\mathcal{D}$ along the finite locally free morphism $1 \times \pi'$ bypassing some of the arguments.

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\section{Properties of algebraic group schemes}
\label{section-algebraic-group-schemes}

\noindent
Recall that a scheme over a field $k$ is (locally) algebraic if it is
(locally) of finite type over $\Spec(k)$, see
Varieties, Definition \ref{varieties-definition-algebraic-scheme}.
This is the sense of algebraic we are using in the title of this section.

\begin{lemma}
\label{lemma-group-scheme-finite-type-field}
Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.
Then $G$ is equidimensional and $\dim(G) = \dim_g(G)$ for all $g \in G$.
For any closed point $g \in G$ we have $\dim(G) = \dim(\mathcal{O}_{G, g})$.
\end{lemma}

\begin{proof}
Let us first prove that $\dim_g(G) = \dim_{g'}(G)$ for any
pair of points $g, g' \in G$. By
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-after-base-change}
we may extend the ground field at will. Hence we may assume that
both $g$ and $g'$ are defined over $k$. Hence there exists an
automorphism of $G$ mapping $g$ to $g'$, whence the equality.
By
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}
we have
$\dim_g(G) = \dim(\mathcal{O}_{G, g}) + \text{trdeg}_k(\kappa(g))$.
On the other hand, the dimension of $G$ (or any open subset of $G$)
is the supremum of the dimensions of the local rings of $G$, see
Properties, Lemma \ref{properties-lemma-codimension-local-ring}.
Clearly this is maximal for closed points $g$ in which case
$\text{trdeg}_k(\kappa(g)) = 0$ (by the Hilbert Nullstellensatz, see
Morphisms, Section \ref{morphisms-section-points-finite-type}).
Hence the lemma follows.
\end{proof}

\noindent
The following result is sometimes referred to as Cartier's theorem.

\begin{lemma}
\label{lemma-group-scheme-characteristic-zero-smooth}
Let $k$ be a field of characteristic $0$. Let $G$ be a
locally algebraic group scheme over $k$. Then the structure
morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth
group scheme.
\end{lemma}

\begin{proof}
By
Lemma \ref{lemma-group-scheme-module-differentials}
the module of differentials of $G$ over $k$ is free.
Hence smoothness follows from
Varieties, Lemma \ref{varieties-lemma-char-zero-differentials-free-smooth}.
\end{proof}

\begin{remark}
\label{remark-when-reduced}
Any group scheme over a field of characteristic $0$ is reduced, see
\cite[I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4]{Perrin-thesis}
and also
\cite[Proposition 4.2.8]{Perrin}.
This was a question raised in
\cite[page 80]{Oort}.
We have seen in
Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}
that this holds when the group scheme is locally of finite type.
\end{remark}

\begin{lemma}
\label{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}
Let $k$ be a perfect field of characteristic $p > 0$ (see
Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}
for the characteristic zero case).
Let $G$ be a locally algebraic group scheme over $k$.
If $G$ is reduced then the structure
morphism $G \to \Spec(k)$ is smooth, i.e., $G$ is a smooth
group scheme.
\end{lemma}

\begin{proof}
By
Lemma \ref{lemma-group-scheme-module-differentials}
the sheaf $\Omega_{G/k}$ is free. Hence the lemma follows from
Varieties, Lemma \ref{varieties-lemma-char-p-differentials-free-smooth}.
\end{proof}

\begin{remark}
\label{remark-reduced-smooth-not-true-general}
Let $k$ be a field of characteristic $p > 0$.
Let $\alpha \in k$ be an element which is not a $p$th power.
The closed subgroup scheme
$$G = V(x^p + \alpha y^p) \subset \mathbf{G}_{a, k}^2$$
is reduced and irreducible but not smooth (not even normal).
\end{remark}

\noindent
The following lemma is a special case of
Lemma \ref{lemma-compact-set-in-affine}
with a somewhat easier proof.

