Exposé X. Characterization and classification of groups of multiplicative type

by A. Grothendieck

Version xy of 6 November 2009: Addenda placed in Section 9, reviewed through 8.8.¹

1. Classification of isotrivial groups. Case of a base field

Recall (IX 1.1) that the group of multiplicative type $H$ over the prescheme $S$ is said to be isotrivial if there exists an étale, finite, surjective morphism $S^{\prime }\to S$ such that $H^{\prime }=H{\times }_S{S}^{\prime }$ is diagonalizable. When $S$ is connected, $S^{\prime }$ decomposes as a finite sum of connected components $S_i^{\prime }$, and one can therefore (possibly by replacing $S^{\prime }$ by one of the $S_i^{\prime }$) assume $S^{\prime }$ connected. Finally, one knows that $S^{\prime }$ can be dominated by an $S_1^{\prime }$ finite, étale and connected, which is Galois, i.e. a principal homogeneous bundle over $S$ with group ${\Gamma }_S$, where $\Gamma$ is an ordinary finite group (cf. SGA 1, V N°s 7 & 3 when $S$ is locally noetherian, and EGA IV₄, 18.2.9 in the general case).² We assume $S^{\prime }$ chosen in this way, and we propose to determine the groups of multiplicative type $H$ over $S$ which are "trivialized" by $S^{\prime }$, i.e. such that $H^{\prime }=H{\times }_S{S}^{\prime }$ is diagonalizable.³ By descent theory (cf. SGA 1, VIII 5.4), the category of these $H$ is equivalent to the category of diagonalizable groups $H^{\prime }$ over $S^{\prime }$, equipped with operations of $\Gamma$ on $H^{\prime }$ compatible with the operations of $\Gamma$ on $S^{\prime }$. (N.B. As the groups under consideration are affine over the base, the question of effectivity of a descent datum is answered in the affirmative, cf. SGA 1, VIII 2.1.) Now $S^{\prime }$ being connected, the contravariant functor

$$M\mapsto D_{S^{\prime }}(M)$$

is an anti-equivalence of the category of ordinary commutative groups with the category of diagonalizable groups over $S^{\prime }$ (cf. VIII 1.6), a quasi-inverse functor being $H\mapsto {\mathrm{Hom}}_{S^{\prime }-gr.}(H,G_{m,S^{\prime }})$.⁴

Proposition 1.1. Let $S$ be a connected prescheme, $S^{\prime }$ a connected principal cover of $S$ with group $\Gamma$ (finite). Then the category of groups of multiplicative type over $S$ split⁵ by $S^{\prime }$ is anti-equivalent to the category of $\Gamma$-modules, i.e. of ordinary commutative groups $M$ equipped with a homomorphism $\Gamma \to {\mathrm{Aut}}_{gr.}(M)$.

One concludes from this in a standard manner:

Corollary 1.2. Let $S$ be a connected prescheme, $\xi :\mathrm{Spec}(\Omega )\to S$ a "geometric point" of $S$, i.e. a homomorphism into $S$ of the spectrum of an algebraically closed field $\Omega$; consider the fundamental group of $S$ at $\xi$ (cf. SGA 1 V, N° 7):

$$\pi ={\pi }_1(S,\xi ).$$

Then the category of isotrivial groups of multiplicative type $H$ over $S$ is anti-equivalent to the category of "Galois modules" under $\pi$, i.e. of $\pi$-modules $M$ such that the stabilizer in $\pi$ of every point of $M$ is an open subgroup.

In this correspondence, to the isotrivial group of multiplicative type $H$ is associated the group $M={\mathrm{Hom}}_{\Omega -gr.}(H_{\xi },G_{m,\Omega })$, where $H_{\xi }$ is the fiber of $H$ at $\xi$; this group is naturally equipped with operations of ${\pi }_1(S,\xi )$.

Remark 1.3. We shall see below (cf. 5.16) that if $S$ is normal, or more generally geometrically unibranch, then every group of multiplicative type and of finite type over $S$ is necessarily isotrivial, so that the classification principle 1.2 is applicable to groups of multiplicative type and of finite type over $S$, which correspond to Galois $\pi$-modules that are of finite type over $\mathbb{Z}$. For the moment, let us confine ourselves to the following result:

Proposition 1.4. Let $k$ be a field, $H$ a group of multiplicative type and of finite type over $k$; then $H$ is isotrivial, i.e. there exists a finite separable extension $k^{\prime }$ of $k$ which splits $H$.

Consequently, by 1.2, if $\pi$ is the topological Galois group of an algebraic closure $\Omega$ of $k$, the category of groups of multiplicative type and of finite type $H$ over $k$ is anti-equivalent to the category of Galois modules $M$ under $\pi$ that are of finite type as $\mathbb{Z}$-modules.

It follows first from the fact that $H$ is of finite type over $k$ and from the "principle of the finite extension" (cf. EGA IV₃, 9.1.4) that there exists a finite extension $k^{\prime }$ of $k$ which splits $H$. Let us recall the principle of the proof: by hypothesis there exist a diagonalizable group of finite type $G$ over $k$, a faithfully flat $S^{\prime }$ over $S=\mathrm{Spec}(k)$, and an isomorphism of $S^{\prime }$-groups $H_{S^{\prime }}\simeq G_{S^{\prime }}$. Possibly replacing $S^{\prime }$ by the residue field of a point of $S^{\prime }$, we may suppose that $S^{\prime }$ is the spectrum of an extension $K$ of $k$. The latter is the inductive limit of its finitely generated subalgebras $A_i$, from which it readily follows (cf. EGA IV₃, 8.8.2.4) that $u$ comes from an $A_i$-isomorphism $u_i:H_{A_i}\simeq G_{A_i}$ for $i$ large enough. By the Nullstellensatz, there exists a quotient ring $k^{\prime }$ of $A_i$ which is a finite extension of $k$. The latter therefore splits $H$.

Then $k^{\prime }$ is a radicial extension of a separable extension $k_s^{\prime }$ of $k$. By IX 5.4, the isomorphism $u^{\prime }:H_{k^{\prime }}\simeq G_{k^{\prime }}$ comes from an isomorphism $H_{k_s^{\prime }}\simeq G_{k_s^{\prime }}$, which proves that $k_s^{\prime }$ splits $H$ and establishes 1.4.

Remark 1.5. Statement 1.2 yields in particular a characterization of isotrivial tori over $S$ of relative dimension $n$: setting $\pi ={\pi }_1(S,\xi )$, they correspond to classes (up to "equivalence") of representations $\pi \to GL(n,\mathbb{Z})$ with kernel an open subgroup of $\pi$.

1.6. Even when $S$ is an algebraic curve, there can exist over $S$ tori (of relative dimension 2) that are not locally isotrivial (and a fortiori not isotrivial); there can also exist locally trivial tori that are not isotrivial. (Note however that such phenomena can present themselves only if $S$ is not normal, as already signalled in 1.3.) Let for example $S$ be an irreducible algebraic curve (over an algebraically closed field to fix ideas) having an ordinary double point $a$, let $S^{\prime }$ be its normalization, and $b$ and $c$ the two points of $S^{\prime }$ above $a$. One then constructs a principal homogeneous bundle $P$ over $S$, with structural group $\mathbb{Z}$, connected, by attaching together an infinity of copies of $S^{\prime }$ along the diagram

······         b            b           b            b   ······
                       c            c           c

(N.B. This is a principal bundle in the sense of the étale topology.) Now one has a homomorphism

                                     ⎛ 1  n ⎞
φ : ℤ ⟶ GL(2, ℤ),       φ(n) =       ⎝ 0  1 ⎠,

which permits the construction of a torus $T$ over $S$, of relative dimension 2, from the trivial torus $G_m^2$ over $P$ and from the descent datum on the latter deduced from $\varphi$. (N.B. one will note that the projection $P\to S$ is covering for the étale topology and a fortiori for the canonical topology of (Sch), and that the descent datum under consideration is necessarily effective, since $G_{m,P}^2$ is affine over $P$.) It is not difficult to prove that $T$ is not isotrivial in a neighborhood of $a$⁶ (it is however trivial on $S-a$).

One finds a variant of this construction by taking for $S$ a curve having two irreducible components S_1 and S_2 intersecting in two points $a^{\prime }$ and $a^{\prime \prime }$, which permits the construction of a principal homogeneous bundle $P$ over $S$ with group ${\mathbb{Z}}_S$, connected and locally trivial, hence an associated torus $T$ which is locally trivial, but not isotrivial.

2. Infinitesimal variations of structure

⁷ Let us begin by recalling the following result (cf. SGA 1, I 8.3 in the case $S$ locally noetherian, EGA IV₄, 18.1.2 in general):

Recall 2.0. Let $S$ be a prescheme, S_0 a sub-prescheme having the same underlying set. Then the functor

X ↦ X_0 = X ×_S S_0

is an equivalence between the category of étale preschemes over $S$ and the analogous category over S_0.

