Lemma 39.8.1. 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}).
39.8 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 33.20.1. This is the sense of algebraic we are using in the title of this section.
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
The following result is sometimes referred to as Cartier's theorem.
Lemma 39.8.2. 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 39.6.3 the module of differentials of G over k is free. Hence smoothness follows from Varieties, Lemma 33.25.1. \square
Remark 39.8.3. Any group scheme over a field of characteristic 0 is reduced, see [I, Theorem 1.1 and I, Corollary 3.9, and II, Theorem 2.4, Perrin-thesis] and also [Proposition 4.2.8, Perrin]. This was a question raised in [page 80, Oort]. We have seen in Lemma 39.8.2 that this holds when the group scheme is locally of finite type.
Lemma 39.8.4. Let k be a perfect field of characteristic p > 0 (see Lemma 39.8.2 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 39.6.3 the sheaf \Omega _{G/k} is free. Hence the lemma follows from Varieties, Lemma 33.25.2. \square
Remark 39.8.5. Let k be a field of characteristic p > 0. Let \alpha \in k be an element which is not a pth power. The closed subgroup scheme
is reduced and irreducible but not smooth (not even normal).
The following lemma is a special case of Lemma 39.7.13 with a somewhat easier proof.
Lemma 39.8.6. 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
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
Lemma 39.8.7. 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 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
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
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 nth infinitesimal neighbourhood of the identity element e of G. For every scheme S/k the base change G_{n, S} is the nth 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
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