Proof. This is [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XXIV, Corollaire 7.2.3].

Proof. Let $U \subset X$ be a quasi-compact open subscheme; to show that X is ind-quasi-affine, we must show that U is quasi-affine. Note that $\pi ^{-1}(U) \subset Y$ is quasi-compact, so it is contained in an affine clopen subscheme $V \subset Y$ by assumption. By Chevalley’s theorem, that affineness can be checked after passing to a finite cover [Reference Dieudonne and GrothendieckDG67, II, Théorème 6.7.1], the closed subset $\pi (V) \subset X$ is affine (when considered with its reduced subscheme structure). The schematic closure $\overline {U}$ of U in X is a closed subset of $\pi (V)$ , so $\overline {U}_{\mathrm {red}}$ is affine. Thus, by Chevalley’s theorem again, $\overline {U}$ is affine. Since U is open in $\overline {U}$ , it follows that U is quasi-affine.

If $\pi $ is open, then the closed subset $\pi (V) \subset X$ is also open, so it is affine when considered with the structure of an open subscheme of X.

Proof. First, smoothness of $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is proved in [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XI, Corollaire 4.2]. If S is normal and locally noetherian, then the result follows from [Reference RaynaudRay70, Théorème IX 2.6]. For the remainder, we therefore assume that S is quasi-compact and quasi-separated. By spreading out (using [Reference Thomason and TrobaughTT90, Theorem C.9]), we may assume that S is of finite type over $\operatorname {\mathrm {Spec}} \mathbf {Z}$ . In particular, the normalization $S' \to S$ is finite. Now $\underline {\operatorname {\mathrm {Hom}}}_{S'\textrm {-}\mathrm {gp}}(H_{S'}, G_{S'})$ is a disjoint union of affine S-schemes, and the morphism $\underline {\operatorname {\mathrm {Hom}}}_{S'\textrm {-}\mathrm {gp}}(H_{S'}, G_{S'}) \to \underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is finite and surjective, so ind-quasi-affineness of $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ follows from Lemma 2.5.

Proof. To show ind-quasi-affineness, we may work locally on S and spread out to assume that S is affine, noetherian and excellent. (For the second claim, we may use [Reference Thomason and TrobaughTT90, Theorem C.9] to make the same reduction.) Using Lemma 2.5, we may also pass from S to its normalization to assume that S is normal. Passing to a further étale cover, we may and do assume that H admits a maximal S-torus. By [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XXIV, Corollaire 7.1.9], the natural restriction map

$$\begin{align*}\underline{\operatorname{\mathrm{Hom}}}_{S\textrm{-}\mathrm{gp}}(H, G) \to \underline{\operatorname{\mathrm{Hom}}}_{S\textrm{-}\mathrm{gp}}(T, G) \end{align*}$$

is finitely presented and affine, so the result follows from Lemma 2.6. (Note that a smooth affine S-scheme is automatically finitely presented.)

Proof of Theorem 2.2.

Recall that H is now assumed to be a geometrically reductive smooth affine S-group scheme. In particular, $H^0$ is a reductive group scheme and $H/H^0$ is finite by Theorem 2.1 (and similarly for G). There are two issues: first, we need to show that $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is representable, and then we need to show that if S is normal, quasi-compact and quasi-separated, and H admits a maximal S-torus, then $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is a disjoint union of finitely presented S-affine S-schemes (at which point ind-quasi-affineness in general follows from Lemma 2.5). For both points, by working locally and spreading out, we may assume that S is noetherian and connected.

To prove representability, first assume that $H/H^0$ is constant and that the natural map $H(S) \to (H/H^0)(S)$ is surjective. Let $h_1, \dots , h_n \in H(S)$ be a system of representatives for $(H/H^0)(S)$ . We may and do assume $h_1 = 1$ . There is then a natural morphism of functors

$$\begin{align*}\beta: \underline{\operatorname{\mathrm{Hom}}}_{S\textrm{-}\mathrm{gp}}(H, G) \to \underline{\operatorname{\mathrm{Hom}}}_{S\textrm{-}\mathrm{gp}}(H^0, G) \times G^n, \end{align*}$$

given by $f \mapsto (f|_{H^0}, f(h_1), \dots , f(h_n))$ . We claim that $\beta $ is a closed embedding. To this end, we need to understand when a tuple $(f_0, g_1, \dots , g_n)$ in $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H^0, G)(S') \times G(S')^n$ lies in the image of $\beta $ .

For indices $i, j$ , let $h_i h_j = h_{\delta (i, j)} h_{i,j}$ , where $1 \leq \delta (i, j) \leq n$ and $h_{i,j} \in H^0(S)$ . In any case, there is a unique morphism of S-schemes $f: H \to G$ with $f|_{H^0} = f_0$ and $f(h_i) = g_i$ for all i: for this, note that for any S-scheme $S'$ and any $h \in H(S')$ , there is a unique open decomposition $S' = \bigsqcup _{i=1}^n S^{\prime }_i$ such that $h|_{S^{\prime }_i} = h_i h^{\prime }_i$ for some $h^{\prime }_i \in H^0(S^{\prime }_i)$ . Thus, f is defined uniquely by requiring $f(h_i h') = g_i f_0(h')$ for every S-scheme $S'$ and every $h' \in H^0(S')$ . Now $(f_0, g_1, \dots , g_n)$ lies in the image of $\beta $ if and only if the above-defined f is a homomorphism.

Unraveling, we find that f is a homomorphism if and only if the following conditions are satisfied:

1. 1. $g_1 = 1$ ,

2. 2. $g_i^{-1} f_0(h) g_i = f_0(h_i^{-1} h h_i)$ for all i,

3. 3. $g_i g_j = g_{\delta (i, j)} f_0(h_{i,j})$ .

So indeed, $\beta $ is a finitely presented closed embedding, whence $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is representable, and in fact, it is ind-quasi-affine over S by Lemma 2.7. Moreover, if S is normal, then this shows that $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is a disjoint union of finitely presented S-affine S-schemes.

Now pass to the general case (i.e., no longer assume that $H/H^0$ is constant and that $H(S) \to (H/H^0)(S)$ is surjective). In any case, there is a finite étale cover $S' \to S$ such that $H_{S'}/H_{S'}^0$ is constant and $H(S') \to (H/H^0)(S')$ is surjective (e.g., take $S'$ to be a Galois closure of the finite étale $H/H^0$ ), and so $\underline {\operatorname {\mathrm {Hom}}}_{S'\textrm {-}\mathrm {gp}}(H_{S'}, G_{S'})$ is ind-quasi-affine over $S'$ by the above. Thus, by effectivity of fpqc descent for ind-quasi-affine morphisms [Sta21, Tag 0APK], we see that $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is representable and locally of finite presentation. Now that we have representability, we may assume that S is normal, quasi-compact and quasi-separated. As we have already seen, $\underline {\operatorname {\mathrm {Hom}}}_{S'\textrm {-}\mathrm {gp}}(H_{S'}, G_{S'})$ is representable by a disjoint union of finitely presented $S'$ -affine $S'$ -schemes, so because the morphism $\underline {\operatorname {\mathrm {Hom}}}_{S'\textrm {-}\mathrm {gp}}(H_{S'}, G_{S'}) \to \underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is finite étale, the result follows from Lemma 2.5.