\begin{lemma}
\label{lemma-points-in-affine}
Let $k$ be an algebraically closed field.
Let $G$ be a locally algebraic group scheme over $k$.
Let $g_1, \ldots, g_n \in G(k)$ be $k$-rational points.
Then there exists an affine open $U \subset G$ containing $g_1, \ldots, g_n$.
\end{lemma}

\begin{proof}
We first argue by induction on $n$ that we may assume all $g_i$ are
on the same connected component of $G$. Namely, if not, then we can
find a decomposition $G = W_1 \amalg W_2$ with $W_i$ open in $G$ and
(after possibly renumbering) $g_1, \ldots, g_r \in W_1$ and
$g_{r + 1}, \ldots, g_n \in W_2$ for some $0 < r < n$. By
induction we can find affine opens $U_1$ and $U_2$ of $G$ with
$g_1, \ldots, g_r \in U_1$ and $g_{r + 1}, \ldots, g_n \in U_2$.
Then
$$g_1, \ldots, g_n \in (U_1 \cap W_1) \cup (U_2 \cap W_2)$$
is a solution to the problem. Thus we may assume $g_1, \ldots, g_n$
are all on the same connected component of $G$. Translating by $g_1^{-1}$
we may assume $g_1, \ldots, g_n \in G^0$ where $G^0 \subset G$ is as in
Proposition \ref{proposition-connected-component}. Choose an affine
open neighbourhood $U$ of $e$, in particular $U \cap G^0$ is nonempty.
Since $G^0$ is irreducible we see that
$$G^0 \cap (Ug_1^{-1} \cap \ldots \cap Ug_n^{-1})$$
is nonempty. Since $G \to \Spec(k)$ is locally of finite type, also
$G^0 \to \Spec(k)$ is locally of finite type, hence any nonempty
open has a $k$-rational point. Thus we can pick $g \in G^0(k)$ with
$g \in Ug_i^{-1}$ for all $i$. Then $g_i \in g^{-1}U$ for all $i$
and $g^{-1}U$ is the affine open we were looking for.
\end{proof}

\begin{lemma}
\label{lemma-algebraic-quasi-projective}
Let $k$ be a field. Let $G$ be an algebraic group scheme over $k$.
Then $G$ is quasi-projective over $k$.
\end{lemma}

\begin{proof}
By Varieties, Lemma \ref{varieties-lemma-ample-after-field-extension}
we may assume that $k$ is algebraically closed. Let $G^0 \subset G$
be the connected component of $G$ as in
Proposition \ref{proposition-connected-component}.
Then every other connected component of $G$ has a $k$-rational
point and hence is isomorphic to $G^0$ as a scheme.
Since $G$ is quasi-compact and Noetherian, there are finitely many of these
connected components. Thus we reduce to the case discussed in
the next paragraph.

\medskip\noindent
Let $G$ be a connected algebraic group scheme over an algebraically closed
field $k$. If the characteristic of $k$ is zero, then $G$ is smooth over
$k$ by Lemma \ref{lemma-group-scheme-characteristic-zero-smooth}.
If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$
be the reduction of $G$. By
Divisors, Proposition \ref{divisors-proposition-push-down-ample}
it suffices to show that $H$ has an ample invertible sheaf.
(For an algebraic scheme over $k$ having an ample invertible
sheaf is equivalent to being quasi-projective over $k$, see
for example the very general
More on Morphisms, Lemma \ref{more-morphisms-lemma-quasi-projective}.)
By Lemma \ref{lemma-reduced-subgroup-scheme-perfect}
we see that $H$ is a group scheme over $k$.
By Lemma \ref{lemma-reduced-group-scheme-prefect-field-characteristic-p-smooth}
we see that $H$ is smooth over $k$.
This reduces us to the situation discussed in the next
paragraph.

\medskip\noindent
Let $G$ be a quasi-compact irreducible smooth group scheme over an
algebraically closed field $k$. Observe that the local rings of $G$
are regular and hence UFDs
(Varieties, Lemma \ref{varieties-lemma-smooth-regular} and
More on Algebra, Lemma \ref{more-algebra-lemma-regular-local-UFD}).
The complement of a nonempty affine open of $G$
is the support of an effective Cartier divisor $D$.
This follows from Divisors, Lemma
\ref{divisors-lemma-complement-open-affine-effective-cartier-divisor}.
(Observe that $G$ is separated by
Lemma \ref{lemma-group-scheme-over-field-separated}.)
We conclude there exists an effective Cartier divisor $D \subset G$
such that $G \setminus D$ is affine. We will use below that
for any $n \geq 1$ and $g_1, \ldots, g_n \in G(k)$ the complement
$G \setminus \bigcup D g_i$ is affine. Namely, it is the intersection
of the affine opens $G \setminus Dg_i \cong G \setminus D$
in the separated scheme $G$.

