Exposé VI_B. Generalities on group schemes

by J.-E. Bertin

[^N.D.E-VI_B-0] Version of 13/10/2024.

This Exposé, which corresponds to no oral lecture of the seminar, is intended to bring together a number of technical results, commonly used, concerning group schemes.¹

1. Morphisms of groups locally of finite type over a field

1.1.

Let $A$ be an Artinian local ring, $G$ and $H$ two $A$-groups², and $u:G\to H$ a morphism of $A$-groups. Then $u$ induces a morphism of groups $u(A):G(A)\to H(A)$.³ Since $H(A)$ acts on $H$ by right translation, $u(A)$ defines, by restriction, an action of $G(A)$ on $H$. This action is compatible with the morphism $u$ and with the action of $G(A)$ on $G$ defined by right translation. Since $G(A)$ acts transitively on the strictly rational points of $G$ (see (VI_A, 0.4) for the definition), one sees that these points "all behave in the same way with respect to $u$"; from this spring the following properties.

Proposition 1.2. Let $u:G\to H$ be a quasi-compact morphism between $A$-groups locally of finite type over $A$. Then the set $u(G)$ is closed in $H$ and one has dim G = dim u(G) + dim Ker u.⁴

Since $u$ commutes with the inversion morphisms of $G$ and $H$, the image $u(G)$ is invariant under the inversion morphism of $H$; the same is therefore true of its closure $u(G{)}^{-}$ in $H$. On the other hand, let $L$ denote the set of points of $H{\times }_{A}H$ whose two projections both lie in $u(G)$; it is clear that $L$ is the image of the morphism $u{\times }_{A}u:G{\times }_{A}G\to H{\times }_{A}H$. Hence the multiplication morphism of $H$ sends $L$ into $u(G)$, in other words $u(G)\cdot u(G)=u(G)$. On the other hand, Lemma 1.2.1 below shows that $L^{-}$, the closure of $L$ in $H{\times }_{A}H$, is the set of points of $H{\times }_{A}H$ whose two projections lie in $u(G{)}^{-}$; hence $u(G{)}^{-}\cdot u(G{)}^{-}=u(G{)}^{-}$, so that the reduced subscheme of $H$ whose underlying space is $u(G{)}^{-}$ is naturally equipped with a group structure in the category $(Sch/k{)}_{red}$, where $k$ is the residue field of $A$ (cf. (VI_A, 0.2)).

Let us prove the first assertion of 1.2. Replacing $A$ by the algebraic closure of its residue field $k$, we may assume that $A$ is an algebraically closed field $k$ (cf. EGA IV₂, 2.3.12). Replacing $u$ by u_red : G_red → H_red, we may assume $G$ and $H$ reduced; in this case, as we have just seen, $u(G)$ is the underlying space of a reduced group subscheme of $G$; we may therefore assume $u$ dominant. Then $G(k)$ acts transitively on the set of connected components of $H$, and it suffices to show that $u(G)\cap H^0$ is closed: we are reduced to the case where $H$ is connected, hence irreducible and of finite type (VI_A, 2.4). Then $u$ is of finite type, since it is quasi-compact and locally of finite type; since $H$ is noetherian, $u(G)$ is constructible (EGA IV₁, 1.8.5), so it contains an open subset $V$ of $H$ (EGA 0_III, 9.2.2), and then, by (VI_A, 0.5), we have $H=V\cdot V\subset u(G)\cdot u(G)=u(G)$.

Let us prove the second assertion. Recall first that the functor Ker u (cf. I, 2.3.6.1) is representable by $u^{-1}(e)$, where $e$ denotes the unit element of $H$. When $u$ is locally of finite type, Ker u is therefore locally of finite type over $A$. We reduce as before, this time using EGA IV₂, 4.1.4, to the case where $A$ is an algebraically closed field $k$.

We may further assume $G$ and $H$ irreducible and of finite type and $u$ dominant: indeed, $k$ being algebraically closed, it is clear that the connected components of $G$, on the set of which $G(k)$ acts transitively, all have the same dimension, and that if $u^0$ denotes the restriction of $u$ to $G^0$, then $(Keru{)}^0\subset Ker{u}^0$, and dim Ker u⁰ = dim Ker u. One sees likewise that $u$ is then of finite type over $k$. If $\eta$ denotes the generic point of $H$, one has dim u⁻¹(η) = dim G − dim H (EGA IV₃, 10.6.1 (ii)). By EGA IV₃, 9.2.3 and 9.2.6, the set of $y\in H$ such that $\dim u^{-1}(y)=\dim u^{-1}(\eta )$ contains a non-empty open subset $V$. Since $u$ is dominant, $U=u^{-1}(V)$ is then a non-empty open subset of $G$ and contains a closed point $x$ of $G$, since $G$ is a Jacobson scheme (EGA IV₃, 10.4.7). Then right translation $r_x$ is an isomorphism of Ker u onto $u^{-1}(u(x))$, so that:

dim Ker u = dim u⁻¹(u(x)) = dim u⁻¹(η) = dim G − dim H.

Lemma 1.2.1. Let $f:X^{\prime }\to X$ and $g:Y^{\prime }\to Y$ be two quasi-compact, dominant morphisms of schemes over an Artinian local ring $A$. Then $f{\times }_{A}g:X^{\prime }{\times }_A{Y}^{\prime }\to X{\times }_{A}Y$ is dominant (and quasi-compact).

Indeed, one has $f{\times }_{A}g=(i{d}_X{\times }_{A}g)\circ (f{\times }_{A}i{d}_{Y^{\prime }})$. It therefore suffices to show that $f{\times }_{A}i{d}_{Y^{\prime }}$ and $i{d}_X{\times }_{A}g$ are dominant. For this one may replace $A$ by its residue field $k$. In this case, $X$ and $Y^{\prime }$ are flat over $A=k$, and since $f{\times }_{A}i{d}_{Y^{\prime }}$ (resp. $i{d}_X{\times }_{A}g$) is deduced from $f$ (resp. $g$) by the flat base change $Y^{\prime }\to A$ (resp. $X\to A$), it is dominant (and quasi-compact), by EGA IV₂, 2.3.7.

Counterexample 1.2.2. Let $k$ be a field of characteristic 0, $G$ the constant $k$-group $\mathbb{Z}$, and $H$ the additive $k$-group $G_{a,k}$. Let $u:G\to H$ be a morphism of $k$-groups. If $u\ne 0$, then $u(G)$ is not closed in $H$.

Proposition 1.3. ⁵ Let $A$ be an Artinian local ring, $k$ its residue field, $G$ an $A$-group locally of finite type and flat, $u:G\to H$ a morphism of $A$-groups. The following assertions are equivalent:

(i) $u$ is flat (resp. quasi-finite, resp. unramified, resp. smooth, resp. étale) at some point of $G$.⁶

(ii) $u$ is flat (resp. quasi-finite, resp. unramified, resp. smooth, resp. étale).

Proof. It suffices to show that (i) implies (ii). First, we have the following lemma.

Lemma 1.3.0. Let $A\to B\to C$ be morphisms of commutative rings and let $n$ be a nilpotent ideal of $A$. Suppose that $C/nC$ is a $(B/nB)$-algebra of finite type.