Proof. It is clear that $\iota $ is monic, and the final claim follows from the others by [Reference Dieudonne and GrothendieckDG67, IV₃, Proposition 8.11.5] and Theorem 2.2, which shows that every connected component of $\underline {\operatorname {\mathrm {Hom}}}_{k\textrm {-}\mathrm {gp}}(H, G)$ is of finite type over k. We now verify that $\iota $ satisfies the valuative criterion of properness. Let A be a k-algebra which is a DVR with fraction field K, and let $(g_1, \dots , g_n) \in G(A)^n$ be such that there exists a K-homomorphism $f_1: H_K \to G_K$ satisfying $f_1(x_i) = g_i$ for all i. Let now $\Gamma \subset H \times _{\operatorname {\mathrm {Spec}} A} G$ be the schematic closure of the graph of $f_1$ , so that $\Gamma $ is a flat closed A-subgroup scheme of $H \times _{\operatorname {\mathrm {Spec}} A} G$ whose projection map $\pi _1$ to H is an isomorphism on generic fibers over A. Moreover, since $(x_i, g_i) \in H(A) \times G(A)$ for all i, it follows that $(x_i, g_i) \in \Gamma (A)$ , and thus, $\pi _{1,s}: \Gamma _s \to H_s$ is surjective. Since $H_s$ is smooth, we see that $\pi _{1, s}$ is flat, and by fibral flatness, it follows that $\pi _1$ is flat. Since $(\ker \pi _1)_K = \{1\}$ , it follows from flatness that $\ker \pi _1 = \{1\}$ . Thus, $\pi _{1, s}$ is a closed embedding, and since $\pi _{1, s}$ is surjective and $H_s$ is smooth, it follows that $\pi _{1, s}$ is an isomorphism. By the fibral isomorphism criterion, $\pi _1$ is therefore an isomorphism, and it is the graph of an A-homomorphism $f: H \to G$ whose generic fiber is $f_1$ . This verifies the valuative criterion.

Proof. We may and do pass to a (possibly transcendental) field extension of k to assume that there exist $x_1, \dots , x_n \in H(k)$ which topologically generate H. If $f(H)$ is G-cr, then by [Reference Bate, Martin and RöhrleBMR05, Proposition 2.16, Theorem 3.1], the G-orbit of $(f(x_1), \dots , f(x_n))$ in $G^n$ is closed. Thus, the G-orbit of f is closed in $\underline {\operatorname {\mathrm {Hom}}}_{k\textrm {-}\mathrm {gp}}(H, G)$ since this orbit is simply the preimage of the G-orbit of $(f(x_1), \dots , f(x_n))$ under the map $\iota $ of Lemma 2.10. Conversely, if the G-orbit of f is closed, then $\iota (G\cdot f)$ is closed in $G^n$ by Lemma 2.10, and we conclude with the fact that $\iota (G \cdot f)$ is simply the G-orbit of $(f(x_1), \dots , f(x_n))$ .

Proof. If f is an isomorphism, then certainly $\ker f = \{1\}$ . If $\ker f_s = \{1\}$ for all $s \in S$ , then $\ker f = \{1\}$ : indeed, the identity section $e: S \to \ker f$ is a morphism of S-schemes which is an isomorphism on fibers over S, so because S is S-flat, it follows from the fibral isomorphism criterion [Reference Dieudonne and GrothendieckDG67, IV₄, Corollaire 17.9.5] that e is an isomorphism (i.e., $\ker f = \{1\}$ ). Thus, f is monic, and it follows from the Ax–Grothendieck theorem [Reference Dieudonne and GrothendieckDG67, IV₄, Proposition 17.9.6] that f is an isomorphism.

Proof. Clearly, (3) implies (1) and (2). Conversely, if $f_s$ and $f_\eta $ are both isomorphisms, then f is an isomorphism by the fibral isomorphism criterion [Reference Dieudonne and GrothendieckDG67, IV₄, Corollaire 17.9.5]. Thus, it suffices to show that (1) and (2) are equivalent. We may and do further assume that A is complete with algebraically closed residue field.

First, suppose that G (and hence also H under either (1) or (2), due to finiteness of $H/H^0$ ) has connected fibers. Assume $f_s$ is an isomorphism. By the fibral isomorphism criterion, $f_{A/\mathfrak {m}^n}$ is an isomorphism for all $n \geq 1$ . By the local flatness criterion [Reference MatsumuraMat89, Theorem 22.3], f is flat. In fact, f is étale near $1$ because the étale locus is open, and since it is a homomorphism with G fppf over A, it follows that f is étale. In particular, $\ker f$ is étale. One can check that an étale normal subgroup scheme of a connected group scheme over a field is automatically central, so $\ker f_\eta $ is central, and thus, $\ker f$ is contained in $Z(G)$ . But $Z(G)$ is of multiplicative type, and $\ker f$ is a flat closed A-subgroup scheme of $Z(G)$ , so it is also of multiplicative type by [Reference ConradCon14, Corollary B.3.3]. Since $\ker f_s = \{1\}$ , we conclude that $\ker f_\eta = \{1\}$ , and hence, $f_\eta $ is a closed embedding by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. VI_B, Corollaire 1.4.2]. For dimension reasons, $f_\eta $ is dominant, so it is surjective by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. VI_B, Proposition 1.2], and hence, it is an isomorphism since $H_\eta $ is smooth.

