The Stacks project

Proof. Let us first prove that $\dim _ g(G) = \dim _{g'}(G)$ for any pair of points $g, g' \in G$. By Morphisms, Lemma 29.28.3 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 29.28.1 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 28.10.3. Clearly this is maximal for closed points $g$ in which case $\text{trdeg}_ k(\kappa (g)) = 0$ (by the Hilbert Nullstellensatz, see Morphisms, Section 29.16). Hence the lemma follows. $\square$

Proof. By Lemma 39.6.3 the module of differentials of $G$ over $k$ is free. Hence smoothness follows from Varieties, Lemma 33.25.1. $\square$

Proof. By Lemma 39.6.3 the sheaf $\Omega _{G/k}$ is free. Hence the lemma follows from Varieties, Lemma 33.25.2. $\square$

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

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 39.7.11. Choose an affine open neighbourhood $U$ of $e$, in particular $U \cap G^0$ is nonempty. Since $G^0$ is irreducible we see that

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. $\square$

Proof. By Varieties, Lemma 33.15.1 we may assume that $k$ is algebraically closed. Let $G^0 \subset G$ be the connected component of $G$ as in Proposition 39.7.11. 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 39.8.2. If the characteristic of $k$ is $p > 0$, then we let $H = G_{red}$ be the reduction of $G$. By Divisors, Proposition 31.17.9 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 37.49.1.) By Lemma 39.7.6 we see that $H$ is a group scheme over $k$. By Lemma 39.8.4 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 33.25.3 and More on Algebra, Lemma 15.121.2). The complement of a nonempty affine open of $G$ is the support of an effective Cartier divisor $D$. This follows from Divisors, Lemma 31.16.6. (Observe that $G$ is separated by Lemma 39.7.3.) 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

such that $U \not= \emptyset $, $j : U \to G$ is an open immersion, and $\pi $ is étale, see Morphisms, Lemma 29.36.20. 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

(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 31.15.7 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

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 33.30.5 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 31.14.1) 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 28.26.1). $\square$

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

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 33.20.2). Since every automorphism of $G_ n$ induces in particular an invertible linear map $A_ n \to A_ n$, we obtain transformations of functors

The final group valued functor is representable, see Example 39.5.4, and the last arrow is visibly injective. Thus for every $n$ we obtain a closed subgroup scheme

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 10.51.6. Let $G^0 \subset G$ be as in Proposition 39.7.11. 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 39.7.9). 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

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

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 35.37.2 (and Descent, Lemma 35.23.19). Then $Z$ represents $C$ (small argument omitted) and the proof is complete. $\square$