Proposition 2.1. Let $S$ be a prescheme, S_0 a sub-prescheme having the same underlying set (i.e. defined by a nilideal $I$). Then the functor

H ↦ H_0 = H ×_S S_0

from the category of preschemes in groups of multiplicative type over $S$ to the analogous category over S_0, is fully faithful.

Moreover, it induces an equivalence between the category of quasi-isotrivial groups of multiplicative type over $S$ and the analogous category over S_0.

Let us first prove the full faithfulness, i.e. that if $H$, $G$ over $S$ are of multiplicative type, then

Hom_{S-gr.}(H, G) ⟶ Hom_{S_0-gr.}(H_0, G_0)

is bijective. The question being local on $S$, we may suppose $S$ affine; there then exists a faithfully flat quasi-compact morphism $S^{\prime }\to S$ which splits $H$ and $G$. Let $S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }$; denote by $H^{\prime },G^{\prime }$ resp. $H^{\prime \prime },G^{\prime \prime }$ the groups deduced from H, G by the base change $S^{\prime }\to S$ resp. $S^{\prime \prime }\to S$; define $S_0^{\prime }$ and $S_0^{\prime \prime }$ similarly, the latter being also isomorphic to $S_0^{\prime }{\times }_{S_0}{S}_0^{\prime }$. One then finds a commutative diagram with exact rows:

Hom_{S-gr.}(H, G)      ⟶  Hom_{S′-gr.}(H′, G′)      ⇉  Hom_{S″-gr.}(H″, G″)
        ↓ u                       ↓ u′                       ↓ u″
Hom_{S_0-gr.}(H_0, G_0) ⟶ Hom_{S′_0-gr.}(H′_0, G′_0) ⇉  Hom_{S″_0-gr.}(H″_0, G″_0),

so to prove that $u$ is bijective, it suffices to prove that $u^{\prime }$ and $u^{\prime \prime }$ are, which reduces us to the case where $H$ and $G$ are diagonalizable, hence of the form $D_S(M)$ and $D_S(N)$, where $M$ and $N$ are ordinary commutative groups. One will therefore have likewise $H_0=D_{S_0}(M)$, $G_0=D_{S_0}(N)$. One then has a commutative diagram⁸

Hom_{S-gr.}(N_S, M_S)       ⥲  Hom_{S-gr.}(D_S(M), D_S(N))
        ↓                                  ↓
Hom_{S_0-gr.}(N_{S_0}, M_{S_0}) ⥲ Hom_{S_0-gr.}(D_{S_0}(M), D_{S_0}(N)),

where the horizontal arrows are isomorphisms by VIII 1.4, so we are reduced to proving that the homomorphism

(×)     Hom_{S-gr.}(N_S, M_S) ⟶ Hom_{S_0-gr.}(N_{S_0}, M_{S_0})

is bijective, i.e. to proving that the functor $M_S\mapsto M_{S_0}$, from preschemes in constant commutative groups over $S$ to preschemes in constant commutative groups over S_0, is fully faithful. Now $(\times )$ identifies also with the natural map

Hom_{gr.}(N, Γ(M_S)) ⟶ Hom_{gr.}(N, Γ(M_{S_0}))

deduced from $\Gamma (M_S)\to \Gamma (M_{S_0})$; this latter map is obviously bijective (since $\Gamma (M_S)$ = set of locally constant maps from $S$ to $M$, depends only on the topological space underlying $S$), whence the desired conclusion.

To prove the second assertion of 2.1, it remains to see that every group H_0 of multiplicative type over S_0 which is quasi-isotrivial comes from a quasi-isotrivial group of multiplicative type $H$ over $S$. To see this, let $S_0^{\prime }\to S_0$ be an étale surjective morphism which splits H_0.

One knows (cf. 2.0) that there exists an étale morphism $S^{\prime }\to S$ and an S_0-isomorphism $S^{\prime }{\times }_S{S}_0\simeq S_0^{\prime }$, so that one may suppose that $S_0^{\prime }$ comes from $S^{\prime }$ by reduction. Since $H_0^{\prime }$ is diagonalizable, one sees at once that it is isomorphic to the group deduced by base change $S_0^{\prime }\to S^{\prime }$ from a diagonalizable group $H^{\prime }$ over $S^{\prime }$ (N.B. if $H_0^{\prime }=D_{S_0^{\prime }}(M)$, one takes $H^{\prime }=D_{S^{\prime }}(M)$). Set as usual $S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }$, $S^{\prime \prime \prime }=S^{\prime }{\times }_S{S}^{\prime }{\times }_S{S}^{\prime }$, and define $S_0^{\prime \prime }$, $S_0^{\prime \prime \prime }$ similarly, deduced from the preceding by the base change $S_0\to S$ and isomorphic also to the fibered square resp. cube of $S_0^{\prime }$ over S_0. Using the full faithfulness already proved, in the cases $(S^{\prime \prime },S_0^{\prime \prime })$ and $(S^{\prime \prime \prime },S_0^{\prime \prime \prime })$, one sees that the natural descent datum on $H_0^{\prime }$ relative to $S_0^{\prime }\to S_0$ (cf. IV 2.1) comes from a well-determined descent datum on $H^{\prime }$ relative to $S^{\prime }\to S$. This descent datum is effective since $H^{\prime }$ is affine over $S^{\prime }$ (SGA 1, VIII 2.1), so there exists an $S$-group $H$ such that $H{\times }_S{S}^{\prime }=H^{\prime }=D_{S^{\prime }}(M)$, and $H$ is therefore of quasi-isotrivial multiplicative type.

One then verifies easily, using now the full-faithfulness result for $(S^{\prime },S_0^{\prime })$, that the given isomorphism between $H_0^{\prime }$ and $H^{\prime }{\times }_{S^{\prime }}{S}_0^{\prime }$ comes from an isomorphism between H_0 and $H{\times }_S{S}_0$. (For a more formal exposition of results of this kind, see Giraud's article in preparation⁹ on descent theory.)

Corollary 2.2. Let $H$ be an $S$-group of multiplicative type, and $H_0=H{\times }_S{S}_0$. For $H$ to be quasi-isotrivial (resp. locally isotrivial, resp. isotrivial, resp. locally trivial, resp. trivial), it is necessary and sufficient that H_0 be so.

The "only if" is trivial; the "if" has already been seen in the quasi-isotrivial case, since thanks to the full faithfulness, it suffices to know that every quasi-isotrivial group over S_0 lifts to a quasi-isotrivial group over $S$. The same argument works for "trivial". For the case "isotrivial", one takes up the reasoning establishing the second assertion of 2.1, but taking $S_0^{\prime }\to S_0$ étale surjective and finite. The cases "locally isotrivial" and "locally trivial" follow at once from the cases "isotrivial" and "trivial".

One can generalize 2.2 somewhat when $I$ is nilpotent, without supposing a priori $H$ of multiplicative type:¹⁰

Corollary 2.3. Suppose the ideal $I$ defining S_0 locally nilpotent. Let $H$ be a prescheme in groups over $S$, flat over $S$, and $H_0=H{\times }_S{S}_0$. For $H$ to be of quasi-isotrivial multiplicative type, it is necessary and sufficient that H_0 be so.

Indeed, suppose H_0 is of quasi-isotrivial multiplicative type; let us prove that $H$ is so. The question being local for the étale topology, and the category of étale preschemes over $S$ being equivalent to the category of étale preschemes over S_0 by the functor $S^{\prime }\mapsto S^{\prime }{\times }_S{S}_0$ (cf. 2.0), one is at once reduced to the case where H_0 is diagonalizable, hence isomorphic to a group $D_{S_0}(M)$. Let $G=D_S(M)$; one then has an isomorphism $u_0:H_0\stackrel{\sim }{\to }{G}_0$; I claim it comes from a unique homomorphism $u:H\to G$, which will therefore be an isomorphism since $u_0$ is one ($H$ and $G$ being flat over $S$, and $I$ locally nilpotent¹¹); this will establish 2.3. Now one has (cf. VIII 1 (xxx))

Hom_{S-gr.}(H, G) ≃ Hom_{S-gr.}(M_S, Hom_{S-gr.}(H, G_{m,S})),

and the second member identifies also with

$${\mathrm{Hom}}_{gr.}(M,{\mathrm{Hom}}_{S-gr.}(H,G_{m,S})),$$

so the homomorphism

(×)     Hom_{S-gr.}(H, G) ⟶ Hom_{S_0-gr.}(H_0, G_0)

is isomorphic to the homomorphism

Hom_{gr.}(M, Hom_{S-gr.}(H, G_{m,S})) ⟶ Hom_{gr.}(M, Hom_{S_0-gr.}(H_0, G_{m,S_0}))

deduced from the restriction homomorphism

(××)    Hom_{S-gr.}(H, G_{m,S}) ⟶ Hom_{S_0-gr.}(H_0, G_{m,S_0});

so to prove that $(\times )$ is bijective, it suffices to prove that $(\times \times )$ is. The question is local on $S$; we may therefore suppose $S$ affine. Now $G_{m,S}$ being commutative and smooth over $S$, the situation is governed by IX 3.6, which completes the proof.