Conversely, suppose that $f_\eta $ is an isomorphism. If $g \in G(k(s))$ is a nontrivial semisimple element, then there is a maximal $k(s)$ -torus $T_0 \subset G_s$ such that $g \in T_0(k(s))$ . By [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. IX, Théorème 3.6], since A is complete, we may find a maximal A-torus $T \subset G$ with special fiber $T_0$ . Since $\ker f|_{T_\eta } = \{1\}$ , it follows from [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. IX, Théorème 6.8] that $\ker f|_T = \{1\}$ , so in particular, $g \not \in (\ker f)(k(s))$ . Note that $(\ker f_s)^0_{\mathrm {{red}}}$ is a smooth connected closed normal subgroup of $G_s$ , so it is reductive, and thus, its semisimple locus is dense. But the above argument shows that its semisimple locus is $\{1\}$ , so in fact, $(\ker f_s)^0_{\mathrm {{red}}} = \{1\}$ , whence $\ker f_s$ is finite. Thus, for dimension reasons, $f_s$ is dominant, and since G is smooth, we find that $f_s$ is flat: this follows from generic flatness and a simple translation argument. By the fibral flatness criterion [Reference Dieudonne and GrothendieckDG67, IV₃, Théorème 11.3.10], f is flat, and so $\ker f$ is flat. Since $\ker f_\eta = \{1\}$ , it follows that $\ker f = \{1\}$ : as a flat closed subscheme of G, $\ker f$ is the closure of its generic fiber. So $f_s$ is a surjective closed embedding, and hence an isomorphism.

Now consider the general case – namely, that $G/G^0$ is finite. Suppose that either $f_s$ or $f_\eta $ is an isomorphism. By the reductive case settled above, we find that $f|_{G^0}: G^0 \to H^0$ is an isomorphism. Moreover, the homomorphism $G/G^0 \to H/H^0$ between constant groups is an isomorphism, since this can be checked on either the special or generic fiber. Thus, a diagram chase shows that f is an isomorphism.

Proof. By spreading out, we may and do assume that S is noetherian, so that is locally noetherian by Lemma 2.3. Note that there is a universal S-homomorphism

$$\begin{align*}f: G \times_S \mathscr{H} \to G \times_S \mathscr{H} \end{align*}$$

whose fiber over a given section of $\mathscr {H}$ is the corresponding endomorphism of G. By Lemmas 2.13 and 2.14, since $\mathscr {H}$ is locally noetherian, the locus U of $u \in \mathscr {H}$ such that $\ker f_u = \{1\}$ is open and closed, and $f_U: G \times _S U \to G \times _S U$ is an isomorphism. It follows from this reasoning that U represents $\underline {\operatorname {\mathrm {Aut}}}_{G/S}$ .

Proof. Since formation of $(\operatorname {\mathrm {Aut}}_{G/k})_{\mathrm {{red}}}$ commutes with separable field extensions on k and purely inseparable extensions leave topological spaces unchanged, we may and do assume that k is algebraically closed. To show that $\phi $ is flat, it suffices to show that the map $G^0(k) \to (\operatorname {\mathrm {Aut}}_{G/k})^0(k)$ is surjective. Indeed, then $\phi $ is a dominant map from a smooth finite type k-scheme to a smooth k-scheme, so it is generically flat, and being a group homomorphism translation, arguments show that it is flat.

Let $r: \operatorname {\mathrm {Aut}}_{G/k} \to \operatorname {\mathrm {Aut}}_{G^0/k} \times \operatorname {\mathrm {Aut}}_{(G/G^0)/k}$ denote the natural restriction homomorphism. The map $G^0 \to \operatorname {\mathrm {Aut}}_{G/k}$ induces a map $Z(G^0) \to \ker r$ . We claim that the image of $Z(G^0)$ in $\ker r$ is of finite index. Once this is done, the lemma will follow: indeed, then

$$\begin{align*}\dim \ker r|_{\phi(G^0)} = \dim \ker r|_{\phi(Z(G^0))} = \dim \ker r. \end{align*}$$

Since the group scheme of outer automorphisms of $G^0$ is etale, $\mathrm {pr}_1 \circ r: G^0 \to \operatorname {\mathrm {Aut}}_{G^0/k}^0$ is surjective. Since $\operatorname {\mathrm {Aut}}_{(G/G^0)/k}$ is finite, we find in particular that $\dim r(\phi (G)) = \dim r(\operatorname {\mathrm {Aut}}_{G/k})$ , so

$$\begin{align*}\dim \phi(G^0) = \dim \ker r + \dim r(\operatorname{\mathrm{Aut}}_{G/k}) = \dim \operatorname{\mathrm{Aut}}_{G/k}\!. \end{align*}$$

Therefore, $\phi (G^0)$ is an open subgroup scheme of the smooth k-group scheme $(\operatorname {\mathrm {Aut}}_{G/k})_{\mathrm {{red}}}$ , so it contains $(\operatorname {\mathrm {Aut}}_{G/k})^0$ and hence is equal to it.

So it remains to show that the image of $Z(G^0)$ of finite index in $(\ker r)(k)$ . First, we must understand some properties of $\operatorname {\mathrm {Aut}}_{G/k}$ . Suppose that $f: G \to G$ is a k-homomorphism inducing the identity on $G^0$ and $G/G^0$ (i.e., $f \in (\ker r)(k)$ ). We define a morphism $\lambda : G \to G$ by

$$\begin{align*}\lambda(g) = f(g)g^{-1}. \end{align*}$$

One checks that $\lambda $ is a $1$ -cocycle. Since $f|_{G^0}$ is the identity, we have

$$\begin{align*}\lambda(gh) = f(gh)(gh)^{-1} = f(g)f(h)h^{-1}g^{-1} = \lambda(g) \end{align*}$$

for all functorial points g and h of G and $G^0$ , respectively. Thus, $\lambda $ factors through a morphism $G/G^0 \to G$ . Moreover, for such g and h, we have

$$\begin{align*}\lambda(g) = \lambda(hg) = f(hg)(hg)^{-1} = h\lambda(g)h^{-1}, \end{align*}$$

so in fact, $\lambda $ has image lying in $Z(G^0)$ . Thus, $\lambda $ is a $1$ -cocycle $G/G^0 \to Z(G^0)$ . Note that any such $\lambda $ determines f uniquely.

As in ordinary group cohomology, there is a short exact sequence

$$\begin{align*}0 \to \mathrm{B}^1(G/G^0, Z(G^0)) \to \mathrm{Z}^1(G/G^0, Z(G^0)) \to \mathrm{H}^1(G/G^0, Z(G^0)) \to 0. \end{align*}$$

Notice that $\mathrm {B}^1(G/G^0, Z(G^0)) = \mathrm {B}^1((G/G^0)(k), Z(G^0)(k))$ (and so on) because $G/G^0$ is constant. Under the correspondence between f and $\lambda $ as in the previous paragraph, $\mathrm {B}^1(G/G^0, Z(G^0))$ consists of those k-homomorphisms f induced by conjugation by an element of $Z(G^0)(k)$ . Moreover, if $G/G^0$ is of order n, then $\mathrm {H}^1(G/G^0, Z(G^0))$ is n-torsion, so $\mathrm {H}^1(G/G^0, Z(G^0))$ admits a surjection from $\mathrm {H}^1(G/G^0, Z(G^0)[n])$ and hence is finite. So indeed, the map $Z(G^0)(k) \to (\ker r)(k)$ has finite index image, and we are done.