(i) Then $C$ is a $B$-algebra of finite type.

(ii) Moreover, if $C$ is flat over $A$ and if $C/nC$ is a $(B/nB)$-algebra of finite presentation, then $C$ is a $B$-algebra of finite presentation.

Indeed, let $x_1,\cdots ,x_n$ be elements of $C$ whose images generate $C/nC$ as a $(B/n)$-algebra. By the nilpotent Nakayama lemma, the $x_i$ generate $C$ as a $B$-algebra. This proves (i). Let $\varphi$ be the resulting surjective morphism $B[X_1,\cdots ,X_n]\to C$, and let $I=Ker(\varphi )$.

Suppose now that $C$ is flat over $A$ and that $C/nC$ is of finite presentation over $B/nB$. Then, on the one hand, $I/nI$ is identified with the kernel of $\bar{\varphi }=\varphi {\otimes }_A(A/n)$. On the other hand, by EGA IV₁, 1.4.4, the kernel of $\bar{\varphi }$ is an ideal of finite type. Let then $P_1,\cdots ,P_s$ be elements of $I$ whose images generate $I/nI$ as an ideal; by the nilpotent Nakayama lemma, they generate $I$. This proves (ii).

Let us return to the proof of 1.3. Let $x$ be an arbitrary point of $G$. Since $G$ is flat over $A$, the fiber-wise flatness criterion in the form of EGA IV₃, 11.3.10.2, shows that $u$ is flat at $x$ if $u{\otimes }_{A}k$ is. Likewise, by the preceding lemma, one sees that $u$ is of finite type (resp. of finite presentation) at $x$ if $u{\otimes }_{A}k$ is. Since the other properties are then verified on fibers (cf. EGA IV₄, 17.4.1, 17.5.1, and 17.6.1, for unramified, smooth, and étale), we are reduced to the case $A=k$.

Let now $x$ be a point of $G$ where one of the conditions of 1.3 (i) holds. Since the properties under consideration are preserved by (fpqc) descent (cf. EGA IV, 2.5.1, 2.7.1, and 17.7.1), one reduces, by replacing $k$ by an algebraic closure of $\kappa (x)$, to the case where $k$ is algebraically closed and $x\in G(k)$.

Since $G$ is a Jacobson scheme (cf. EGA IV₃, 10.4.7) and since the set $W$ of points of $G$ where $u$ is flat (resp. quasi-finite, unramified, smooth, or étale) is stable under generization⁷ (resp. open), it suffices to show that every point $y\in G(k)$ belongs to $W$. Now, for such a point $y$, the translation $r_y\circ r_x^{-1}$ sends $x$ to $y$, hence $u$ has the desired property at $y$, i.e. $y\in W$. ∎

Corollary 1.3.1. Let $A$ be an Artinian local ring, $k$ its residue field, $G$ a flat $A$-group. The following assertions are equivalent:⁸

(i) $G$ is locally quasi-finite (resp. unramified, resp. smooth, resp. étale) over $A$ at some point.

(ii) $G$ is locally quasi-finite (resp. unramified, resp. smooth, resp. étale) over $A$.

⁹ Proof. Indeed, if $G$ satisfies one of the conditions of (i) at a point $x$, there exists an open neighborhood $U$ of $x$ which is of finite type over $A$. Consequently, it suffices to apply 1.3 in the case where $H$ is the unit $A$-group, taking into account the following lemma. ∎

Lemma 1.3.1.1. Let $A$ be an Artinian local ring and $G$ an $A$-group. If there exists a non-empty open subset of $G$ of finite type over $A$, then $G$ is locally of finite type over $A$.

By Lemma 1.3.0, we may assume $A$ equal to its residue field $k$. Moreover, by (fpqc) descent, we may assume $k$ algebraically closed (cf. EGA IV₂, 2.7.1). Let $V$ be the open subset of $G$ formed by the points where $G$ is of finite type over $k$; by hypothesis, $V\ne \varnothing$. Since $G$ is a Jacobson scheme, $V$ contains a closed point $x$ and, to show that $V=G$, it suffices to show that every closed point $y$ of $G$ belongs to $V$. Now, for such a point $y$, the translation $r_y\circ r_x^{-1}$ sends $x$ to $y$, whence $y\in V$. ∎

Corollary 1.3.2. Let $A$ be an Artinian local ring, $u:G\to H$ a morphism between $A$-groups locally of finite type. The following assertions are equivalent:

(i) $u$ is universally open,

(ii) $u$ is open,

(iii) $u$ is open at some point of $G$,

(iv) the morphism $u^0:G^0\to H^0$ deduced from $u$ is dominant,

(iv′) $u^0$ is surjective,

(v) there exists a connected component $C$ of $G$ such that, if $D$ denotes the connected component of $H$ containing $u(C)$, the morphism $u^{\prime }:C\to D$ deduced from $u$ is dominant.

Proof. The implications (i) ⇒ (ii) ⇒ (iii) and (iv′) ⇒ (iv) ⇒ (v) are clear. Since $G^0$ is of finite type over $A$ (VI_A, 2.4), and hence noetherian, $u^0$ is quasi-compact, so $u^0(G^0)$ is closed in $H^0$ by 1.2; hence (iv) ⇒ (iv′). On the other hand, since $G^0$ (resp. $C$) is open in $G$ (VI_A, 2.3) and $H^0$ (resp. $D$) is irreducible (VI_A, 2.4.1), one sees that (ii) implies (iv) (resp. (iii) implies (v)). It remains to show that (v) implies (i).

The open subset $C$ (resp. $D$) of $G$ (resp. $H$) will be endowed with its induced scheme structure, and $u^{\prime }$ will denote the morphism $C\to D$ deduced from $u$. Let $k$ be the residue field of $A$.¹⁰ Since the base change $\mathrm{Spec}k\to \mathrm{Spec}A$ is a universal homeomorphism, we may assume $A=k$. By hypothesis, $u^{\prime }$ is dominant and, since $C$ is of finite type over $k$ (VI_A, 2.4.1) and hence noetherian, $u^{\prime }$ is quasi-compact. By EGA IV₂, 2.3.7, $u^{\prime }{\otimes }_k\bar{k}$ is again quasi-compact and dominant, where $\bar{k}$ denotes an algebraic closure of $k$. Then, since $C{\otimes }_k\bar{k}$ is a union of connected components of $G{\otimes }_k\bar{k}$, the morphism $u{\otimes }_k\bar{k}:G{\otimes }_k\bar{k}\to H{\otimes }_k\bar{k}$ satisfies assertion (v). We are thus reduced to the case where $A=k$ is an algebraically closed field, taking into account EGA IV₂, 2.6.4.

In this case, we may further replace $u$ by $u_{red}$, and we are reduced to the case where $H$ is reduced. Let then $\xi$ (resp. $\eta$) be the generic point of $C$ (resp. $D$). Since $u^{\prime }$ is quasi-compact and dominant, $u^{\prime }(\xi )=\eta$ (cf. EGA IV₁, 1.1.5). On the other hand, since $H$ is reduced, the local ring $O_{H,\eta }$ is a field, hence $u^{\prime }$ is flat at the point $\xi$.¹¹ Hence, by 1.3, $u$ is flat; moreover, since $u$ is locally of finite type and $H$ is locally noetherian, $u$ is locally of finite presentation. Therefore, by EGA IV₂, 2.4.6, $u$ is universally open. ∎

Proposition 1.4. Let $A$ be an Artinian local ring, and $u:G\to H$ a quasi-compact morphism between $A$-groups locally of finite type. The following assertions are equivalent:

(i) $u$ is proper,

(ii) there exists $h\in H$ such that the fiber $u^{-1}(h)$ is non-empty and proper over $\kappa (h)$,

(iii) Ker u is proper over $A$.