Corollary 2.4. Let $A$ be an artinian local ring with residue field $k$, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(k)$.

(i)¹² Let $H$ be an $S$-prescheme in groups, flat and locally of finite type, such that $H_0=H{\times }_S{S}_0$ is of multiplicative type. Then $H$ is of multiplicative type, of finite type and isotrivial. In particular, every $S$-prescheme in groups $H$ of multiplicative type and of finite type is isotrivial.

(ii) The functor $H\mapsto H_0$ is an equivalence between the category of groups of multiplicative type of finite type over $A$ and the analogous category over $k$.

Indeed, let $H$ be as in (i). Then H_0 is of multiplicative type and locally of finite type, hence of finite type, hence isotrivial by 1.4. Therefore, by 2.3 and 2.2, $H$ is of multiplicative type (hence of finite type) and isotrivial. Assertion (ii) then follows from 2.1.

Corollary 2.5. Let $S^{\prime }\to S$ be a faithfully flat and radicial morphism.

(i) The functor $H\mapsto H^{\prime }=H{\times }_S{S}^{\prime }$, from the category of groups of multiplicative type over $S$ to the analogous category over $S^{\prime }$, is fully faithful.

Moreover, it induces an equivalence between the subcategories formed by the quasi-isotrivial groups of multiplicative type.

(ii) If $H$ is of multiplicative type, for it to be quasi-isotrivial (resp. locally isotrivial, resp. isotrivial, resp. locally trivial, resp. trivial) it is necessary and sufficient that $H^{\prime }$ be so.

Indeed, let $S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }$ and $S^{\prime \prime \prime }=S^{\prime }{\times }_S{S}^{\prime }{\times }_S{S}^{\prime }$; then the hypothesis that $S^{\prime }\to S$ is radicial implies that the diagonal immersions $S^{\prime }\to S^{\prime \prime }$ and $S^{\prime }\to S^{\prime \prime \prime }$ are surjective, so the base change by either of these immersions is governed by 2.1 and 2.2. Taking into account that $S^{\prime }\to S$ is a morphism of effective descent for the fibered category of groups of multiplicative type over preschemes (since it is faithfully flat and quasi-compact¹³), our assertion follows formally from 2.1 and 2.2 (cf. for a formal argument the already-cited Giraud article).

3. Finite variations of structure: complete base ring

Lemma 3.1. Let $A$ be a noetherian ring, equipped with an ideal $I$ such that $A$ is separated and complete for the $I$-adic topology, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(A/I)$, $G$ and $H$ two $S$-group schemes, with $G$ of multiplicative type and isotrivial, $H$ affine over $S$, flat over $S$ at the points of $H_0=H{\times }_S{S}_0$, H_0 of quasi-isotrivial multiplicative type. Then the natural map

Hom_{S-gr.}(G, H) ⟶ Hom_{S_0-gr.}(G_0, H_0)

is bijective.

For every integer $n⩾0$, let $S_n=\mathrm{Spec}(A/I^{n+1})$, and let $G_n$, $H_n$ be the groups deduced from $G$, $H$ by the base change $S_n\to S$. Since $G/S$ is of isotrivial multiplicative type and $H$ affine over $S$, then, by IX 7.1, the natural homomorphism

Hom_{S-gr.}(G, H) ⟶ lim_n Hom_{S_n-gr.}(G_n, H_n)

is bijective. On the other hand, by 2.3, the $H_n$ are of quasi-isotrivial multiplicative type, and by 2.1, the transition homomorphisms in the projective system $({\mathrm{Hom}}_{S_n-gr.}(G_n,H_n){)}_n$ are isomorphisms, whence 3.1 at once.

Theorem 3.2. Let $A$ be a noetherian ring equipped with an ideal $I$ such that $A$ is separated and complete for the $I$-adic topology, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(A/I)$. Then:

(i) The functor

H ↦ H_0 = H ×_S S_0

is an equivalence between the category of isotrivial groups of multiplicative type over $S$ and the analogous category over S_0.

(ii) Let $H$ be an $S$-group of multiplicative type and of finite type; for $H$ to be isotrivial, it is necessary and sufficient that H_0 be so.

Proof. For (i) one can either take up the proof of 2.1, or use 1.2, in either case using the fact that the functor

S′ ↦ S′_0 = S′ ×_S S_0

from the category of finite étale schemes over $S$ to the category of finite étale schemes over S_0 is an equivalence (SGA 1, I 8.4), which one can also state (reducing to the case $S$ connected, i.e. S_0 connected), by choosing a geometric point $\xi$ of S_0, by saying that the canonical homomorphism

π₁(S_0, ξ) ⟶ π₁(S, ξ)

is an isomorphism.

Let us prove (ii), i.e. that if H_0 is isotrivial, then $H$ is so. By (i), there exists an isotrivial group of multiplicative type $G$ over $S$ and an S_0-isomorphism

$$u_0:G_0\stackrel{\sim }{\to }{H}_0.$$

¹⁴ Since $H$ is of finite type, so are H_0 and G_0; therefore, by IX 2.1 b), the type of $G$ at each point of $S$ is an abelian group of finite type, and so $G$ is of finite type over $S$. On the other hand, by 3.1, $u_0$ comes from a homomorphism of $S$-groups

$$u:G⟶H.$$

Finally, since $G$, $H$ are of multiplicative type and of finite type over $S$, and since $u_0$ is an isomorphism, then, by IX 2.9, $u$ is an isomorphism (taking into account that every neighborhood of S_0 in $S$ equals $S$).

Corollary 3.3. Let $A$ be a complete noetherian local ring with residue field $k$.

(i) Every group of multiplicative type and of finite type over $A$ is isotrivial.

(ii) The functor $H\mapsto H{\otimes }_{A}k=H_0$ is an equivalence between the categories of groups of multiplicative type and of finite type over $A$ and over $k$.

¹⁴ First, (i) follows from 3.2 (ii) and 1.4; then (ii) follows from 3.2 (i), taking into account the fact that $H$ is of finite type if and only if H_0 is (cf. the proof of 3.2 (ii)).

Remark 3.3.1. One will note that 3.3 yields, by 1.4, a complete classification of groups of multiplicative type and of finite type over $A$ in terms of the topological Galois group of an algebraic closure $\Omega$ of $k$.

Remarks 3.4. Under the hypotheses of 3.2 (i.e. $A$ noetherian, separated and complete for the $I$-adic topology, but without further hypothesis on $A/I$), it will follow from N° 5 that the functor $H\mapsto H_0$, from the category of groups of multiplicative type and of finite type over $S$ to the analogous category over S_0, is again fully faithful (without hypotheses of isotriviality).¹⁵

However, it is not in general essentially surjective; in fact, there can exist an S_0-group H_0 of multiplicative type and of finite type, locally trivial if one wishes (but not isotrivial), which does not come by reduction from a group of multiplicative type $H$ over $S$.

To see this, let us take up either of the examples 1.6 of a non-isotrivial group of multiplicative type over a non-normal curve. One may obviously take this curve affine, say S_0, and assume that it is embedded in the affine space of dimension 2, hence defined by an ideal $J$ in k[X, Y]. We shall take for $A$ the completion of this ring for the $J$-preadic topology, so that $A$ is a regular ring of dimension 2, a fortiori normal. We shall see in 5.16 that it follows that every group of multiplicative type and of finite type over $S=\mathrm{Spec}(A)$ is isotrivial; therefore H_0, which is of finite type and not isotrivial, does not come from a group of multiplicative type $H$ over $S$ (since $H$ would necessarily be of finite type, hence isotrivial).

Lemma 3.5.¹⁶ Let $S$ be a prescheme, $u:G\to H$ a homomorphism of $S$-preschemes in groups, locally of finite presentation and flat over $S$, $U$ the set of $s\in S$ such that the induced homomorphism on fibers $u_s:G_s\to H_s$ is flat (resp. smooth, resp. unramified, resp. étale, resp. quasi-finite).

Then $U$ is open, and the restriction $u|U:G|U\to H|U$ is a flat (resp. smooth, resp. unramified, resp. étale, resp. quasi-finite) morphism.