\medskip\noindent
We may choose the top row of the diagram
$$\xymatrix{ G & U \ar[l]_j \ar[r]^\pi & \mathbf{A}^d_k \\ & W \ar[r]^{\pi'} \ar[u] & V \ar[u] }$$
such that $U \not = \emptyset$, $j : U \to G$ is an open immersion, and
$\pi$ is \'etale, see
Morphisms, Lemma \ref{morphisms-lemma-smooth-etale-over-affine-space}.
There is a nonempty affine open $V \subset \mathbf{A}^d_k$ such that
with $W = \pi^{-1}(V)$ the morphism $\pi' = \pi|_W : W \to V$ is finite \'etale.
In particular $\pi'$ is finite locally free, say of degree $n$.
Consider the effective Cartier divisor
$$\mathcal{D} = \{(g, w) \mid m(g, j(w)) \in D\} \subset G \times W$$
(This is the restriction to $G \times W$ of the pullback of $D \subset G$
under the flat morphism $m : G \times G \to G$.)
Consider the closed subset\footnote{Using the material
in Divisors, Section \ref{divisors-section-norms}
we could take as effective Cartier
divisor $E$ the norm of the effective Cartier divisor $\mathcal{D}$
along the finite locally free morphism $1 \times \pi'$ bypassing
some of the arguments.}
$T = (1 \times \pi')(\mathcal{D}) \subset G \times V$.
Since $\pi'$ is finite locally free, every irreducible component
of $T$ has codimension $1$ in $G \times V$. Since $G \times V$
is smooth over $k$ we conclude these components are effective Cartier
divisors (Divisors, Lemma \ref{divisors-lemma-weil-divisor-is-cartier-UFD}
and lemmas cited above)
and hence $T$ is the support of an effective Cartier divisor
$E$ in $G \times V$. If $v \in V(k)$, then
$(\pi')^{-1}(v) = \{w_1, \ldots, w_n\} \subset W(k)$ and we see that
$$E_v = \bigcup\nolimits_{i = 1, \ldots, n} D j(w_i)^{-1}$$
in $G$ set theoretically. In particular we see that $G \setminus E_v$
is affine open (see above).
Moreover, if $g \in G(k)$, then there exists a $v \in V$
such that $g \not \in E_v$. Namely, the set $W'$ of $w \in W$ such that
$g \not \in Dj(w)^{-1}$ is nonempty open and it suffices to pick $v$
such that the fibre of $W' \to V$ over $v$ has $n$ elements.

\medskip\noindent
Consider the invertible sheaf
$\mathcal{M} = \mathcal{O}_{G \times V}(E)$ on $G \times V$.
By Varieties, Lemma \ref{varieties-lemma-rational-equivalence-for-Pic}
the isomorphism class $\mathcal{L}$ of the restriction
$\mathcal{M}_v = \mathcal{O}_G(E_v)$ is independent of $v \in V(k)$.
On the other hand, for every $g \in G(k)$ we can find a $v$
such that $g \not \in E_v$ and such that $G \setminus E_v$
is affine. Thus the canonical section
(Divisors, Definition
\ref{divisors-definition-invertible-sheaf-effective-Cartier-divisor})
of $\mathcal{O}_G(E_v)$
corresponds to a section $s_v$ of $\mathcal{L}$ which does not
vanish at $g$ and such that $G_{s_v}$ is affine.
This means that $\mathcal{L}$ is ample by definition
(Properties, Definition \ref{properties-definition-ample}).
\end{proof}

\begin{lemma}
\label{lemma-algebraic-center}
Let $k$ be a field. Let $G$ be a locally algebraic group scheme over $k$.
Then the center of $G$ is a closed subgroup scheme of $G$.
\end{lemma}

\begin{proof}
Let $\text{Aut}(G)$ denote the contravariant functor on the category of
schemes over $k$ which associates to $S/k$ the set of automorphisms
of the base change $G_S$ as a group scheme over $S$. There is a natural
transformation
$$G \longrightarrow \text{Aut}(G),\quad g \longmapsto \text{inn}_g$$
sending an $S$-valued point $g$ of $G$ to the inner automorphism of $G$
determined by $g$. The center $C$ of $G$ is by definition the kernel of
this transformation, i.e., the functor which to $S$ associates those
$g \in G(S)$ whose associated inner automorphism is trivial. The statement
of the lemma is that this functor is representable by a closed subgroup
scheme of $G$.