Proof. It is clear that (i) implies (iii), and that (iii) implies (ii). On the other hand, it follows from the hypotheses that $u$ is of finite type and, since $G$ is separated (VI_A, 0.3), so is $u$ (EGA I, 5.5.1). It therefore remains to show that assertion (ii) implies that $u$ is universally closed, so that we may assume $A$ equal to its residue field $k$.¹² Let $k^{\prime }$ be an algebraic closure of $\kappa (h)$, $u^{\prime }:G^{\prime }\to H^{\prime }$ the morphism deduced from $u$ by base change, and $h^{\prime }$ a point of $H^{\prime }$ above $h$; then the fiber $u^{\prime -1}(h^{\prime })=u^{-1}(h){\times }_{\kappa (h)}{k}^{\prime }$ is non-empty and proper, and it suffices to show that $u^{\prime }$ is proper (EGA IV₂, 2.6.4). We may therefore assume that $k$ is algebraically closed and $h\in H(k)$.

We have seen (1.2) that $u(G)$ is then the underlying set of a closed reduced group subscheme of $H$; since every closed immersion is proper (EGA II, 5.4.2), we may assume that $u$ is surjective and that $H$ is reduced. Since $u$ is surjective, the group $G(k)$ acts transitively on the set of closed points of $H$; whatever the closed point $y$ of $H$, $u^{-1}(y)$ is therefore proper over $\kappa (y)$. By EGA IV₃, 9.6.1, the set of $y\in H$ such that $u^{-1}(y)$ is not proper over $\kappa (y)$ is locally constructible; since it contains no closed point, it is empty (cf. EGA IV₃, 10.3.1 and 10.4.7).

¹³ Consider now the generic point $\eta$ of $H^0$; by what precedes, the fiber $u^{-1}(\eta )=G{\times }_H\mathrm{Spec}\kappa (\eta )$ is proper over $\kappa (\eta )$. On the other hand, since $H$ is reduced, $\kappa (\eta )$ equals $O_{H,\eta }$. Since $O_{H,\eta }$ is the direct limit of the rings $O_H(V)$, as $V$ runs through the non-empty open subsets of $H^0$, it follows from EGA IV₃, 8.1.2 a) and 8.10.5 (xii), that there exists a non-empty open subset $V$ of $H^0$ such that the restriction of $u$ over $V$ is proper. It is then clear that the $g\cdot V$, for $g\in G(k)$, form an open cover of $H$ such that, for every $g\in G(k)$, the restriction of $u$ over the open $g\cdot V$ is proper; one deduces that $u$ is proper (cf. EGA II, 5.4.1). ∎

Corollary 1.4.1. Let $A$ be an Artinian local ring, and $u:G\to H$ a morphism between $A$-groups locally of finite type. The following assertions are equivalent:

(i) $u$ is locally quasi-finite,

(ii) $u$ is quasi-finite at some point,

(iii) Ker u is discrete,

(iv) the restriction of $u$ to each connected component of $G$ is finite.

If in addition $u$ is quasi-compact, these assertions are equivalent to the following:

(v) $u$ is finite.

Proof. It is clear that (iv) implies (iii), that (iii) implies (ii) (EGA I, 6.4.4), and that, in the case where $u$ is quasi-compact, assertions (iv) and (v) are equivalent. We have already seen in 1.3 that (i) and (ii) are equivalent.

Let us show finally that (i) implies (iv). Let $C$ be a connected component of $G$; since $C$ is of finite type over $A$ (VI_A, 2.4.1) and $G$ and $H$ are separated (VI_A, 0.2), by EGA I, 5.5.1 and 6.3.4, the restriction $u^{\prime }$ of $u$ to $C$ is separated and of finite type.¹⁴ Since the fibers of $u^{\prime }$ are discrete, it follows that $u^{\prime }$ is quasi-finite (cf. EGA II, 6.2.2). Since every quasi-finite proper morphism is finite (cf. EGA III₁, 4.4.2), it therefore suffices to show that $u^{\prime }$ is universally closed.

For this, we may assume that $A$ is equal to its residue field $k$. Then, by (fpqc) descent (cf. EGA IV₂, 2.6.4), it suffices to show that $u^{\prime }{\otimes }_k\bar{k}$ is universally closed, where $\bar{k}$ denotes an algebraic closure of $k$. Moreover, since $C$ is of finite type over $k$, $C{\otimes }_k\bar{k}$ is the sum of finitely many connected components $C_1^{\prime },\cdots ,C_n^{\prime }$ of $G{\otimes }_k\bar{k}$, and it suffices to show the assertion for each $C_i^{\prime }$. One is thus reduced to the case where $k$ is algebraically closed.

Let then $g$ be a closed point of $C$. If $u^0:G^0\to H$ is the restriction of $u$ to $G^0$, one has $u^{\prime }=r_{u(g)}\circ u^0\circ r_g^{-1}$, where $r_g$ denotes right translation by $g$. Therefore, to show that $u^{\prime }$ is proper, it suffices to show that $u^0$ is. By hypothesis, $u$ is locally quasi-finite, so the fiber Ker u is discrete (and non-empty); we have seen that $u^0$ is of finite type, so the fiber $Ker{u}^0$ is finite (cf. EGA II, 6.2.2), hence proper and non-empty. Therefore $u^0$ is proper by 1.4. ∎

Corollary 1.4.2. Let $A$ be an Artinian local ring, and $u:G\to H$ a quasi-compact morphism between $A$-groups locally of finite type. The following assertions are equivalent:¹⁵

(i) $u$ is a closed immersion,

(ii) $u$ is a monomorphism,

(iii) Ker u is trivial, i.e. isomorphic to the unit $k$-group.

In particular, every group subscheme¹⁶ of $H$ is closed.

Proof. It is clear that (i) implies (ii), and if one considers the functors represented respectively by $G$ and $H$, it is immediate that conditions (ii) and (iii) are equivalent. Finally, if Ker u is the unit $k$-group, Ker u is a proper non-empty fiber, so $u$ is a proper monomorphism by 1.4, of finite presentation since $H$ is locally noetherian (EGA IV₁, 1.6.1), and hence a closed immersion (EGA IV₃, 8.11.5).

The last assertion follows from the fact that, since $H$ is locally noetherian, every immersion $G\to H$ is quasi-compact (EGA I, 6.6.4). ∎

Counterexample 1.4.3. Let $k$ be a field of characteristic 0, $G$ the constant $k$-group $\mathbb{Z}$, and $H$ the $k$-group $G_{a,k}$. Let $u:G\to H$ be a morphism of $k$-groups. If $u\ne 0$, then $Keru=0$, but $u$ is not a closed immersion (cf. 1.2.2).