Indeed, let $V$ be the set of points at which $u$ is flat (resp. …). One knows that $V$ is open (cf. SGA 1, I to IV in the locally noetherian case, EGA IV in general¹⁷), and that for $x\in G$ above $s\in S$, one has $x\in V$ if and only if $u_s:G_s\to H_s$ is flat (resp. …) at $x$ (same reference; in the flat, smooth, étale cases, one uses here the flatness of $G$ and $H$ over $S$). Since $u_s$ is a homomorphism of locally algebraic groups, this also means that $u_s$ is flat (resp. …) everywhere (Exp. VI_B, 1.3), i.e. $s\in U$. Therefore $U$ is the inverse image of the open set $V$ by the unit section of $G$, hence open, and $V=G|U$, so $G|U\to H|U$ is flat (resp. …), which completes the proof. (N.B. In the "unramified" or "quasi-finite" case, the flatness hypothesis on $G$ and $H$ is unnecessary.)

Lemma 3.6. Let $H$ be a commutative algebraic group over a field $k$, admitting an open subgroup $G$ of multiplicative type. Then the family of subschemes ${}_{n}H$ ($n>0$) of $H$ is schematically dense; in particular, if one has ${}_{n}H={}_{n}G$ for every $n>0$, then $H=G$.

Here ${}_{n}H$ denotes the kernel of $n\cdot i{d}_H$.¹⁸ Let $\bar{k}$ be an algebraic closure of $k$; it suffices to show that the family $({}_n{H}_{\bar{k}}{)}_{n>0}$ is schematically dense in $H_{\bar{k}}$, for then the family $({}_{n}H{)}_{n>0}$ will be so in $H$ (cf. IX 4.5). Thus, one may suppose $k$ algebraically closed, hence $G$ of the form $D_k(M)$, $M$ a finitely generated ordinary commutative group.

Let $M_0=M/Torsion(M)$; then $T=D_k(M_0)$ is the largest torus contained in $G$, and $H/T$ is finite, so $H(k)/T(k)$ is annihilated by an integer $\nu >0$. One can find a finite number of elements $g_i\in H(k)$ such that

H = ∐_i g_i · G,

and one will have $g_i^{\nu }\in T(k)$. Since $k$ is algebraically closed, $\nu \cdot i{d}_T$ is surjective on $T(k)\simeq k^{\ast d}$, so up to replacing the $g_i$ by $g_i{t}_i^{-1}$, where $t_i\in T(k)$ is such that $t_i^{\nu }=g_i^{\nu }$, one may suppose that $g_i^{\nu }=1$. If then $n$ is a multiple of $\nu$, one will have

_n H ⊇ g_i · _n G,

and since (for $n>0$ variable) the family of ${}_{n}G$ is schematically dense in $G$ by IX 4.7, conclusion 3.6 follows.

Theorem 3.7. Let $A$ be a noetherian ring equipped with an ideal $I$ such that $A$ is separated and complete for the $I$-adic topology, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(A/I)$, and let $H$ be an affine $S$-group scheme of finite type, flat over $S$ at the points of H_0, such that $H_0=H{\times }_S{S}_0$ is of multiplicative type and isotrivial.

Then there exists an open and closed subgroup $G$ of $H$, of isotrivial multiplicative type (and of finite type), such that $G_0=H_0$.

By 3.2 (i), there exists an isotrivial group of multiplicative type $G$ over $S$ and an isomorphism

$$u_0:G_0\stackrel{\sim }{\to }{H}_0.$$

By 3.1, $u_0$ comes from a unique homomorphism of $S$-groups

$$u:G⟶H.$$

Using IX 6.6, one sees that $u$ is a monomorphism (since if $U$ is the set of $s\in S$ such that $u_s:G_s\to H_s$ is a monomorphism, then $U$ is an open neighborhood of S_0 hence identical to $S$, and $G|U\to H|U$ is a monomorphism). By IX 2.5, $u$ is even a closed immersion.

¹⁹ Therefore $G$ is of finite type, hence of finite presentation over $S$. Then, by lemma 3.5 in the "étale" case, one sees that there exists an open set $U$ neighborhood of S_0, hence identical to $S$, such that $G|U\to H|U$ is étale, so $u$ is étale, hence an open immersion (since it is an étale monomorphism²⁰), which completes the proof.

Corollary 3.8. Let $A$ be a noetherian ring equipped with an ideal $I$ such that $A$ is separated and complete for the $I$-adic topology, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(A/I)$, and let $H$ be an $S$-prescheme in groups of finite type, affine and flat over $S$.

For $H$ to be of multiplicative type and isotrivial, it is necessary and sufficient that H_0 be so, and that $H$ satisfy one of the following conditions a) and b) (which are therefore equivalent given the preceding conditions):

a) The fibers of $H$ are of multiplicative type, and of constant type on each connected component of $S$.

b) $H$ is commutative and the ${}_{n}H$ ($n>0$) are finite over $S$.

These last conditions are also implied by the following:

c) The fibers of $H$ are connected.

Of course, if $H$ is of multiplicative type (and isotrivial), conditions a) and b) are verified by IX 1.4.1 a) and 2.1 c), so only the "if" requires proof. We shall use the subgroup $G$ indicated in 3.7. When condition c) is satisfied, one obviously has $G=H$ and we are done.

In case b), one notes that the immersion $u:G\to H$ induces an open immersion

_n u : _n G ⟶ _n H

which induces an isomorphism $({}_{n}G{)}_0\stackrel{\sim }{\to }({}_{n}H{)}_0$; since ${}_{n}H$ is finite over $S$, this implies at once that ${}_{n}u$ is an isomorphism (the complement of its image is finite over $S$ and reduces to $\varnothing$). By 3.6 it follows that the morphisms induced on fibers $u_s:G_s\to H_s$ are isomorphisms, so $u$ is surjective, hence an isomorphism.

Finally, in case a), one may assume $S$ connected,²¹ and it follows that for every $s\in S$, $u_s:G_s\to H_s$ is a monomorphism of algebraic groups of multiplicative type and of the same type over $\kappa (s)$.²² I claim that such a homomorphism is necessarily an isomorphism (which will again complete the proof²³). Indeed, one may suppose, possibly extending the base field, that the two groups over $\kappa (s)$ are diagonalizable, and then this follows from VIII 3.2 b) and from the fact that a surjective homomorphism of isomorphic finitely generated $\mathbb{Z}$-modules $M\to N$ is necessarily bijective.

Corollary 3.9. Let $A$ be a noetherian ring equipped with an ideal $I$ such that $A$ is separated and complete for the $I$-adic topology, and let $H$ be an $S$-prescheme in groups of finite type, affine and flat over $S$, with connected fibers.

For $H$ to be an isotrivial torus, it is necessary and sufficient that H_0 be so.

4. Case of an arbitrary base. Quasi-isotriviality theorem

Let $A$ be a local ring. Recall that one says, after Nagata, that $A$ is henselian if every algebra $B$ finite over $A$ is a product of local algebras $B_i$ finite over $A$.

Recall 4.0.²⁴ Let $A$ be a henselian local ring, $k$ its residue field, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(k)$, and $\xi$ a geometric point of S_0. Then, the functor

X ↦ X_0 = X ×_S S_0

is an equivalence between the category of étale covers of $S$ and the analogous category over S_0 (cf. EGA IV₄, § 18.5). Consequently (cf. SGA 1, V), one has ${\pi }_1(S_0,\xi )={\pi }_1(S,\xi )$.

Suppose moreover $A$ noetherian and denote by $A^{\prime }$ its completion, $S^{\prime }=\mathrm{Spec}(A^{\prime })$. Then $A^{\prime }$ is a complete noetherian local ring, hence henselian (loc. cit., 18.5.14), and the functor

X ↦ X′ = X ×_S S′,

from the category of étale covers of $S$ to the analogous category over $S^{\prime }$, fits into a commutative diagram

                Rev.ét.(S)  ⟶  Rev.ét.(S′)
                       ↘        ↙
                      ≃   ↘   ↙  ≃
                            Rev.ét.(S_0)

so is also an equivalence of categories, whence ${\pi }_1(S_0,\xi )={\pi }_1(S^{\prime },\xi )$.

Remark 4.0.1. Since $S$ is connected ($A$ being local), it follows from 1.2 that the category of isotrivial groups of multiplicative type over $S$ is equivalent to the analogous category over S_0 (and also over $S^{\prime }$ if moreover $A$ is noetherian).

Lemma 4.1. Let $A$ be a henselian local ring with residue field $k$, $S=\mathrm{Spec}(A)$, $S_0=\mathrm{Spec}(k)$.

(i) The functor

H ↦ H_0 = H ×_S S_0

is an equivalence between the category of finite groups of multiplicative type over $S$ and the analogous category over S_0.