Proof. Let $\mathfrak {g} = \operatorname {\mathrm {Lie}} G$ . We claim that $\mathrm {H}^2(G_s, \mathfrak {g}_s) = 0$ for all $s \in S$ . There is a Hochschild–Serre spectral sequence

$$\begin{align*}\mathrm{E}_2^{pq} = \mathrm{H}^p(G_s/G_s^0, \mathrm{H}^q(G_s^0, \mathfrak{g}_s)) \Rightarrow \mathrm{H}^{p+q}(G_s, \mathfrak{g}_s), \end{align*}$$

and since $G_s/G_s^0$ has order prime to for all $s \in S$ , we find

$$\begin{align*}\mathrm{H}^n(G_s, \mathfrak{g}_s) = \mathrm{H}^0(G_s/G_s^0, \mathrm{H}^n(G_s^0, \mathfrak{g}_s)) \end{align*}$$

for all n. By [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XXIV, Corollaire 1.13(ii)] (in which reference reductive groups over fields are also required to be connected), we have $\mathrm {H}^2(G_s^0, \mathfrak {g}_s) = 0$ , so indeed, $\mathrm {H}^2(G_s, \mathfrak {g}_s) = 0$ for all $s \in S$ . The same argument, using [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XXIV, Corollaire 1.15.1], shows that $\mathrm {H}^1(G_s, \mathfrak {g}_s) = 0$ for all $s \in S$ if $Z(G^0)$ is smooth.

To show that $\underline {\operatorname {\mathrm {Aut}}}_{G/S}$ is smooth, it suffices to verify the infinitesimal criterion of smoothness, and for this [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. III, Corollaire 2.9(ii)] shows that it suffices to show $\mathrm {H}^2(G_s, \mathfrak {g}_s) = 0$ for all $s \in S$ , which we showed above. Moreover, [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. III, Corollaire 2.9(i)] shows that if $\mathrm {H}^1(G_s, \mathfrak {g}_s) = 0$ for all $s \in S$ , then the morphism $\phi $ satisfies the infinitesimal criterion of smoothness, so it is smooth. In particular, the previous paragraph shows that if $Z(G^0)$ is smooth, then $\phi $ is smooth.

Finally, to show that $\phi $ is flat in general, we may use the fibral flatness criterion [Reference Dieudonne and GrothendieckDG67, IV₃, Théorème 11.3.10] to assume that $S = \operatorname {\mathrm {Spec}} k$ for a field k. Thus, since $\operatorname {\mathrm {Aut}}_{G/k}$ is smooth, we may conclude using Lemma 2.17.

Proof. Working locally and spreading out, we may and do assume that S is locally noetherian. Consider the morphism $\phi : G \to \operatorname {\mathrm {Aut}}_{G/S}$ . By Lemma 2.18, $\phi $ is flat, so $Z(G) = \ker \phi $ is a flat closed S-subgroup scheme of G, and it is smooth provided that $Z(G^0)$ is smooth. So we need only show that $Z(G)$ is of multiplicative type.

We define ; note that $Z(G^0)$ is of multiplicative type. By [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. IX, Théorème 6.8], it follows that $C_{G^0}(G)$ is of multiplicative type. Moreover, there is a short exact sequence

$$\begin{align*}1 \to C_{G^0}(G) \to Z(G) \to Z(G)/C_{G^0}(G) \to 1. \end{align*}$$

Since $Z(G)/C_{G^0}(G)$ is a closed commutative flat and finitely presented S-subgroup scheme of $G/G^0$ , it is finite étale commutative of order invertible on S, and in particular, it is of multiplicative type. Thus, $Z(G)$ is a commutative extension of multiplicative type group schemes, so it is of multiplicative type by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XVII, Prop. 7.1.1].

Proof. We may and do assume that k is algebraically closed. Let $\phi : H \to \operatorname {\mathrm {Aut}}_{H/k}$ be the natural map. By Lemma 2.17, $\phi (H)$ is an open subset of $\operatorname {\mathrm {Aut}}_{H/k}$ , so $\phi (H)$ is an open subgroup scheme of $(\operatorname {\mathrm {Aut}}_{H/k})_{\mathrm {{red}}}$ and the quotient (which exists as a scheme by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. VI_A, Théorème 3.2]) is locally of finite type. Moreover, $A_{\mathrm {{red}}}$ is étale: since $\phi (H)$ is (topologically) open in $\operatorname {\mathrm {Aut}}_{H/k}$ , A is discrete, and any reduced discrete scheme locally of finite type over an algebraically closed field is étale. The natural map $N_G(H) \to A$ has kernel $HC_G(H)$ , so $N_G(H)/HC_G(H)$ is a finite type closed subscheme of A. Since $A_{\mathrm {{red}}}$ is étale, it follows that $N_G(H)/HC_G(H)$ is finite. If $H/H^0$ has order prime to , so H is weakly reductive, then already A is étale by Lemma 2.18, and thus, $N_G(H)/HC_G(H)$ is étale.

Proof. We may and do assume that k is algebraically closed. Let $g \in N_G(H)(k)$ ; we need to show that under our hypotheses, $g^n$ acts by an inner automorphism on H for some integer n prime to p. Let T be a maximal torus of H, and note that conjugacy of maximal tori in $H^0$ implies that after translation by $H^0(k)$ , we may assume $g \in N_G(T)(k)$ . Since $N_G(T)/C_G(T)$ is a subquotient of W, it follows that $g^{|W|} \in C_G(T)(k)$ . Thus, after replacing g by $g^{|W|}$ , we may and do assume that g centralizes T. In particular, g acts trivially on the Dynkin diagram of $(H^0, T)$ , so g acts on $H^0$ by an inner automorphism. Thus, after further translation by $H^0(k)$ , we may and do assume that g centralizes $H^0$ . Since $p \geq |H/H^0|$ , we may pass to a further prime-to-p power of g to assume that g acts trivially on $H/H^0$ . (It is a general fact, easily checked, that if $p \mid |\operatorname {\mathrm {Aut}}(A)|$ for a finite group A, then $p < |A|$ .)

Now that g acts trivially on $H^0$ and $H/H^0$ , the argument of Lemma 2.17 shows that $\operatorname {\mathrm {Ad}}(g)$ corresponds to a $1$ -cocycle $\eta : H/H^0 \to Z(H^0)$ (which lands in $Z(H^0)(k)$ since $H/H^0$ is constant). The corresponding class in $\mathrm {H}^1(H/H^0, Z(H^0)(k))$ is killed by $|H/H^0|$ , so further passing from g to $g^{|H/H^0|}$ , we may and do assume that $\operatorname {\mathrm {Ad}}(g)$ is cohomologically trivial, from which is follows that g acts on H by conjugation by an element of $h \in Z(H^0)(k)$ . Thus, g now acts on H by inner automorphisms, and we are done.