We shall use later the two following results, which should have appeared in Exposé VI_A:

Lemma 1.5. Let $k$ be a field and $G$ a $k$-group locally of finite type. Then:

(i) All irreducible components of $G$ have the same dimension.

(ii) For every $g\in G$, one has ${\dim }_{g}G=\dim G$.

¹⁷ Assertion (i) has already been proved in (VI_A, 2.4.1), and assertion (ii) follows from it. Indeed, let $g$ be a point of $G$ and $C$ the connected component of $G$ containing $g$. By definition (EGA 0_IV, 14.1.2), ${\dim }_{g}G$ is the infimum of the integers $\dim U$, as $U$ runs through the open neighborhoods of $g$; one therefore has ${\dim }_{g}G=\dim U_0$ for some U_0, which one may assume contained in $C$ (since $\dim V\le \dim U$ if $V\subset U$). Then, since $C$ is irreducible and of finite type over $k$ (VI_A, 2.4.1), one has dim U_0 = dim C = tr.deg_k κ(ξ), where $\xi$ is the generic point of $C$, by EGA IV₂, 5.2.1. Hence dim_g G = dim C = dim G. ∎

Proposition 1.6. ¹⁸ Let $S$ be a scheme of characteristic zero and $G$ an $S$-group scheme, locally of finite presentation over $S$ at every point of the unit section $ϵ(S)$. For $G$ to be smooth over $S$ at every point of the unit section, it is necessary and sufficient that the O_S-module ${\omega }_{G/S}=ϵ\ast ({\Omega }_{G/S}^1)$ (called the conormal module to the unit section of $G$) be locally free.

Recall that a scheme $S$ is said to be of characteristic zero if for every closed point $x$ of $S$, the field $\kappa (x)$ has characteristic zero.

Recall also (II 4.11) that, if $\pi$ denotes the structural morphism $G\to S$, one has ${\Omega }_{G/S}^1=\pi \ast ({\omega }_{G/S})$, so that it comes to the same thing to say that the O_S-module ${\omega }_{G/S}$ is locally free, or that the O_G-module ${\Omega }_{G/S}^1$ is locally free.

If there exists an open neighborhood $U$ of $ϵ(S)$ which is smooth over $S$, then, by EGA IV₄, 17.2.3, ${\Omega }_{U/S}^1$ is locally free of finite type, as well as ${\omega }_{G/S}=ϵ\ast ({\Omega }_{U/S}^1)$.

Conversely, if ${\omega }_{G/S}$ is locally free, the same holds for ${\Omega }_{G/S}^1=\pi \ast ({\omega }_{G/S})$. Since $S$ is of characteristic 0, the Jacobian criterion (EGA IV₄, 16.12.2) therefore implies that $G$ is differentially smooth over $S$. Then it follows from EGA IV₄, 17.12.5, that $G$ is smooth over $S$ at every point of the unit section. ∎

Corollary 1.6.1 (Cartier). Given a field $k$ of characteristic zero, every $k$-group locally of finite type over $k$ is smooth over $k$.

Indeed, it is then clear that the $k$-module ${\omega }_{G/k}$ is locally free, so, by 1.6, $G$ is smooth over $k$ at the unit point $e$, and hence smooth over $k$ by 1.3.1. ∎

2. "Open properties" of groups and group morphisms locally of finite presentation

2.0.

In all that follows, $S$ will denote an arbitrary scheme; an $S$-group scheme will be called an $S$-group. Given an $S$-group $G$, we shall denote by $ϵ$ the unit section, by $c:G\to G$ the inversion morphism, and by µ the multiplication morphism $G{\times }_{S}G\to G$. For every $S$-scheme $X$, we shall denote by $\pi$ or ${\pi }_X$ the structural morphism $X\to S$.

Given a property $P(u)$ of a morphism of $S$-schemes $u:X\to Y$, we shall say that $P(u)$ is stable under base change if, whenever $u$ satisfies $P(u)$, so does the morphism $u_{(Y^{\prime })}$, for any $S$-morphism $Y^{\prime }\to Y$. We say that $P(u)$ is local in nature for the topology $T$ (cf. Exp. IV, §§ 4 and 6) if $P(u)$ satisfies the following two conditions:

a) $P(u)$ is stable under base change,

b) whenever there exists a covering family of $S$-morphisms $Y_i\to Y$ for the topology $T$ such that each morphism $u_{(Y_i)}$ satisfies $P(u)$, then $u$ satisfies $P(u)$.

Proposition 2.1. Let $P(u)$ be a property of a morphism of $S$-schemes, local in nature for the (fpqc) topology. Let $u:G\to H$ be a morphism of $S$-groups. Assume $G$ flat and universally open over $S$.

Let $W$ be the largest open subset of $H$ above which $u$ satisfies the property $P(u)$, and let $V=u^{-1}(W)$. Then $U={\pi }_G(V)$ is open in $S$ and $V$ is an open group subscheme of $G{|}_U$.

The existence of a largest open subset $W$ of $H$ above which $u$ satisfies $P(u)$ follows from the fact that $P(u)$ is local in nature for the Zariski topology. Since ${\pi }_G$ is universally open, ${\pi }_G(V)$ is an open subset $U$ of $S$. It suffices to show that $V$ is a group subscheme of $G{|}_U$. We may therefore assume $U=S$.

Set then $G^{\prime }=G{\times }_{S}V$, $H^{\prime }=H{\times }_{S}V$, $V^{\prime }=V{\times }_{S}V$, $W^{\prime }=W{\times }_{S}V$, and $u^{\prime }=u_{(V)}$; let $W_1^{\prime }$ be the largest open subset of $H^{\prime }$ above which $u^{\prime }$ satisfies $P(u)$; since $V$ is flat and universally open over $S$, so is $H^{\prime }$ over $H$, and Lemma 2.1.1 below shows that $W_1^{\prime }=W^{\prime }$. Now consider the automorphism of $V$-schemes $a$ (resp. $b$) of $G^{\prime }$ (resp. $H^{\prime }$), namely right translation by the inverse of the diagonal section $\delta$ (resp. by the inverse of $u(\delta )$), defined by

a(g, v) = (g · v⁻¹, v),    (resp. b(h, v) = (h · u(v⁻¹), v)),

for any morphism $T\to S$, $g\in G(T)$, $v\in V(T)$, and $h\in H(T)$. It is clear that $u^{\prime }\circ a=b\circ u^{\prime }$, which shows that $W^{\prime }$ is stable under $b$, hence $V^{\prime }$ is stable under $a$, so that $V$ is a group subscheme of $G$. ∎

Lemma 2.1.1. Let $P(u)$ be a property of an $S$-morphism $u$, local in nature for the (fpqc) topology. Consider a cartesian square of morphisms of $S$-schemes:

       f′
X′ ─────────→ Y′
│              │
│              │ g
▼              ▼
X  ─────────→ Y,
       f

where $g$ is flat and open. Let $W$ (resp. $W_1^{\prime }$) be the largest open subset of $Y$ (resp. $Y^{\prime }$) above which $f$ (resp. $f^{\prime }=f_{(Y^{\prime })}$) satisfies $P(u)$. Then $W_1^{\prime }=W{\times }_Y{Y}^{\prime }$.