(ii) If moreover $A$ is noetherian, denoting by $A^{\prime }$ its completion and $S^{\prime }=\mathrm{Spec}(A^{\prime })$, the functor

H ↦ H′ = H ×_S S′

is an equivalence between the category of finite groups of multiplicative type over $S$ and the analogous category over $S^{\prime }$.

As in 4.0, the second assertion is a consequence of the first; let us prove the latter. One already knows that the functor under consideration is essentially surjective, since every group of multiplicative type H_0 over $S_0=\mathrm{Spec}(k)$, finite hence of finite type over $k$, is isotrivial by 1.4, hence comes from an isotrivial group of multiplicative type over $S$ by 4.0.1.

It remains to prove full faithfulness, i.e. that for two finite groups $G$, $H$ of multiplicative type over $S$, the map below is bijective:

Hom_{S-gr.}(G, H) ⟶ Hom_{S_0-gr.}(G_0, H_0),

i.e.,²⁵ denoting by $F$ the $S$-functor ${\mathrm{Hom}}_{S-gr.}(G,H)$, that the natural map

Hom_S(S, F) ⟶ Hom_{S_0}(S_0, F_0)

induced by the base change $S_0\to S$ is bijective. For this, given recall 4.0, it suffices to prove:

Lemma 4.2. Let $G$, $H$ be two finite groups of multiplicative type over a prescheme $S$. Then $F={\mathrm{Hom}}_{S-gr.}(G,H)$ is representable by a finite étale scheme over $S$.

²⁶ Let $f:S^{\prime }\to S$ be a faithfully flat quasi-compact morphism such that $G^{\prime }$ and $H^{\prime }$ are diagonalizable. It suffices to show that $F_{S^{\prime }}$ is representable by a preschem $X^{\prime }$ étale and finite (hence affine) over $S^{\prime }$, since the descent datum on $X^{\prime }$ relative to $f$ (cf. VIII 1.7.2) will then be effective (by SGA 1 VIII, 2.1), whence the existence of an $S$-prescheme $X$ such that $X{\times }_S{S}^{\prime }=X^{\prime }$, which represents $F$, and is étale and finite over $S$ (cf. SGA 1, V 5.7 and EGA IV₄, 17.7.3 (ii)).

One may therefore suppose that $G=D_S(M)$ and $H=D_S(N)$, where $M$ and $N$ are finite commutative groups (cf. VIII 2.1 c)). Then, $K={\mathrm{Hom}}_{gr.}(N,M)$ is a finite abelian group and, by VIII 1.5, one has an isomorphism

$${\mathrm{Hom}}_{S-gr.}(G,H)\simeq K_S,$$

which completes the proof of 4.2 and hence of 4.1.

Proposition 4.3.0.²⁷ Let $S$ be a henselian local scheme, $s$ its closed point, $g:X\to S$ a morphism locally of finite type, $x$ an isolated point of the fiber $X_s$.

(i) Then $O_{X,x}$ is finite over $O_{S,s}$. (In particular, if the residue extension $\kappa (x)/\kappa (s)$ is trivial, then $O_{S,s}\to O_{X,x}$ is surjective, by Nakayama's lemma.)

(ii) If moreover $g$ is separated, then $X^{\prime }=\mathrm{Spec}(O_{X,x})$ is an open and closed subscheme of $X$, i.e. one has a decomposition as a disjoint sum $X=X^{\prime }\bigsqcup X^{\prime \prime }$.

Proof. By the local form of Zariski's main theorem given in [Pes66], or [Ray70, Ch. IV, Th. 1], $x$ has an affine open neighborhood $U=\mathrm{Spec}(B)$ of finite type and quasi-finite over $A=O_{S,s}$, and there exists an open immersion $U\hookrightarrow Y=\mathrm{Spec}(C)$, where $C$ is a finite $A$-algebra. (N.B. to reach this conclusion, one may also use Chevalley's semi-continuity theorem (EGA IV₃, 13.1.4), then the form of Zariski's main theorem given in loc. cit., 8.12.8.)

Since $A$ is henselian, $Y$ is the disjoint sum of local schemes $Y_1,\cdots ,Y_n$, each finite over $S$, and the points of $Y$ above $s$ are the closed points $y_1,\cdots ,y_n$. Therefore $x=y_i$ for some $i$, and $O_{X,x}=O_{U,x}=C_i$ is finite over $A$. Moreover, $X^{\prime }=C_i$ is an open subscheme of $U$ hence of $X$.

Suppose moreover $g$ separated. Then, since the morphism $X^{\prime }\to S$ is finite ($C_i$ being finite over $A$), so is the immersion $X^{\prime }\to X$ (cf. EGA II, 6.1.5 (v)), so $X^{\prime }$ is also closed in $X$.

Lemma 4.3. Let $A$ be a noetherian henselian local ring, $A^{\prime }$ its completion, $S=\mathrm{Spec}(A)$, $S^{\prime }=\mathrm{Spec}(A^{\prime })$, $s$ the closed point of $S$, $G$ and $H$ two $S$-preschemes in groups, with $G$ of multiplicative type and of finite type over $S$, $H$ locally of finite type and separated over $S$, $H_s$ of multiplicative type, and $H$ flat over $S$ at the points of $H_s$.

Let $G^{\prime }$, $H^{\prime }$ be deduced from $G$, $H$ by the base change $S^{\prime }\to S$. Then the natural map below is bijective:

Hom_{S-gr.}(G, H) ⥲ Hom_{S′-gr.}(G′, H′).

Since $S^{\prime }$ is faithfully flat and quasi-compact over $S$, one knows by SGA 1, VIII 5.2 that this map is a bijection of the first member onto the part of the second formed by the $u^{\prime }:G^{\prime }\to H^{\prime }$ such that the two inverse images $u_1^{\prime \prime }$, $u_2^{\prime \prime }:G^{\prime \prime }\to H^{\prime \prime }$ of $u^{\prime }$ on $S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }$ (by the projections $p{r}_1$, $p{r}_2$ of $S^{\prime \prime }$ onto $S^{\prime }$) are equal.

Therefore everything reduces to proving that for every homomorphism of $S^{\prime }$-groups $u^{\prime }:G^{\prime }\to H^{\prime }$, one has $u_1^{\prime \prime }=u_2^{\prime \prime }$. By the density theorem IX 4.8 it suffices to prove that $u_1^{\prime \prime }$ and $u_2^{\prime \prime }$ coincide on ${}_n{G}^{\prime \prime }$ for every integer $n>0$. (N.B. one needs here in an essential way the density theorem in a case where the base prescheme, here $S^{\prime \prime }$, is not locally noetherian.)

This reduces us, replacing $G$ by ${}_{n}G$, to the case where there exists an integer $n>0$ such that $n\cdot i{d}_G=0$, hence where $G$ is finite over $S$. Let likewise ${}_{n}H$ be the kernel of the $n$-th power morphism ${\varphi }_n$ in $H$. (N.B. we have not assumed $H$ commutative, so ${\varphi }_n$ (resp. ${}_{n}H$) is not necessarily a homomorphism of groups (resp. a subgroup of $H$).) Since ${}_{n}H$ is defined as the fibered product of ${\varphi }_n:H\to H$ and the unit section $ϵ:S\to H$, its formation commutes with every base change $T\to S$, i.e. one has $({}_{n}H{)}_T={}_n(H_T)$.

²⁸ We denote by an index $m$ on the right the reductions modulo $m^{m+1}$, where $m$ is the maximal ideal of $A$. Then $H_m$ is flat over $S_m$ (since $H$ is so over $S$ at the points of H_0); therefore, by 2.4, since H_0 is of multiplicative type and of finite type, so is $H_m$. Therefore each $({}_{n}H{)}_m={}_n(H_m)$ is a subgroup of $H_m$, of multiplicative type, finite and flat over $S_m$.

In particular, $({}_{n}H{)}_0={}_n(H_0)$ is finite over S_0; since $S$ is henselian local and ${}_{n}H$ is (like $H$) locally of finite type and separated over $S$, then, by 4.3.0, the local rings of ${}_{n}H$ at the points of $({}_{n}H{)}_0$ are finite over $S$ and one has a decomposition as a disjoint sum of open sets

(∗)    _n H = _n H⁺ ⊔ _n H⁻,

where $Z={}_n{H}^{+}$ is finite over $S$, and ${}_n{H}^{-}$ lies above $S-s$.

Note that, for every finite $S$-scheme $Y$ (such as $G$), every $S$-morphism $Y\to {}_{n}H$ factors through $Z$, and that the formation of the decomposition $(\ast )$ commutes with the base change $S^{\prime }\to S$ (where $S^{\prime }=\mathrm{Spec}(A^{\prime })$, $A^{\prime }$ the completion of $A$).