Proof. Omitted.

Proof. By spreading out and étale-localizing around a point of S, we may and do assume that $S = \operatorname {\mathrm {Spec}} A$ for a strictly henselian noetherian local ring A. In particular, since H is finite étale, it is a constant group and there exists a scheme-theoretic section $H \to E$ . We may moreover pushforward by an inclusion of M into a torus to assume that M is a torus. The existence of a section implies that for every S-scheme $S'$ , the sequence

$$\begin{align*}1 \to M(S') \to E(S') \to H(S') \to 1 \end{align*}$$

is exact.

First, note that as an extension of H by M which admits a section, E corresponds to a cohomology class in $\mathrm {H}^2(H, M)$ , the Hochschild cohomology group (see, for example, [Reference Demazure and GabrielDG70, II, §3, Prop. 2.3] or [Reference DemarcheDem15, Prop. 2.3.6]). Concretely, this is the space of $2$ -cocycles $H \times H \to M$ modulo coboundaries. Since H is a constant group scheme, we have $\mathrm {H}^2(H, M) = \mathrm {H}^2(H(S), M(S))$ . Since $H(S)$ is finite of order n, $\mathrm {H}^2(H(S), M(S))$ is killed by n by the classical theory. Since n is invertible on the strictly henselian S, via the correspondence between extensions and classes in $\mathrm {H}^2$ , the image of the extension corresponding to multiplication by n on M is

$$\begin{align*}1 \to M(S) \xleftarrow[n]{\cong} M(S)/M[n](S) \to E(S)/M[n](S) \to H(S) \to 1. \end{align*}$$

Thus, this extension is split (i.e., there is a section $\alpha _0: H(S) \to E(S)/M[n](S)$ which is a homomorphism). This is equivalent to a section $H \to E/M[n]$ , which we also denote by $\alpha _0$ .

Since S is strictly henselian, $\alpha _0$ lifts to a section $\alpha : H \to E$ . Now $\alpha _0(x)^n = \alpha _0(x^n) = 1$ for all $x \in H(S)$ , so we have $\alpha (x)^n \in M[n](S)$ , and thus,

$$\begin{align*}\alpha(x)^N = \alpha(x)^{n^2} = 1. \end{align*}$$

Moreover, since $\alpha _0$ is a homomorphism, we have $\alpha (xx')^{-1}\alpha (x)\alpha (x') \in M[n](S)$ for all $x, x' \in H(S)$ . In other words,

$$\begin{align*}\alpha(x)\alpha(x') = \alpha(xx') g_{x, x'} \end{align*}$$

for some $g_{x, x'} \in M[n](S)$ .

Now we conclude the proof that $[Nd]: E \to E$ is a homomorphism. We shall check this on $S'$ -points for every S-scheme $S'$ . Any element of $G(S')$ is Zariski-locally of the form $\alpha (x)g$ for some $x \in H(S')$ and $g \in M(S')$ , and using centrality of M, we compute, for $x, x' \in H(S')$ and $g, g' \in M(S')$ ,

$$\begin{align*}(\alpha(x)g\alpha(x')g')^{Nd} = \alpha(xx')^{Nd}g_{x, x'}^{Nd} g^{Nd} g^{\prime}{Nd} = g^{Nd} g^{\prime}{Nd} = (\alpha(x) g)^{Nd} (\alpha(x') g')^{Nd}, \end{align*}$$

so indeed $[Nd]: E \to E$ is a homomorphism.

Finally, we show that $E[N] \to H$ is faithfully flat when M is a torus. For this, let h be a local section of H. After fppf localization, we may lift h to a section e of E. Note that $e^n$ is a local section of M, so because M is a divisible group scheme, we may pass to a further fppf localization to assume $e^n = m^n$ for some local section m of M. Because M is central in E, we see that $(em^{-1})^n = 1$ , and thus, $em^{-1}$ is a local section of $E[N]$ mapping to h. So faithful flatness follows from Lemma 2.25.

Proof of Proposition 2.27.

The claims in the second paragraph of the proposition follow directly from the universal property of the first paragraph. First, note that if G has connected fibers, then [Reference ConradCon14, Thm. 5.3.1] shows that $G^{\mathrm {ab}}$ exists and is a torus. In general, any S-homomorphism $f: G \to H$ as in the statement of the proposition induces an S-homomorphism $G^0 \to H$ , so f kills ${\mathscr D}(G^0)$ and hence factors through an S-homomorphism $G/{\mathscr D}(G^0) \to H$ , where $G/{\mathscr D}(G^0)$ is a smooth affine S-group scheme with torus identity component. Thus, we may and do assume that $G^0$ is a torus.

Working étale-locally on S (using effectivity of étale descent for affine S-schemes), we may choose representatives $h_1, \dots , h_n \in G(S)$ for $G/G^0$ . Define the map $\phi : (G^0)^n \to G^0$ by $\phi (g_1, \dots , g_n) = \prod _{i=1}^n h_ig_ih_i^{-1}g_i^{-1}$ . Note that $\phi $ is an S-homomorphism, so by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. IX, Thm. 6.8], there is some S-subgroup scheme M of $G^0$ of multiplicative type through which $\phi $ factors and such that the factored map $\phi : (G^0)^n \to M$ is faithfully flat. Since f vanishes on all commutators, it annihilates M, and thus, f must factor through $G/M$ . By definition, if $g \in G^0(S')$ is a section, then $h_igh_i^{-1} \in gM$ , so since $G^0$ is commutative, it follows that $G^0/M$ is central in $G/M$ . Thus, replacing G by $G/M$ , we may and do assume that $G^0$ is central in G.

Working Zariski-locally on S, we may and do assume that the index of $G^0$ in G is constant, say equal to n. By Lemma 2.26, is an S-subgroup scheme of G where $N = n^2$ . By the same lemma, the map $G[N] \to G/G^0$ is faithfully flat, so there is a short exact sequence

$$\begin{align*}1 \to G^0[N] \to G[N] \to G/G^0 \to 1. \end{align*}$$

Since $G^0[N]$ and $G/G^0$ are both finite étale (the former because N is invertible on the base), it follows that $G[N]$ is also finite étale.