Set $W^{\prime }=W{\times }_Y{Y}^{\prime }$; since $P(u)$ is stable under base change, one has $W^{\prime }\subset W_1^{\prime }$. Since $g$ is open, $W_1=g(W_1^{\prime })$ is an open subset of $Y$. Set $V_1=f^{-1}(W_1)$ and $V_1^{\prime }=V_1{\times }_{W_1}{W}_1^{\prime }$; it is clear that $V_1^{\prime }=f^{\prime -1}(W_1^{\prime })$. Since $g$ is flat and open, the morphism $W_1^{\prime }\to W_1=g(W_1^{\prime })$ deduced from $g$ is faithfully flat and open, hence covering for the (fpqc) topology (cf. IV 6.3.1 (iv)). Since the morphism $V_1^{\prime }\to W_1^{\prime }$ deduced from $f^{\prime }$ satisfies $P(u)$, the same holds for the morphism $V_1\to W_1$ deduced from $f$; hence $W_1\subset W$, and $W_1^{\prime }\subset g^{-1}(W_1)\subset g^{-1}(W)=W^{\prime }$; therefore $W^{\prime }=W_1^{\prime }$. ∎

Remark 2.1.2. A great many properties of a morphism are local in nature for the (fpqc) topology; let us mention the properties of being flat, (universally) open, (locally) of finite type, of finite presentation, quasi-finite (cf. EGA IV₂, 2.5.1, 2.6.1 and 2.7.1), smooth, étale, unramified (EGA IV₄, 17.7.3).

The proof of 2.1 in fact uses only base changes by flat morphisms; the proposition therefore applies to a property satisfying condition b) of 2.0 relative to the (fpqc) topology, and stable under base change by flat morphisms (for example, that of being quasi-compact and dominant).

Of course, one can state an analogous proposition concerning properties local in nature for a topology $T$ finer than the Zariski topology, the condition to verify on $G$ being then that ${\pi }_G$ be universally open and covering for the topology $T$.

In particular, if $G$ is flat and locally of finite presentation over $S$, one has an analogous statement for properties stable under base change by flat morphisms locally of finite presentation, satisfying condition b) of 2.0 relative to the (fpqc) topology (e.g., the properties of being regular, reduced, Cohen–Macaulay, etc. (EGA IV₂, 6.8)).

Proposition 2.2. Let $G$ and $H$ be two $S$-groups and $u:G\to H$ a morphism of $S$-groups. Then:¹⁹

(i) Suppose $G$ or $H$ flat over $S$, and $G$ or $H$ locally of finite presentation over $S$, and let $V$ be the largest open subset of $G$ such that the restriction of $u$ to $V$ is flat and locally of finite presentation (resp. smooth, resp. étale). Then $U={\pi }_V(V)$ is open in $S$, and $V$ is an open group subscheme of $G{|}_U$.

(ii) Suppose $G$ or $H$ universally open over $S$, and let $V$ be the largest open subset of $G$ such that the restriction of $u$ to $V$ is universally open. Then $U={\pi }_V(V)$ is open in $S$, and $V$ is an open group subscheme of $G{|}_U$.

Proof. We first show (i). Let us show that the restriction ${\pi }_V$ of ${\pi }_G$ to $V$ is flat and locally of finite presentation.

a) If ${\pi }_G$ is flat (resp. locally of finite presentation), so is ${\pi }_V$.

b) If ${\pi }_H$ is flat (resp. locally of finite presentation), so is ${\pi }_V$, as the composition of the restriction of $u$ to $V$ and ${\pi }_H$.

So in the four cases envisaged in the statement, ${\pi }_V$ is flat and locally of finite presentation, hence universally open (EGA IV₂, 2.4.6). Set $U={\pi }_V(V)$; $U$ is therefore open in $S$. It then suffices to show that $V$ is an open group subscheme of $G{|}_U$; we may therefore assume $U=S$.

Set then $G^{\prime }=G{\times }_{S}V$, $H^{\prime }=H{\times }_{S}V$, $V^{\prime }=V{\times }_{S}V$ and $u^{\prime }=u_{(V)}$. Then, since $V$ is flat and locally of finite presentation over $S$, so is $H^{\prime }$ over $H$. By EGA IV₄, 17.7.4, $V^{\prime }$ is then the largest open subset of $G^{\prime }$ such that the restriction of $u^{\prime }$ to $V^{\prime }$ is flat and locally of finite presentation (resp. smooth, resp. étale). With the automorphisms $a$ and $b$ defined as in the proof of 2.1, it is then clear that $V^{\prime }$ is stable under $a$, hence $V$ is a group subscheme of $G$.

Let us show (ii). The restriction ${\pi }_V$ of ${\pi }_G$ to $V$ is a universally open morphism, either because so is ${\pi }_G$, or as the composition of the restriction of $u$ to $V$ and ${\pi }_H$ in the case where ${\pi }_H$ is universally open. Set $U={\pi }_V(V)$; $U$ is then open in $S$. It suffices to show that $V$ is an open group subscheme of $G{|}_U$. We may therefore assume $U=S$.

Set as before $G^{\prime }=G{\times }_{S}V$, $H^{\prime }=H{\times }_{S}V$, $V^{\prime }=V{\times }_{S}V$ and $u^{\prime }=u_{(V)}$. Then ${\pi }_V:V\to S$ is surjective and universally open, the same is true of $G^{\prime }\to G$, so that $V^{\prime }$ is the largest open subset of $G^{\prime }$ such that the restriction of $u^{\prime }$ to $V^{\prime }$ is universally open, by virtue of (EGA IV₃, 14.3.4 (i) and (ii)). With the automorphisms $a$ and $b$ defined as before, it is then clear that $V^{\prime }$ is stable under $a$, hence $V$ is a group subscheme of $G$. ∎

Corollary 2.3. Let $G$ be an $S$-group and $V$ the largest open subset of $G$ which is flat and locally of finite presentation (resp. smooth, étale, universally open) over $S$. Then $U={\pi }_V(V)$ is open in $S$, and $V$ is an open group subscheme of $G{|}_U$.

It suffices to apply 2.2 in the case where $H$ is the unit $S$-group and $u$ is the unique morphism of $S$-groups $G\to H$, since then ${\pi }_H$ is an isomorphism and ${\pi }_G={\pi }_H\circ u$. ∎

Corollary 2.4. Let $G$ be an $S$-group; if there exists a neighborhood $X$ of the unit section having one (or several) of the following properties:

$X$ is flat and locally of finite presentation (resp. smooth, étale, universally open) over $S$,

then there exists an open group subscheme of $G$ having the same properties.

It suffices to apply 2.3, remarking that here, with the notations of 2.2, one has $ϵ(S)\subset V$, hence $U=S$. ∎

Proposition 2.5. ²⁰ Let $u:G\to H$ be a morphism of $S$-groups.

(i) Suppose that $G$ (resp. $H$) is of finite presentation and flat (resp. of finite type) over $S$ at the points of its unit section. Then the sets:

$$U_{flat}\supset U_{smooth}\supset U_{\acute{e}}t,$$

formed of the points $s\in S$ such that $u_s$ is flat (resp. smooth, étale), are open in $S$.