\medskip\noindent
Choose an integer $n \geq 1$. Let $G_n \subset G$ be the $n$th infinitesimal
neighbourhood of the identity element $e$ of $G$. For every scheme $S/k$
the base change $G_{n, S}$ is the $n$th infinitesimal neighbourhood of
$e_S : S \to G_S$. Thus we see that there is a natural transformation
$\text{Aut}(G) \to \text{Aut}(G_n)$ where the right hand side is the
functor of automorphisms of $G_n$ as a scheme ($G_n$ isn't in general
a group scheme). Observe that $G_n$ is the spectrum of an artinian
local ring $A_n$ with residue field $k$ which has finite dimension
as a $k$-vector space
(Varieties, Lemma \ref{varieties-lemma-algebraic-scheme-dim-0}).
Since every automorphism of $G_n$ induces in particular an invertible
linear map $A_n \to A_n$, we obtain transformations of functors
$$G \to \text{Aut}(G) \to \text{Aut}(G_n) \to \text{GL}(A_n)$$
The final group valued functor is representable, see
Example \ref{example-general-linear-group}, and the
last arrow is visibly injective.
Thus for every $n$ we obtain a closed subgroup scheme
$$H_n = \Ker(G \to \text{Aut}(G_n)) = \Ker(G \to \text{GL}(A_n)).$$
As a first approximation we set $H = \bigcap_{n \geq 1} H_n$
(scheme theoretic intersection). This is a closed subgroup scheme
which contains the center $C$.

\medskip\noindent
Let $h$ be an $S$-valued point of $H$ with $S$ locally Noetherian.
Then the automorphism $\text{inn}_h$ induces the identity on all
the closed subschemes $G_{n, S}$. Consider the kernel
$K = \Ker(\text{inn}_h : G_S \to G_S)$.
This is a closed subgroup scheme of $G_S$ over $S$ containing the
closed subschemes $G_{n, S}$ for $n \geq 1$.
This implies that $K$ contains an open neighbourhood of
$e(S) \subset G_S$, see
Algebra, Remark \ref{algebra-remark-intersection-powers-ideal}.
Let $G^0 \subset G$ be as in Proposition \ref{proposition-connected-component}.
Since $G^0$ is geometrically irreducible, we conclude that $K$ contains
$G^0_S$ (for any nonempty open $U \subset G^0_{k'}$ and any field extension
$k'/k$ we have $U \cdot U^{-1} = G^0_{k'}$, see proof of
Lemma \ref{lemma-irreducible-group-scheme-over-field-qc}).
Applying this with $S = H$ we find that $G^0$ and $H$
are subgroup schemes of $G$ whose points commute: for any scheme $S$
and any $S$-valued points $g \in G^0(S)$, $h \in H(S)$ we have
$gh = hg$ in $G(S)$.

\medskip\noindent
Assume that $k$ is algebraically closed. Then we can pick a $k$-valued
point $g_i$ in each irreducible component $G_i$ of $G$. Observe that in
this case the connected components of $G$ are the irreducible components
of $G$ are the translates of $G^0$ by our $g_i$. We claim that
$$C = H \cap \bigcap\nolimits_i \Ker(\text{inn}_{g_i} : G \to G) \quad (\text{scheme theoretic intersection})$$
Namely, $C$ is contained in the right hand side. On the other hand, every
$S$-valued point $h$ of the right hand side commutes with $G^0$
and with $g_i$ hence with everything in $G = \bigcup G^0g_i$.

\medskip\noindent
The case of a general base field $k$ follows from the result for the
algebraic closure $\overline{k}$ by descent. Namely, let
$A \subset G_{\overline{k}}$ the closed subgroup scheme representing
the center of $G_{\overline{k}}$. Then we have
$$A \times_{\Spec(k)} \Spec(\overline{k}) = \Spec(\overline{k}) \times_{\Spec(k)} A$$
as closed subschemes of $G_{\overline{k} \otimes_k \overline{k}}$ by
the functorial nature of the center. Hence we see that $A$ descends
to a closed subgroup scheme $Z \subset G$ by
Descent, Lemma \ref{descent-lemma-closed-immersion}
(and Descent, Lemma \ref{descent-lemma-descending-property-closed-immersion}).
Then $Z$ represents $C$ (small argument omitted) and the proof is complete.
\end{proof}

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