By working étale locally on S, we may assume that $G[N]$ is a constant group scheme. In this case, the functor on S-schemes $S' \mapsto [G[N](S'), G[N](S')]$ is represented by the constant S-group scheme ${\mathscr D}(G[N])$ . Moreover, if ${\mathscr D}(G)$ denotes the sheafification of the functor $S' \mapsto [G(S'), G(S')]$ , then we have ${\mathscr D}(G) = {\mathscr D}(G[N])$ : indeed, for an S-scheme $S'$ and $g, h \in G(S')$ , centrality of $G^0$ in G shows that the commutator $ghg^{-1}h^{-1}$ depends only on the images of g and h in $(G/G^0)(S')$ . In particular, since the map $G[N] \to G/G^0$ is an epimorphism of fppf sheaves, we may pass to an fppf cover of $S'$ to assume $g, h \in G[N](S')$ . So indeed, ${\mathscr D}(G) = {\mathscr D}(G[N])$ , which is a finite étale group scheme killed by N, so its order is invertible on S.

Finally, note that is a smooth S-group scheme by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. V, Thm. 4.1]. It is commutative by the previous paragraph. Moreover, since $G^0$ is a torus the image of $G^0$ in $G^{\mathrm {ab}}$ (which makes sense by [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. IX, Thm. 6.8]) is also a torus, which must be equal to the relative identity component $(G^{\mathrm {ab}})^0$ for dimension reasons. Moreover, $G/G^0 \to G^{\mathrm {ab}}/(G^{\mathrm {ab}})^0$ is an epimorphism of sheaves, so the component group of $G^{\mathrm {ab}}$ is of order invertible on S. Thus, it follows from [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. XVII, Prop. 7.1.1] that $G^{\mathrm {ab}}$ is a group scheme of multiplicative type, as desired.

Proof. By Lemma 2.25, to show that $\overline {f}$ is faithfully flat, it suffices to show that $\overline {f}$ is an epimorphism of fppf sheaves. To this end, let $t_2$ be a local section of $T_2$ . Since $T_2$ is a torus, after localizing, we may assume there is some section $t_2'$ such that $t_2^{\prime }n = t_2$ . Further localizing, there exists a section $t_1$ of $T_1$ such that $f(t_1) = t_2'$ . We then have

$$\begin{align*}f\left(\prod_{\gamma \in \Gamma} (\gamma \cdot t_1)\right) = \prod_{\gamma \in \Gamma} (\gamma \cdot t_2') = t_2 \cdot \prod_{\gamma \in \Gamma} (\gamma \cdot t_2') t_2^{\prime}{-1}, \end{align*}$$

which has the same image in as $t_2$ . Since $\prod _{\gamma \in \Gamma } (\gamma \cdot t_1)$ lies in $\ker \phi $ , we see that indeed $\overline {f}$ is an epimorphism of fppf sheaves.

Now suppose that $t_1$ is a local section of $\ker \phi $ such that $\overline {f}(t_1) = 1$ . We claim that then $f(t_1)^n = 1$ . We may write $f(t_1) = \prod _{\gamma \in \Gamma } (\gamma \cdot t_{2, \gamma }) t_{2, \gamma }^{-1}$ locally. Using $\Gamma $ -equivariance of f and the fact that $t_1$ is fixed by $\Gamma $ , we have

$$\begin{align*}f(t_1)^n = \prod_{\gamma' \in \Gamma} (\gamma' \cdot f(t_1)) = \prod_{\gamma, \gamma' \in \Gamma} (\gamma' \gamma \cdot t_{2, \gamma})(\gamma' \cdot t_{2, \gamma}^{-1}) = \prod_{\gamma \in \Gamma} \left(\prod_{\gamma' \in \Gamma} (\gamma'\gamma \cdot t_{2, \gamma})(\gamma' \cdot t_{2, \gamma})^{-1}\right) = 1, \end{align*}$$

where the final equality follows by reindexing. So indeed, $\ker \overline {f}$ lies in $[n]^{-1}(\ker f)$ , and in particular, it is quasi-finite. Since it is of multiplicative type, it is finite, and hence, $\overline {f}$ is an isogeny.

Proof. Propositions 2.19 and 2.27 show that $Z(G)^0$ and $G^{\mathrm {ab}, 0}$ are S-tori under our assumptions, so it suffices to check the result on geometric fibers. In other words, we may and do assume that $S = \operatorname {\mathrm {Spec}} k$ for an algebraically closed field k. Note that the natural $Z(G^0) \to G^{0, \mathrm {ab}}$ (whose target is not $G^{\mathrm {ab}, 0}$ ) is an isogeny of multiplicative type groups with kernel $Z({\mathscr D}(G^0))$ (irrespective of whether $Z(G^0)$ is smooth): a maximal central torus $T_0$ of $G^0$ is of finite index in $Z(G^0)$ , and there is the standard central isogeny $T_0 \times {\mathscr D}(G^0) \to G^0$ .

Now $G/G^0$ acts on $Z(G^0)$ and $G^0/{\mathscr D}(G^0)$ by conjugation, so we can apply Lemma 2.29 with $T_1 = Z(G^0)$ , $T_2 = G^{0, \mathrm {ab}}$ , $\Gamma = G/G^0$ , and $f: T_1 \to T_2$ the natural map. We will also use the notation $\phi $ , $\psi $ , $\overline {f}$ of Lemma 2.29. In this case, $\ker \phi = C_{G^0}(G)$ , which admits $Z(G)^0$ as an open and closed S-subgroup scheme: indeed, $C_{G^0}(G) = G^0 \cap Z(G)$ , and $Z(G)^0$ (resp. $G^0$ ) is open and closed in $Z(G)$ (resp. G). Moreover, . Thus, to establish the proposition, it suffices to show that $\overline {f}$ is an isogeny, which is smooth provided $Z({\mathscr D}(G^0))$ is smooth. The fact that $\overline {f}$ is an isogeny follows directly from Lemma 2.29. The last statement of Lemma 2.29 shows that $\ker \overline {f}$ lies in $[n]^{-1}(Z({\mathscr D}(G^0)))$ , where n is the order of $G/G^0$ . Since n is invertible on S by hypothesis, smoothness of $Z({\mathscr D}(G^0))$ implies smoothness of $[n]^{-1}(Z({\mathscr D}(G^0)))$ .

Proof. Affineness of a morphism can be checked Zariski-locally on the target, so we may freely shrink X to assume that the special fiber of X is connected. In this case, $X_0$ is either empty or all of $X_s$ . If $X_0$ is empty, then the lemma is obvious; otherwise, $X_0 = V(\pi )$ , where $\pi $ is a uniformizer of A, and the lemma is again clear.

Proof. By (1) and deformation theory [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. III, Corollaire 2.8], the orbit map $\phi : G \to \underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is smooth, and thus, $C_G(H)$ is smooth and affine. To show that $C_G(H)$ is geometrically reductive, it suffices by Theorem 2.1 to assume that $S = \operatorname {\mathrm {Spec}} A$ for a complete DVR A with algebraically closed residue field. In this case, Theorem 2.2 shows that $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is a disjoint union of finite type S-affine S-schemes.