Then $Z^{\prime }=Z{\times }_S{S}^{\prime }$ is a finite scheme over $S^{\prime }$, as is $P^{\prime }=Z^{\prime }{\times }_{S^{\prime }}{Z}^{\prime }$. Denote by $\nu$ the restriction to $P^{\prime }$ of the multiplication of $H^{\prime }$ and by $\sigma$ the automorphism of $P^{\prime }$ which exchanges the two factors. Since $P^{\prime }$ is finite over $S^{\prime }$ and $H^{\prime }$ separated and locally of finite type over $S^{\prime }$, then, by EGA II 5.4.3 and IV₁ 1.1.3, $Y=\nu (P^{\prime })$ is a closed subscheme of $H^{\prime }$, universally closed and quasi-compact, hence of finite type, hence proper over $S^{\prime }$. Moreover, $Y\to S^{\prime }$ has finite fibers (since $P^{\prime }\to S^{\prime }$ does). Therefore, since $S^{\prime }$ is noetherian, the morphism $Y\to S^{\prime }$ is finite (cf. EGA III, 4.4.2). Since $Z^{\prime }\subseteq Y$ and $Z_m^{\prime }=Y_m^{\prime }$ for every $m$, then $Z^{\prime }=Y$, by lemma IX 5.0, so $Z^{\prime }$ is a subgroup of $H^{\prime }$. Likewise, the kernel $K=Ker(\nu ,\nu \circ \sigma )$ is a closed subscheme of $P^{\prime }$, such that $K_m=P_m^{\prime }$ for every $m$ (since $Z_m^{\prime }={}_n{H}_m$ is commutative), so $K=P^{\prime }$, i.e. $Z^{\prime }$ is a commutative subgroup of $H^{\prime }$.

On the other hand, since each reduction $Z_m^{\prime }={}_n{H}_m$ is flat over $S_m$, then, by the "local criterion of flatness" (cf. EGA 0_III, 10.2.2 or [BAC], III § 5, Example 1 and Th. 1), $Z^{\prime }$ is flat over $S^{\prime }$. Since moreover $Z_0^{\prime }={}_n{H}_0$ is a finite group of multiplicative type over S_0, hence isotrivial by 1.4, then $Z^{\prime }$ is of multiplicative type (and isotrivial) over $S^{\prime }$, by 3.8 b). Since $S^{\prime }\to S$ is faithfully flat and quasi-compact, one deduces that the multiplication $Z{\times }_{S}Z\to H$ factors through $Z$ and makes $Z$ a subgroup of $H$, finite over $S$ and of multiplicative type (since $Z^{\prime }$ is).

Finally, by the remarks made above on decomposition $(\ast )$, $Z^{\prime }$ is the "finite part" of ${}_n{H}^{\prime }$, and so the morphism $u^{\prime }:G^{\prime }\to {}_n{H}^{\prime }$ takes its values in $Z^{\prime }$. Since $Z$ is of multiplicative type and finite over $S$, as is $G={}_{n}G$, then, by 4.1, $u^{\prime }$ comes from a $u:G\to Z$, and therefore $u_1^{\prime \prime }=u_2^{\prime \prime }$. This completes the proof of 4.3.

Lemma 4.4.0.²⁹ Let $A$ be a ring, $S=\mathrm{Spec}(A)$, $H$ an $S$-group scheme of multiplicative type and of finite type, quasi-isotrivial (resp. isotrivial), $(A_i)$ a filtered family of subrings of $A$ of which $A$ is the inductive limit, $S_i=\mathrm{Spec}(A_i)$.

Then there exist an index $i$ and an $S_i$-group scheme $H_i$ of multiplicative type and of finite type, quasi-isotrivial (resp. isotrivial), such that $H=H_i{\times }_{S_i}S$.

Theorem 4.4. Let $S$ be a prescheme, $H$ an $S$-prescheme in groups affine and of finite presentation over $S$, and $s\in S$. Assume:

α) $H$ is flat over $S$ at the points of $H_s$.

β) $H_s$ is of multiplicative type.

Then there exist an étale morphism $S^{\prime }\to S$, a point $s^{\prime }$ of $S^{\prime }$ above $s$ such that the extension $\kappa (s^{\prime })/\kappa (s)$ is trivial, and an open and closed subgroup $G^{\prime }$ of $H^{\prime }=H{\times }_S{S}^{\prime }$, of multiplicative type and of finite type, isotrivial, such that $G_{s^{\prime }}^{\prime }=H_{s^{\prime }}^{\prime }$.

³⁰ a) Let us provisionally denote $T=\mathrm{Spec}(O_{S,s})$ and show that, by "descent of conclusions", one can reduce to the case where $S=T$. Indeed, suppose we have found an étale morphism $g:T^{\prime }\to T$, a point $s^{\prime }\in T^{\prime }$ and a subgroup $G^{\prime }$ of $H^{\prime }=H_{T^{\prime }}$ satisfying the conditions of the statement; replacing $T^{\prime }$ by an affine open neighborhood of $s^{\prime }$, one may suppose $T^{\prime }$ of finite presentation over $T$.

Since $G^{\prime }$ is isotrivial, there exists an étale finite surjective morphism $f:T\to T^{\prime }$ such that $G=G^{\prime }{\times }_{T^{\prime }}T$ is isomorphic to $D_T(M)$, where $M$ is a finitely generated abelian group. Since $T^{\prime }$, $H$, and $T$, $G^{\prime }$ are of finite presentation over $T=\mathrm{Spec}(O_{S,s})$, and $O_{S,s}$ is the inductive limit of the subalgebras $A_i=O_S(S_i)$, where $S_i$ runs through the affine open neighborhoods of $s$ in $S$, then, by EGA IV₃, 8.8.2 (and Exp. VI_B, 10.2 and 10.3), there exist an index $i$, $S_i$-preschemes (resp. an $S_i$-prescheme in groups) of finite presentation $S_i^{\prime }$ and $T_i$ (resp. $G_i^{\prime }$), and morphisms $g_i:S_i^{\prime }\to S_i$ and $f_i:{\tilde{T}}_i\to S_i^{\prime }$ (resp. a morphism of $S_i$-preschemes in groups $u_i:G_i^{\prime }\to H{\times }_S{S}_i^{\prime }$), from which $f$ and $g$ (resp. $u:G^{\prime }\to H^{\prime }$) come by the base change $T^{\prime }\to S_i$. Moreover, taking $i$ large enough, $g_i$ will be étale, $f_i$ étale finite surjective, and $u_i$ an open and closed immersion (cf. EGA IV, 8.10.5 and 17.7.8).

Then, $G$ comes from the groups $G_i=G_i{\times }_{S_i}{T}_i$ and $D_{T_i}(M)$; therefore, by EGA IV₃, 8.8.2 (i) (and VI_B, 10.2), there exists an index $j⩾i$ such that $G_j\simeq D_{T_j}(M)$, so $G_j$ is of isotrivial multiplicative type. Denote by $s_j^{\prime }$ the image of $s^{\prime }$ in $S_j^{\prime }$. Then the étale morphism $S_j^{\prime }\to S_j\hookrightarrow S$, the point $s_j^{\prime }$ and the open and closed subgroup $G_j^{\prime }$ of $H{\times }_S{S}_j^{\prime }$, verify the conditions of the statement. This shows that one may suppose $S=\mathrm{Spec}(A)$ local, with closed point $s$.

b) Then, $A$ is the inductive limit of local rings $A_i$ which are localizations of $\mathbb{Z}$-algebras of finite type; denote $S_i=\mathrm{Spec}(A_i)$. Let us show that the hypotheses α) and β) "descend" to some $A_i$. Since $H\to S$ is of finite presentation, the set of $x\in H$ such that $H$ is flat over $S$ at $x$ is an open set $W$, which contains $H_s$ by hypothesis, hence contains a quasi-compact open set $V$ containing $H_s$ (since $H_s$ being affine, one can cover it by a finite number of affine open sets contained in $W$). Then the open immersion $\tau :V\hookrightarrow H$ is of finite presentation, so $V\to S$ is too.

Therefore, by EGA IV₃, 8.8.2 (and Exp. VI_B, 10.2 and 10.3), there exist an index $i$, an $S_i$-prescheme $V_i$ (resp. an $S_i$-prescheme in groups $H_i$) of finite presentation over $S_i$, and an $S_i$-morphism ${\tau }_i:V_i\to H_i$ from which $V$, $H$ and $\tau$ come by base change $S\to S_i$; moreover, taking $i$ large enough, ${\tau }_i$ will be an open immersion and $V_i$ will be flat over $S_i$, by EGA IV₃, 8.10.5 and 11.2.6. Denote by $s_i$ the image of $s$ in $S_i$; since the open immersion $(V_i{)}_{s_i}\hookrightarrow (H_i{)}_{s_i}$ gives, by the base change $\kappa (s_i)\to \kappa (s)$, the equality $V_s=H_s$, one already has $(V_i{)}_{s_i}=(H_i{)}_{s_i}$, i.e. $(H_i{)}_{s_i}\subseteq V_i$, so $H_i$ is flat over $S_i$ at the points of $(H_i{)}_{s_i}$. Finally, $(H_i{)}_{s_i}$ is of multiplicative type, since $H_s=(H_i{)}_{s_i}{\otimes }_{\kappa (s_i)}\kappa (s)$ is. Therefore the triple $(S_i,H_i,s_i)$ verifies the hypotheses of 4.4, and if the desired assertion is verified for this triple, it will also be verified, by base change, for $(S,H,s)$. This reduces us to the case where $A$ is local and noetherian. Let us now distinguish two cases.