If moreover $G$ (resp. $H$) is locally of finite presentation and flat (resp. locally of finite type) over $S$, then the set $V_{flat}$ (resp. $V_{smooth}$, $V_{\acute{e}}t$) of points of $G$ where $u$ is flat (resp. smooth, resp. étale) is the inverse image under ${\pi }_G$ of $U_{flat}$ (resp. $U_{smooth}$, $U_{\acute{e}}t$).

(ii) Suppose that, for every $s\in S$, $G_s$ is locally of finite type over $\kappa (s)$ (a condition satisfied if $G$ is of finite type over $S$ at the points of the unit section (1.3.1.1)), and that $u$ is locally of finite type (resp. locally of finite presentation) at the points of the unit section of $G$. Then the sets:

$$U_l.q.f.\supset U_n.r.$$

formed of the $s\in S$ such that $u_s$ is locally quasi-finite (resp. unramified) are open in $S$.

If moreover $u$ is locally of finite type (resp. locally of finite presentation), then the set $V_l.q.f.$ (resp. $V_n.r.$) of points of $G$ where $u$ is quasi-finite (resp. unramified) is the inverse image under ${\pi }_G$ of $U_l.q.f.$ (resp. $U_n.r.$).

Proof. Let us show (i). First note (1.3.1.1) that, for every $s\in S$, $G_s$ is locally of finite type over $\kappa (s)$. Let $Y$ be the open subset of $H$ formed by the points where ${\pi }_H$ is of finite type, and let X_1 be the open subset of $u^{-1}(Y)$ formed by the points where ${\pi }_G$ is of finite presentation. By EGA IV₃, 11.3.1, the set $X$ of points of X_1 where ${\pi }_G$ is of finite presentation and flat is open in X_1, hence in $G$.

Let $u_X:X\to Y$ be the morphism deduced from $u$. Since $Y$ (resp. $X$) contains the unit section of $H$ (resp. $G$), the restriction ${\pi }_X$ of ${\pi }_G$ to $X$ is surjective, and the morphism $S\to X$ deduced from ${ϵ}_G$ is an $S$-section of $X$.

Consider the sets $W_{flat}\supset W_{smooth}\supset W_{\acute{e}}t$, formed by the points of $X$ where $u_X$ is flat (resp. smooth, resp. étale). It is clear that $W_{smooth}$ and $W_{\acute{e}}t$ are open in $X$, cf. N.D.E. (6). On the other hand, since ${\pi }_Y$ is locally of finite type and ${\pi }_X={\pi }_Y\circ u_X$ is locally of finite presentation, by EGA IV₁, 1.4.3 (v), $u_X$ is locally of finite presentation. Consequently, the flat locus W_X of $u_X$ is open in $X$ (EGA IV₃, 11.3.1).

Let $x\in X$ and set $s={\pi }_X(x)$; then by EGA IV₂, 11.3.10 (combined with EGA IV₄, 17.5.1, resp. 17.6.1), $x$ belongs to $W_{flat}$ (resp. $W_{smooth}$, $W_{\acute{e}}t$) if and only if $u_s$ is flat (resp. smooth, étale) at the point $x$, or, what comes to the same by 1.3, if and only if $u_s$ is flat (resp. smooth, étale).

Consequently, for "♭ = flat, smooth or étale", one has $U_{♭}={ϵ}_G^{-1}(W_{♭})$ and $W_{♭}={\pi }_X^{-1}(U_{♭})$. Since $W_{♭}$ is open in $X$, hence in $G$, the first equality shows that $U_{♭}$ is open in $S$.

The second assertion of (i) follows from what has just been seen, since then $Y=H$, $X=G$, and $V_{♭}=W_{♭}={\pi }_G^{-1}(U_{♭})$.

Let us show assertion (ii). Let $V_l.t.f.\supset V_l.q.f.$ be the open subsets of $G$ formed by the points at which $u$ is of finite type (resp. quasi-finite). Set $X=V_l.t.f.$ and denote by $u_X$ the restriction of $u$ to $X$. By hypothesis, $X$ contains $ϵ(S)$. Let $x\in X$ and set $s={\pi }_X(x)$.

If $u$ is quasi-finite at $x$, then, by base change, $u_s$ is quasi-finite at $x$, and so, since $G_s$ is assumed locally of finite type, $u_s$ is locally quasi-finite by 1.3.1.

Conversely, if $u_s$ is locally quasi-finite, then $u_X^{-1}(u_X(x))\subset u_s^{-1}(u_s(x))$ is a finite set. Since $u_X:X\to H$ is locally of finite type, it follows from Chevalley's semicontinuity theorem (EGA IV₃, 13.1.3) that there exists an open neighborhood $W$ of $x$ in $X$ such that the fiber $u_X^{-1}(u_X(x^{\prime }))$ is discrete for every $x^{\prime }\in W$. Hence, by EGA II, 6.2.2 (and EGA III₂, Err_III 20), $u_X$ is quasi-finite at $x$.

Consequently, one has $U_l.q.f.={ϵ}_G^{-1}(V_l.q.f.)$ and $V_l.q.f.={\pi }_X^{-1}(U_l.q.f.)$, and the first equality shows that $U_l.q.f.$ is open in $S$.

If moreover $G$ is locally of finite type over $S$, then $X=G$, and so $V_l.q.f.$ is the inverse image under ${\pi }_G$ of $U_l.q.f.$. This proves the first half of (ii).

Consider now the open subsets $V_l.p.f.\supset V_n.r.$, formed by the points where $u$ is of finite presentation (resp. unramified), and suppose that $Z=V_l.p.f.$ contains the unit section of $G$. Let $z\in Z$; set $s={\pi }_Z(z)$ and $h=u_Z(z)$.

If $u$ is unramified at $z$, then, by base change, $u_s$ is unramified at $z$, and so, since $G_s$ is assumed locally of finite type, $u_s$ is unramified by 1.3.1.

Conversely, if $u_s$ is unramified at the point $z$, then the fiber $u_s^{-1}(h)=u^{-1}(h)$ is unramified over $\kappa (h)$ at the point $z$. Since $Z$ is an open subset of $G$, the fiber $u_Z^{-1}(h)=u_s^{-1}(h)\cap Z$ is unramified over $\kappa (h)$ at the point $z$. Since $u_Z$ is locally of finite presentation, this entails, by EGA IV₄, 17.4.1, that $u_Z$ is unramified at the point $z$.

One therefore has $U_n.r.={ϵ}_G^{-1}(V_n.r.)$ and $V_n.r.={\pi }_Z^{-1}(U_n.r.)$, and the first equality shows that $U_n.r.$ is open in $S$.

If moreover $G$ is locally of finite presentation over $S$, then $Z=G$, and so $V_n.r.$ is the inverse image under ${\pi }_G$ of $U_n.r.$. This completes the proof of assertion (ii). ∎

Corollary 2.6. Let $u:G\to H$ be a morphism of $S$-groups which is a radicial morphism (which is the case if $u$ is a monomorphism (EGA I, 3.5.4)). Suppose $G$ (resp. $H$) locally of finite presentation and flat (resp. locally of finite type) over $S$. Then the set $U$ of $s\in S$ such that $u_s$ is an open immersion is open in $S$, and the restriction of $u$ over $U$ is an open immersion.