Let C be the schematic closure of the G-orbit of $f_\eta $ in $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ , so C is a G-stable closed subscheme of $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ through which the orbit map of f factors. Let $C_1$ be the G-stable open subscheme of C obtained by deleting all of the components of $C_s$ not containing a G-translate of $f_s$ , so $C_1$ is affine by Lemma 3.1. Since $\phi $ is smooth, $\phi $ has open image; moreover, each fiber of $\phi $ has closed image by (2) and Lemma 2.10. Consequently, each fiber of $C_1$ is the (open) orbit of f in that fiber, and the map $i: C_1 \to \underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G)$ is an open embedding on both fibers. Since $C_1$ is flat, it follows that i is étale and radicial, and thus, [Reference Dieudonne and GrothendieckDG67, IV₄, Théorème 17.9.1] shows that i is an open embedding.

Now we show that $C_G(H)$ is geometrically reductive; for this, it is equivalent to show that $G/C_G(H)$ is affine by Theorem 2.1. We will show, in fact, that the natural map $G/C_G(H) \to C_1$ is an isomorphism. By definition, $\phi $ factors through $C_1$ , and the previous paragraph shows that this factored map is surjective. Note that the quotient $G/C_G(H)$ exists as a smooth separated algebraic space of finite type by work of Artin [Reference ArtinArt74, Corollary 6.3]. Moreover, the induced morphism $G/C_G(H) \to C_1$ is a monomorphism, so by [Reference KnutsonKnu71, II, 6.15], it follows that $G/C_G(H)$ is a scheme. We claim that the morphism $G/C_G(H) \to C_1$ is an isomorphism. Indeed, from the above, we see that it is a smooth surjective monomorphism. Thus, by [Reference Dieudonne and GrothendieckDG67, IV₄, Théorème 17.9.1], it is a surjective open embedding and thus an isomorphism. As remarked above, this shows that $C_G(H)$ is geometrically reductive.

For the final claim, we may and do assume that $S = \operatorname {\mathrm {Spec}} k$ for an algebraically closed field k of characteristic $p> 0$ . If $\pi _0 C_G(H)$ has any p-torsion, then a simple argument with the Jordan decomposition shows that $C_G(H)(k)$ admits unipotent elements not lying in $C_G(H)^0(k)$ . This does not happen in good characteristic by the argument of [Reference Springer and SteinbergSS70, III, 3.15].

Proof. Note that $\underline {\operatorname {\mathrm {Hom}}}_{k\textrm {-}\mathrm {gp}}(H, G)$ admits a disjoint union decomposition into pieces on which the multiset of weights for T on $\mathfrak {g}$ is constant; let U be the piece containing f. By hypothesis and [Reference Gille, Polo, Demazure, Grothendieck, Artin, Bertin, Gabriel, Raynaud and SerreGP11, Exp. III, Corollaire 2.8], every orbit map $G \to U$ is smooth, and thus, every orbit is an open subscheme of U. Thus, every orbit is also closed. By Lemma 2.11, it follows that $f(H)$ is G-cr.

Proof. By passing separately from H to ${\mathscr D}(H^0)$ , $H^0/{\mathscr D}(H^0)$ , and $H/H^0$ (and from G to centralizers of these), we can assume that H is either semisimple, a torus or finite étale of order invertible on S. Assume first that H is either a torus or finite étale. In this case, let $G_1 = G \ltimes H$ , a geometrically reductive smooth affine S-group scheme which is weakly reductive whenever G is. By Theorem 3.2 (whose hypotheses always hold in the current setting), the centralizer $C_{G_1}(H)$ is smooth, affine and geometrically reductive, and it is weakly reductive whenever G is. There is a natural projection map $f: C_{G_1}(H) \to H$ , and $C_G(H) = f^{-1}(1)$ . If H is a torus, then in particular, it is commutative, and so f is split by the natural inclusion $H \to G_1$ . Thus, $C_{G_1}(H) = C_G(H) \times H$ and $C_G(H)$ is geometrically reductive, smooth and affine. Moreover, $C_{G^0}(H)$ has connected fibers by the classical theory, so $C_G(H)$ is weakly reductive if G is weakly reductive. Now assume that H is finite étale of order invertible on S. Note f factors through a constant map $C_{G_1}(H)/C_{G_1}(H)^0 \to H$ , so $C_G(H)$ is an open and closed S-subgroup scheme of $C_{G_1}(H)$ , from which the result follows in this case. If G is weakly reductive, then to show that $C_G(H)$ is weakly reductive, it suffices to show that $C_{G^0}(H)$ is weakly reductive, which follows from the above and [Reference Fargues and ScholzeFS24, Proposition VIII 5.11] applied on S-fibers.

Now assume that H is semisimple. By Lemma 3.3, Theorem 3.4, [Reference Bate, Martin and RöhrleBMR05, Corollary 3.42], and our bounds on the residue characteristics, $H_s$ is $G_s$ -cr for all $s \in S$ . The action of H on $G^0$ is given by a map $f: H \to G^0/Z(G^0)$ , and Theorem 3.2 combines with Theorem 3.4 to show that $C_{G^0/Z(G^0)}(H)$ is geometrically reductive, smooth and affine. The assumption on residue characteristics implies (by considering the simple types) that $Z(G^0)$ is smooth, so as H acts faithfully, we conclude that $C_{G^0}(H)$ is geometrically reductive, smooth and affine. If X is the stabilizer of f in G, then $C_{G^0}(H) = X \cap G^0$ is an open S-subgroup scheme of X, so X is smooth affine, and we claim that X is geometrically reductive. For this, we may and do assume $S = \operatorname {\mathrm {Spec}} A$ for a DVR A. Now the quotient $G/X$ is affine: first, $G^0/C_{G^0}(H)$ is isomorphic to an affine open subscheme U of $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G^0/Z(G^0))$ which is also a closed subscheme on S-fibers, as follows from the proof of Theorem 3.2. The quotient $G/X$ is a finite union of $G/G^0$ -translates of U in $\underline {\operatorname {\mathrm {Hom}}}_{S\textrm {-}\mathrm {gp}}(H, G^0/Z(G^0))$ , so it is an open subscheme which is closed on fibers. By Theorem 2.2 and Lemma 3.1, it follows that $G/X$ is affine and thus X is geometrically reductive by Theorem 2.1. We may now pass from G to X to assume that H acts trivially on $G^0$ .