1°) $A$ is local noetherian and henselian. Let Â be its completion, $\hat{S}=\mathrm{Spec}(\hat{A})$, and $\hat{H}=H{\times }_S\hat{S}$. Applying theorem 3.7, one finds an Ŝ-group Ĝ of multiplicative type, isotrivial and of finite type, and a homomorphism of Ŝ-groups

$$\hat{u}:\hat{G}⟶\hat{H}$$

which is an open immersion and a closed immersion, such that û induces an isomorphism ${\hat{u}}_0:{\hat{G}}_0\stackrel{\sim }{\to }{\hat{H}}_0$.

By remark 4.0.1, the base change functor by $\hat{S}\to S$ induces an equivalence between the category of isotrivial groups of multiplicative type over $S$, and over Ŝ; in particular Ĝ "comes from" an $S$-group of multiplicative type $G$, isotrivial and of finite type. By 4.3, û comes from a homomorphism

$$u:G⟶H;$$

moreover $u$ is an open and closed immersion and induces an isomorphism $u_0:G_0\to H_0$, since this is so after the faithfully flat quasi-compact base change $\hat{S}\to S$. This therefore proves 4.4 in this case (taking of course, in the conclusion of 4.4, $S^{\prime }=S$ and $s^{\prime }=s$).

2°) $A$ is local noetherian. The reduction to case 1°) is immediate, by applying 1°) to the "henselized" ring $A^h$ of $A$. More precisely, one sees easily (using SGA 1, I § 5³¹) that the local rings $O_{S^{\prime },s^{\prime }}$ of the $S$-preschemes étale $S^{\prime }$ equipped with a point $s^{\prime }$ above $s$ such that $\kappa (s^{\prime })/\kappa (s)$ is trivial, form a filtered inductive system, whose inductive limit is a noetherian henselian local ring $A^h$ (called the "henselized" ring of $A$); for details of this construction (due to Nagata in the normal case), cf. SGA 4, VIII § 4³¹. The sorites of EGA IV₃ § 8 then allow, as in part a) of the proof, to deduce from a known result on the inductive limit $A^h$ of the $O_{S^{\prime },s^{\prime }}$, an analogous result on one of the $(S^{\prime },s^{\prime })$, which proves 4.4.

Corollary 4.5. Let $S$ be a prescheme, $H$ an $S$-prescheme in groups of multiplicative type and of finite type. Then $H$ is quasi-isotrivial, i.e. is split by an étale surjective morphism $S^{\prime }\to S$.

Indeed, let $s\in S$. By 4.4, there exist an étale morphism $S^{\prime }\to S$, an $s^{\prime }\in S^{\prime }$ above $s$ such that $\kappa (s)=\kappa (s^{\prime })$, and a subgroup $G^{\prime }$ of $H^{\prime }$, of isotrivial multiplicative type and of finite type, such that $G_{s^{\prime }}^{\prime }=H_{s^{\prime }}^{\prime }$. Since $G^{\prime }$ and $H^{\prime }$ are of multiplicative type and of finite type, then, by IX 2.9, there exists an open neighborhood $U^{\prime }$ of $s^{\prime }$ such that $G^{\prime }|U^{\prime }=H^{\prime }|U^{\prime }$.

³² Suppose moreover $S$ local henselian, with closed point $s$; then, by EGA IV₄, 18.5.11 b), there exists a section $\sigma$ of $S^{\prime }\to S$ such that $\sigma (s)=s^{\prime }$. (N.B. one can see directly that $B=O_{S^{\prime },s^{\prime }}$ equals $A=O_{S,s}$ as follows: by 4.3.0 (i), one has $B\simeq A/I$, and since $B$ is a finitely presented and flat $A$-algebra, $I$ is a finitely generated ideal (cf. EGA IV₁, 1.4.7), and $I=Im$ (where $m$ is the maximal ideal of $A$), whence $I=0$.) Therefore $H$ is already isotrivial. One thus obtains:

Corollary 4.6. Let $A$ be a henselian local ring, $k$ its residue field, and $\pi$ the topological Galois group of an algebraic closure of $k$.

(i) Every group of multiplicative type and of finite type over $S=\mathrm{Spec}(A)$ is isotrivial.

(ii) The category of these groups over $S$ is equivalent (via the functor $H\mapsto H{\otimes }_{A}k$) to the analogous category over $S_0=\mathrm{Spec}(k)$; it is therefore anti-equivalent, by 1.4, to the category of Galois modules under $\pi$ which are of finite type as $\mathbb{Z}$-modules.

Corollary 4.7. Under the conditions of 4.4 suppose moreover that one of the following conditions is verified:

a) For every generization $t$ of $s$, $H_t$ is of multiplicative type and of the same type as $H_s$.

b) $H$ is commutative and the ${}_{n}H$ ($n>0$) are finite over $S$ in a neighborhood of $s$.

c) For every generization $t$ of $s$, the fiber $H_t$ is connected.

Then there exists an open neighborhood $U$ of $s$ such that $H|U$ is of multiplicative type.

Taking into account that an étale morphism is open, one is reduced to proving that there exists (with the notations of the conclusion of 4.4) an open neighborhood $U^{\prime }$ of $s^{\prime }$ such that $G^{\prime }|U^{\prime }=H^{\prime }|U^{\prime }$. Set $S^{\prime \prime }=\mathrm{Spec}(O_{S^{\prime },s^{\prime }})$; since $G^{\prime }$ and $H^{\prime }$ are of finite presentation over $S^{\prime }$, it suffices to show, by EGA IV₃, 8.8.2, that $G^{\prime \prime }=H^{\prime \prime }$. We may therefore suppose $S=S^{\prime \prime }$, and then hypotheses (a), (b), (c) become those of the lemma below, whose proof is the same as that of 3.8:

Lemma 4.7.1. Let $S$ be a local scheme, $s$ its closed point, $H$ an $S$-group scheme of finite type, $G$ an open and closed subgroup of $H$, of multiplicative type, such that $G_s=H_s$. Suppose moreover one of the following conditions verified:

a) The fibers of $H$ are of multiplicative type and of the same type as $H_s$.

b) $H$ is commutative and the ${}_{n}H$ ($n>0$) are finite over $S$.

c) The fibers of $H$ are connected.

Then $G=H$.

Corollary 4.8. Let $S$ be a prescheme, $H$ an $S$-prescheme in groups affine, flat and of finite presentation over $S$, with fibers of multiplicative type.

For $H$ to be of multiplicative type, it is necessary and sufficient that it satisfy one of the two following conditions (equivalent given the preceding conditions):

a) The type of $H_s$ (cf. IX 1.4) is a locally constant function of $s\in S$.

b) $H$ is commutative, and the ${}_{n}H$ ($n>0$) are finite over $S$.

Moreover, these conditions a), b) are implied by the following:

c) The fibers of $H$ are connected.

In particular, one finds:

Corollary 4.9. Let $S$ be a prescheme, $H$ an $S$-prescheme in groups flat and of finite presentation over $S$. Suppose moreover $H$ affine over $S$ and with connected fibers.³³ If $s\in S$ is such that $H_s$ is a torus, there exists an open neighborhood $U$ of $s$ such that $H|U$ is a torus.

In particular, if all the fibers of $H$ are tori, $H$ is a torus.

5. Scheme of homomorphisms from one group of multiplicative type to another. Twisted constant groups and groups of multiplicative type

Definition 5.1. a) Let $S$ be a prescheme, $R$ a prescheme in groups over $S$. One says that $R$ is a twisted constant group over $S$ if it is locally isomorphic, in the sense of the faithfully flat quasi-compact topology, to a constant group scheme, i.e. of the form M_S with $M$ an ordinary group.

b) One says that the twisted constant group $R$ over $S$ is quasi-isotrivial, resp. isotrivial, resp. locally isotrivial, resp. locally trivial, resp. trivial, if in the definition above one can replace the faithfully flat quasi-compact topology by the étale topology, resp. the global finite étale topology, resp. the finite étale topology, resp. the Zariski topology, resp. the coarsest (or "chaotic") topology, cf. IX 1.1 and 1.2, and IV 6.6.

