The Stacks project

Lemma 39.8.8. 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 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

\[ G \to \text{Aut}(G) \to \text{Aut}(G_ n) \to \text{GL}(A_ n) \]

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

\[ H_ n = \mathop{\mathrm{Ker}}(G \to \text{Aut}(G_ n)) = \mathop{\mathrm{Ker}}(G \to \text{GL}(A_ n)). \]

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 10.51.6. 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

\[ C = H \cap \bigcap \nolimits _ i \mathop{\mathrm{Ker}}(\text{inn}_{g_ i} : G \to G) \quad (\text{scheme theoretic intersection}) \]

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


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BF8. Beware of the difference between the letter 'O' and the digit '0'.