Since H has connected fibers, it also acts trivially on the finite étale S-group scheme $G/G^0$ . It would now be enough to show that H acts trivially on G. Thus, there is a natural map $\varphi : H \times G \to G^0$ given by $\varphi (h, g) = (hgh^{-1})g^{-1}$ . Since H acts trivially on $G^0$ , it follows that $\varphi $ factors through a map $H \times G/G^0 \to Z(G^0)$ . For fixed $g \in (G/G^0)(S)$ , the map $\varphi (-, g): H \to Z(G^0)$ is an S-homomorphism. Since $Z(G^0)$ is of multiplicative type and H is semisimple, $\varphi (-, g)$ is trivial, as desired. Finally, if G is weakly reductive, then to show that $C_G(H)$ is weakly reductive, it is easy to pass from G to $G^0/Z(G^0)$ and thus reduce to Theorem 3.2.

Proof. By [Reference Andersen and JantzenAJ84, 4.4], every symmetric power $\operatorname {\mathrm {Sym}}^n \mathfrak {g}^*$ admits a good filtration: in other words, there is a filtration of $\operatorname {\mathrm {Sym}}^n \mathfrak {g}^*$ such that every subquotient is isomorphic to $\mathrm {H}^0(\lambda )$ for some dominant weight $\lambda $ . Since p is pretty good for G, [Reference HerpelHer13, Theorem 5.2] shows that $\mathfrak {g} \cong \mathfrak {g}^*$ as G-representations and thus, in particular, $\mathfrak {g}$ admits a good filtration. Now if $H = C_G(\Lambda )$ , then the above discussion and [Reference Fargues and ScholzeFS24, Proposition VIII.5.12] show that $\mathcal {O}(G/H)$ also admits a good filtration. Using the equality $\mathrm {H}^i(H, \mathfrak {g}) = \mathrm {H}^i(G, \mathfrak {g} \otimes _k \mathcal {O}(G/H))$ (which relies on the fact that $G/H$ is affine), it is enough to show that $\mathrm {H}^i(G, \mathrm {H}^0(\lambda ) \otimes _k \mathrm {H}^0(\mu )) = 0$ for all $i> 0$ and all dominant weights $\lambda $ and $\mu $ . This is proved in [Reference JantzenJan03, II, 4.13].

Proof. The first claim follows immediately from Corollary 3.5. Let $H = C_G(\Lambda )$ . By Theorem 3.2, it is enough to show that $\mathrm {H}^1(H_s, \mathfrak {g}_s) = 0$ and $H_s$ is $G_s$ -cr for all $s \in S$ . The first condition holds by Theorem 3.6, and the second condition holds by [Reference Bate, Martin and RöhrleBMR05, Corollary 3.17].

Proof. If G is connected, flatness comes from [Reference CotnerCot22a, Theorem 1.1]. The smoothness of the fibers follows, for example, from [Reference HerpelHer13, Theorem 1.1].

In general, note that $\overline {u}$ has order a power of p, and hence, u lies in the identity component of G. It suffices to show that every component of G which contains a point in the special fiber centralizing u has an A-point centralizing u. For then, the centralizer $C_G(u)$ is a union of copies of the smooth $C_{G^0}(u)$ .

Given $g \in G(A)$ such that $\overline {g}$ centralizes $u_s$ in the special fiber, as u is pure, [Reference CotnerCot22a, Theorem 5.11] shows that $gu g^{-1}$ and u are $G^0(A)$ -conjugate. Thus, there is $h \in G^0(A)$ such that $h u h^{-1} = g u g^{-1}$ , and hence, $h^{-1} g \in G(A)$ centralizes u and lies in the desired component of G.

Fact 3.11. If p is a good prime for H, the nilpotent orbits for H are in bijection with $H(k)$ -conjugacy classes of pairs $(L,P)$ where L is a Levi factor of a parabolic subgroup of H and P is a distinguished parabolic of L. The nilpotent orbit for H associated to $(L,P)$ is the unique one meeting $\operatorname {\mathrm {Lie}}(P)$ in its Richardson orbit for L.

Proof. We will prove this for a nilpotent element $\overline {X}$ ; the unipotent case follows using an integral Springer isomorphism [Reference CotnerCot22b, Theorem 1.1]. Let $\tau _s : (\mathbf {G}_m)_s \to G_s$ be a cocharacter associated to $\overline {X}$ , which exists since $\textrm {char}(k)$ is good for G. This lifts to a cocharacter $\tau : \mathbf {G}_m \to G$ by smoothness of the scheme of maximal tori [Reference ConradCon14, Theorem 3.2.6]. This cocharacter defines

$$\begin{align*}\bigoplus_{n \geq 2} \mathfrak{g}(\tau,n) \subset \mathfrak{g}. \end{align*}$$

Over the special fiber, $\overline {X}$ is in $\mathfrak {g}_s(\tau ,2)$ , and the $\operatorname {\mathrm {Ad}}_{P(\tau _s)}$ -orbit of $\overline {X}$ is open and dense. Pick a fiberwise nilpotent $X \in \mathfrak {g}(\tau ,2)$ lifting $\overline {X}$ and consider the parabolic $P_G(\tau ) \subset G$ . It naturally acts on $\bigoplus _{n \geq 2} \mathfrak {g}(\tau ,n) $ and the stabilizer of X is $C_G(X)$ . As we know, $\dim C_G(X)_\eta \leq \dim C_G(X)_s$ , and since the orbit is open and dense in the special fiber, it follows that X is pure. Then argue as in [Reference CotnerCot22a, Lemma 5.2] to show that $P_G(\tau )$ gives the instability parabolic for X and $\tau $ is an associated cocharacter for X in the special and generic fibers. By [Reference CotnerCot22a, Proposition 2.13(6)], this information determines the Bala-Carter data in the special and generic fibers.

Proof. Note that (1) implies (2) as the images of $u_{\sigma }$ in $G(K)$ and $G(k)$ have the same Bala-Carter labels. Since the dimensions of the centralizer of a unipotent element over an algebraically closed field depend only on the Bala-Carter label and not the characteristic, (2) implies (3) and (4). If u is pure and generically has Bala-Carter label $\sigma $ , then u and $u_\sigma $ are $G(K')$ -conjugate for some extension $K'$ of K. By [Reference CotnerCot22a, Theorem 5.11], u and $u_\sigma $ are $G(\mathcal {O}_{K'})$ -conjugate. Thus, (3) implies (1). Similarly, (4) implies (1).

Proof. Over an algebraically closed field, if X is nilpotent, then all nonzero multiples of X lie in the same nilpotent orbit [Reference JantzenJan04, 2.10 Lemma]. Thus, the same is true for powers of unipotents. In particular, $u^n$ is therefore pure, and $u^n$ has the same Bala-Carter labels as u generically and in the special fiber.