By 2.5 (i), the set $U^{\prime }$ of points $s\in S$ such that $u_s$ is étale is open in $S$. Since $u$ is radicial, so is $u_s$, for every $s\in S$, hence by EGA IV₄, 17.9.1, one has $U=U^{\prime }$, which shows that $U$ is open. Finally, by 2.5 (i), the restriction of $u$ over $U$ is étale; since $u$ is radicial, this restriction is an open immersion (EGA IV₄, 17.9.1). ∎

Proposition 2.7. Let $G$ be an $S$-group. The following conditions are equivalent:

(i) $G$ is unramified over $S$ at the points of the unit section,

(ii) the unit section is an open immersion,

(iii) $G$ is of finite presentation over $S$ at the points of the unit section, and for every $s\in S$, $G_s$ is unramified over $\kappa (s)$.

If, moreover, one assumes $G$ locally of finite presentation over $S$, then each of the three preceding conditions is equivalent to the following:

(iv) $G$ is unramified over $S$.

The equivalence of assertions (i) and (ii) follows from the more general Lemma 2.7.1 below. Note (1.3.1.1) that either of conditions (i) or (iii) implies that, for every $s\in S$, $G_s$ is locally of finite type over $\kappa (s)$. Then (EGA IV₄, 17.4.1), assertion (i) is equivalent to the fact that, for every $s\in S$, $G_s$ is unramified over $\kappa (s)$ at the point $e_s$, unit of $G_s$, or again (1.3.1), to the fact that $G_s$ is unramified over $\kappa (s)$, so assertions (i) and (iii) are equivalent. Finally, if $G$ is locally of finite presentation over $S$, assertions (iii) and (iv) are equivalent (EGA IV₄, 17.4.1). ∎

Lemma 2.7.1. Let $G$ be an $S$-scheme equipped with a section $ϵ$. In order that $G$ be unramified over $S$ at the points of this section, it is necessary and sufficient that $ϵ$ be an open immersion.

The condition is necessary, by EGA IV₄, 17.4.1 a) ⇒ b″). Conversely, if $ϵ$ is an open immersion, then the restriction to $ϵ(S)$ of the structural morphism $G\to S$ is an isomorphism, hence $G$ is unramified over $S$ at the points of $ϵ(S)$. ∎

Corollary 2.8. Let $u:G\to H$ be a morphism of $S$-groups. Suppose that, for every $s\in S$, $G_s$ is locally of finite type over $\kappa (s)$.²¹

a) If $u$ is locally of finite type, the following conditions are equivalent:

(i) $u$ is locally quasi-finite,

(ii) for every $s\in S$, $u_s:G_s\to H_s$ is locally quasi-finite,

(iii) Ker u is locally quasi-finite over $S$,

(iv) the fibers of Ker u are discrete.

b) If $u$ is locally of finite presentation, the following conditions are equivalent:

(v) $u$ is unramified,

(vi) for every $s\in S$, $u_s:G_s\to H_s$ is unramified,

(vii) Ker u is unramified,

(viii) the unit section $S\to Keru$ is an open immersion.

The equivalences (i) ⇔ (ii) and (v) ⇔ (vi) follow from 2.5 (ii), and Reminder 2.8.1 below shows that (i) ⇒ (iii) and (v) ⇒ (vii).

For every $s\in S$, denote by $e_s$ the unit element of $H_s$. Then (iii) (resp. (vii)) implies that, for every $s\in S$,

(Ker u)_s = Ker u_s = u_s⁻¹(e_s)

is locally quasi-finite (resp. unramified) over $\kappa (s)=\kappa (e_s)$, hence $u_s$ is quasi-finite (resp. unramified) at the unit point of $G_s$, hence, by 1.3, $u_s$ is locally quasi-finite (resp. unramified). So (iii) ⇒ (ii) and (vii) ⇒ (vi). Finally, (ii) ⇔ (iv) by 1.4.1, and (vii) ⇔ (viii) by 2.7. ∎

Reminder 2.8.1. Recall (cf. I 2.3.6.1) that, given a morphism $u:G\to H$ of $S$-groups, the kernel of $u$, denoted Ker u, is the group subfunctor of $G$ defined by setting, for any morphism $f:T\to S$,

(Ker u)(T) = {a ∈ G(T) | u ∘ a = ε_H ∘ f}.

By loc. cit. (or EGA I, 4.4.1), this functor is representable by the $S$-group $G{\times }_{H}S=u^{-1}({ϵ}_H(S))$, simply denoted Ker u. In particular, the structural morphism $Keru\to S$ is deduced from $u$ by base change.

Lemma 2.9. Let $\pi :X\to S$ be a morphism admitting an $S$-section $ϵ$.

(i) If $\pi$ is injective, it is integral.²²

(ii) If $\pi$ is locally of finite type, and if, for every $s\in S$, ${\pi }_s$ is an isomorphism, then $\pi$ is an isomorphism.²³

Let us first note that, by Lemma 2.9.1 below, $\pi$, being a homeomorphism, is an affine morphism.

If $\pi$ is injective, $ϵ$ is surjective. Since $ϵ$ is a surjective immersion, $ϵ(S)$ is isomorphic to the closed subscheme of $X$ defined by a nil-ideal $I$ of O_X. Since $ϵ$ is an $S$-section of the morphism $\pi$, one has a direct sum decomposition $O_X=O_S\oplus I$ of O_S-modules. Since $I$ is a nil-ideal of O_X, $I$ is evidently integral over O_S, hence O_X is integral over O_S, and $\pi$ is integral.

Suppose now that $\pi$ is locally of finite type. Since $\pi \circ ϵ=i{d}_S$, $ϵ$ is locally of finite presentation (cf. EGA IV₁, 1.4.3 (v)), hence $I$ is an ideal of finite type of O_X (EGA IV₁, 1.4.1). For every $s\in S$, one has $O_{X_s}=\kappa (s)\oplus I{\otimes }_{O_S}\kappa (s)$. By hypothesis, ${ϵ}_s$ is an isomorphism, hence $I{\otimes }_{O_S}\kappa (s)=0$ for every $s\in S$, hence a fortiori $I{\otimes }_{O_X}\kappa (x)=0$ for every $x\in X$, which implies, by Nakayama's lemma, that $I=0$, hence $\pi$ is an isomorphism. ∎

Lemma 2.9.1. Let $f:X\to Y$ be a morphism of schemes which is a homeomorphism; then $f$ is an affine morphism.²⁴

It suffices to show that every point $y\in Y$ has an open neighborhood $W$ such that the restriction of $f$ over $W$ is an affine morphism. So let $y\in Y$, and let $V$ be an affine open neighborhood of $y$ in $Y$. Let $V^{\prime }=f^{-1}(V)$. Then $V^{\prime }$ is an open neighborhood of $x=f^{-1}(y)$ in $X$. There exists an open affine neighborhood $W^{\prime }$ of $x$ in $X$ contained in $V^{\prime }$. Set then $W=f(W^{\prime })$. Then $W$ is an open neighborhood of $y$ in $Y$ contained in the affine scheme $V$, hence $W$ is separated. Since $W^{\prime }$ is an affine scheme, the restriction of $f$ over $W$ is then an affine morphism (EGA II, 1.2.3). ∎

Corollary 2.10. Let $G$ be an $S$-group locally of finite type. Suppose that, for every $s\in S$, $G_s$ is the unit $\kappa (s)$-group; then $G$ is the unit $S$-group.