To say that $R$ is quasi-isotrivial (resp. isotrivial) thus means that there exists an étale surjective (resp. and finite) morphism $S^{\prime }\to S$ such that $R^{\prime }=R{\times }_S{S}^{\prime }$ is a constant group over $S$; to say that it is trivial means that $R$ is a constant group.

c) One defines as in VIII 1.4 the type of a twisted constant group $R$ over $S$ at an $s\in S$; it is a class of ordinary groups up to isomorphism, which for variable $s$ is a locally constant function of $s$, hence constant if $S$ is connected. One will also say that $R$ is "of type $M$" if all the fibers of $R$ are of type $M$.

Beware that $R$ is quasi-compact over $S$ only if it is finite over $S$, i.e. if its type at every $s\in S$ is a finite group.³⁴

d) The case most interesting for us is that where $R$ is commutative. We shall then say that $R$ is "finitely generated" if its type at each point $s\in S$ is given by a finitely generated $\mathbb{Z}$-module, a notion which should not be confused with the schematic notion "$R$ of finite type over $S$" (cf. above).

Remark 5.2. We shall also have to consider $S$-preschemes $X$ which are locally isomorphic (for the faithfully flat quasi-compact topology) to constant preschemes, independently of any group structure. We shall then say that $X$ is a twisted constant bundle over $S$, and we extend to these preschemes the terminology introduced in 5.1. Of course one will note that when $X$ is endowed with an $S$-group structure, the meaning of the expressions "twisted constant", "isotrivial" etc. changes, depending on whether one takes into account or not the group structure over $S$. The same holds if one considers on $X$ any other species of algebraic structure (for example that of Galois principal bundle that will be considered in the following number).

Proposition 5.3. Let $R$ be a commutative twisted constant group over $S$.

(i) The functor $H=D_S(R)$ (cf. VIII 1) is representable and is a group of multiplicative type over $S$.

(ii) For every $s\in S$, the type of $R$ at $s$ equals that of $H$ at $s$.

(iii) For $R$ to be quasi-isotrivial (resp. isotrivial, resp. trivial, resp. locally isotrivial, resp. locally trivial), it is necessary and sufficient that $H$ be so.

Remark 5.3.1.³⁵ One may worry about seeing in 5.3 the term "type" used in two different senses depending on whether we speak of $R$ or $H$; fortunately, when an $S$-prescheme in groups $G$ is at the same time a twisted constant group and of multiplicative type, its type in either sense is the same, thanks to the fact that a finite ordinary commutative group is isomorphic to its dual!

Proof. Since the families covering for the faithfully flat quasi-compact topology are families of effective descent for the fibered category of affine preschemes in groups over variable base preschemes (SGA 1, VIII 2.1), one sees that $H$ is representable (and is affine over $S$), since it is so "locally"³⁶ (because it is so when $R$ is constant, and then $H$ is a diagonalizable group).

The fact that $H$ is of multiplicative type is then evident by definition, as is the fact that the type of $R$ and $H$ at $s\in S$ is the same. Finally, since $H_{S^{\prime }}=D_{S^{\prime }}(R_{S^{\prime }})$, the last assertion is reduced to the "trivial" case, i.e. to verifying that $R$ is trivial if and only if $H$ is, which follows at once from the biduality theorem VIII 1.2.

To finish making precise the correspondence between twisted constant groups and groups of multiplicative type, one must start from a group of multiplicative type $H$, and study $R=D_S(H)$. If the latter is representable, it is obviously a twisted constant group, and one will have $H\simeq D_S(R)$. In other words:

Scholie 5.4.0. The functor $R\mapsto D_S(R)$ is an anti-equivalence³⁷ between the category of twisted constant groups over $S$ and that of groups of multiplicative type $H$ over $S$ such that $D_S(H)$ is representable.

I do not know whether this condition on $H$ is always satisfied; we shall see however that it is satisfied when $H$ is quasi-isotrivial, in particular when $H$ is of finite type.

Lemma 5.4. Let $S^{\prime }\to S$ be a faithfully flat morphism locally of finite presentation, $X^{\prime }$ an $S^{\prime }$-prescheme separated, locally of finite presentation and locally quasi-finite over $S^{\prime }$.

Then every descent datum on $X^{\prime }$ relative to $S^{\prime }\to S$ is effective, i.e. there exists an $S$-prescheme $X$, and an $S^{\prime }$-isomorphism $X{\times }_S{S}^{\prime }\simeq X^{\prime }$ compatible with the descent datum.

We have already noted that the hypothesis on $S^{\prime }\to S$ implies that it is a morphism covering for the faithfully flat quasi-compact topology (IV 6.3.1 (iv)), a fortiori a morphism of effective descent.

When $X^{\prime }\to S^{\prime }$ is quasi-compact, hence of finite presentation and quasi-finite, then it is a quasi-affine morphism (cf. SGA 1, VIII 6.2³⁸), and effectivity follows in this case from SGA 1, VIII 7.9. In the general case, one reduces at once to the case where $S$ and $S^{\prime }$ are affine. One covers $X^{\prime }$ by affine open sets $U_i^{\prime }$; let $V_i^{\prime }$ be the saturation of $U_i^{\prime }$ for the equivalence relation in $X^{\prime }$ defined by the descent datum, i.e. $V_i^{\prime }=q_2(q_1^{-1}(U_i^{\prime }))$, where $q_1$, $q_2$ are the two projections of $X_1^{\prime \prime }=X^{\prime }{\times }_{S^{\prime }}(S^{\prime \prime },p_1)$ onto $X^{\prime }$ ($q_1=p{r}_1$, and $q_2$ is deduced from the first projection of $X_2^{\prime \prime }=X^{\prime }{\times }_{S^{\prime }}(S^{\prime \prime },p_2)$ thanks to the given descent isomorphism $X_1^{\prime \prime }\simeq X_2^{\prime \prime }$). Since $S^{\prime }\to S$ is faithfully flat quasi-compact locally of finite presentation, the same holds for $p_1:S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }\to S^{\prime }$, hence also for $q_1$ and $q_2$, which are consequently open morphisms (SGA 1 IV 6.6³⁹). Consequently, $V_i^{\prime }$ is an open and quasi-compact part of $X^{\prime }$. By what we have already seen, the descent data induced on the $V_i^{\prime }$ are effective, whence it follows that the descent datum on $X^{\prime }$ is so (SGA 1, VIII 7.2).

Corollary 5.5. A faithfully flat morphism of finite presentation $S^{\prime }\to S$ is a morphism of effective descent for the fibered category of twisted constant groups (over variable base preschemes).

Indeed, this amounts to asserting the effectivity of a descent datum under the conditions of 5.4, when $X^{\prime }$ is a constant $S^{\prime }$-prescheme.

Theorem 5.6. Let $S$ be a prescheme, $G$ and $H$ two $S$-preschemes in groups of quasi-isotrivial multiplicative type, with $G$⁴⁰ of finite type.

Then ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable by, and is a quasi-isotrivial twisted constant group over $S$;⁴¹ for every $s\in S$, if the type at $s$ of $G$ (resp. $H$) is $M$ (resp. $N$), that of ${\mathrm{Hom}}_{S-gr.}(H,G)$ is ${\mathrm{Hom}}_{gr.}(M,N)$.

One proceeds as in 4.2, using the fact that the assertion is established (VIII 1.5) when $G$ and $H$ are trivial. The necessary effectivity criterion is furnished by 5.5 (in the case of an étale surjective morphism $S^{\prime }\to S$).

In particular, taking $G=G_{m,S}$, one finds:

Corollary 5.7. (i) Let $H$ be a quasi-isotrivial $S$-group of multiplicative type; then the $S$-group $D_S(H)$ is representable and is a quasi-isotrivial twisted constant group over $S$.

(ii) The functors $R\mapsto D_S(R)$ and $H\mapsto D_S(H)$ are anti-equivalences, quasi-inverse to one another, between the category of quasi-isotrivial twisted constant $S$-groups and that of quasi-isotrivial $S$-groups of multiplicative type.

(iii) These functors induce anti-equivalences between the subcategories formed by the isotrivial, resp. locally isotrivial, resp. locally trivial, resp. trivial groups.

The last assertion is included only for the record, being already contained in 5.3.

Moreover, since every $S$-group of multiplicative type and of finite type is quasi-isotrivial by 4.5, one deduces from 5.6:

Corollary 5.8. Let $S$ be a prescheme, $G$ and $H$ two $S$-preschemes in groups of multiplicative type and of finite type. Then ${\mathrm{Hom}}_{S-gr.}(H,G)$ is representable and is a finitely generated quasi-isotrivial twisted constant $S$-group.