More generally:

Corollary 2.11. Let $f:X\to Y$ be an $S$-morphism locally of finite type. In order that $f$ be a monomorphism, it is necessary and sufficient that $f_s$ be a monomorphism for every $s\in S$.

It is clear that the condition is necessary; let us show that it is sufficient. If, for every $s\in S$, $f_s$ is a monomorphism, then a fortiori for every $y\in Y$, $f_y$ is a monomorphism; we may therefore assume $Y=S$.

By EGA I, 5.3.8, to show that $f$ is a monomorphism, it suffices to show that ${\Delta }_f:X\to X{\times }_{S}X$ is an isomorphism, or, what comes to the same, that the first projection $p:X{\times }_{S}X\to X$ is an isomorphism. But, if $f_s$ is a monomorphism, it likewise follows from EGA I, 5.3.8 that the first projection $X_s{\times }_{\kappa (s)}{X}_s\to X_s$ (which is identified with $p_s$) is an isomorphism. Now $p$ has the $S$-section ${\Delta }_f$, hence Lemma 2.9 affirms that if for every $s\in S$, $p_s$ is an isomorphism, then so is $p$. ∎

Corollary 2.12. Let $G$ be an $S$-scheme having an $S$-section $ϵ$ and such that the structural morphism $\pi :G\to S$ is closed. Let $s\in S$ be such that $\pi$ is of finite presentation at the point $ϵ(s)$ and that ${ϵ}_s:\mathrm{Spec}(\kappa (s))\to G_s$ is an isomorphism (or, what comes to the same, that ${\pi }_s:G_s\to \mathrm{Spec}\kappa (s)$ is an isomorphism). Then there exists an open neighborhood $U$ of $s$ in $S$ such that $ϵ{|}_U:U\to G{\times }_{S}U$ is an isomorphism.

Let $V$ be the set of points of $G$ where $\pi$ is unramified; it is known that $V$ is open (cf. N.D.E. (6)) and contains $ϵ(s)$. Hence $W={ϵ}^{-1}(V)$ is an open subset of $S$ containing $s$, such that for every $t\in W$, $\pi$ is unramified at $ϵ(t)$. Since $\pi$ is closed, so is $\pi {|}_W$, hence we may assume $W=S$.

Then, by 2.7.1, $X=G-ϵ(S)$ is a closed subset of $G$ not meeting $G_s$, hence, since $\pi$ is closed, $F=\pi (X)$ is a closed subset of $S$ not containing $s$; set $U=S-F$; then $U$ is an open subset of $S$ such that $ϵ{|}_U$ is an isomorphism of $U$ onto $G{|}_U$. ∎

3. Neutral component of a group locally of finite presentation

3.0.

Given a part $A$ (resp. $B$) of an $S$-scheme $X$ (resp. $Y$), by abuse of notation, $A{\times }_{S}B$ will denote the part of $X{\times }_{S}Y$ formed by the points whose first projection lies in $A$ and second in $B$.

Given a part $A$ of an $S$-group $G$, we shall say that $A$ is stable under the group law of $G$ if $c(A)\subset A$ and µ(A ×_S A) ⊂ A.

Definition 3.1. Let $G$ be an $S$-functor in groups satisfying the following condition:

(+)    for every s ∈ S, the functor G_s = G ⊗_S κ(s) is representable.

One then denotes by $G_s^0$ the connected component of the unit element of the $\kappa (s)$-group $G_s$.²⁵ One defines a sub-$S$-functor in groups of $G$, called the neutral component of $G$, denoted $G^0$, by setting, for every morphism $T\to S$:

G⁰(T) = {u ∈ G(T) | ∀ s ∈ S, u_s(T_s) ⊂ G⁰_s}.

One has thus defined the functor $G\mapsto G^0$ from $(Sch/S)$-gr. to $(Sch/S)$-gr.

Remark 3.2. (i) Let $G$ be an $S$-functor in groups satisfying condition (+); then $Lie(G^0/S)=Lie(G/S)$, by virtue of Exposé II.²⁶ Indeed, for every $S$-scheme $T$, denote by I_T the $T$-scheme of dual numbers over $T$ and by $\tau :T\to I_T$ the zero section (cf. II, 2.1). By definition, one has

Lie(G/S)(T) = {u ∈ G(I_T) | u ∘ τ = e},

where $e:T\to G$ denotes the composition of $T\to S$ and the unit section $S\to G$, and likewise

Lie(G⁰/S)(T) = {u ∈ G⁰(I_T) | u ∘ τ = e}
              = {u ∈ G(I_T) | u ∘ τ = e and u_s((I_T)_s) ⊂ G⁰_s, ∀ s ∈ S}.

Now, for every $s\in S$, $T_s$ and $(I_T{)}_s=I_{T_s}$ have the same underlying set, hence if $u\in Lie(G/S)(T)$, one has $u_s(I_{T_s})=e_s$, where $e_s$ denotes the unit point of $G_s$, whence $u\in Lie(G^0/S)(T)$. So the inclusion $Lie(G^0/S)(T)\subset Lie(G/S)(T)$ is an equality for every $T$, whence $Lie(G^0/S)=Lie(G/S)$.

(ii) Let $G$ and $G^{\prime }$ be two $S$-functors in groups satisfying (+); then:

a) if $G\subset G^{\prime }$, then $G^0\subset G^{\prime 0}$,

b) if $G\subset G^{\prime }$ and $G^{\prime 0}\subset G$, then $G^0=G^{\prime 0}$,

c) if for every $s\in S$, $G_s$ is locally of finite type over $\kappa (s)$, then $G^0$ satisfies property (+), by (VI_A, 2.3), and one has $(G^0{)}^0=G^0$.

Proposition 3.3. Let $G$ be an $S$-functor in groups satisfying condition (+) and let $S^{\prime }$ be an $S$-scheme. Then $(G{\times }_S{S}^{\prime }{)}^0=G^0{\times }_S{S}^{\prime }$; in other words, the functor $G\mapsto G^0$ commutes with base change, i.e. the following diagram is commutative:

(Sch/S)-gr. ───── G ↦ G⁰ ────→ (Sch/S)-gr.
     │                              │
     ▼                              ▼
(Sch/S′)-gr. ─────────────────→ (Sch/S′)-gr.

It suffices indeed to check that, for every $s^{\prime }\in S^{\prime }$ with image $s$ in $S$, $((G{\times }_S{S}^{\prime }){\otimes }_{S^{\prime }}\kappa (s^{\prime }){)}^0=(G_s{\otimes }_{\kappa (s)}\kappa (s^{\prime }){)}^0$ equals $G_s^0{\otimes }_{\kappa (s)}\kappa (s^{\prime })$; this follows from (VI_A, 2.1.2). Note, for later use in 4.2, that we have not used the group structure of $G_s$, only the fact that $G_s^0$ has a rational point, namely $ϵ(s)$, hence is geometrically connected (see also EGA IV₂, 4.5.14). ∎

Special case 3.4. Let $G$ be an $S$-group scheme; denote by $G^0$ the subset of $G$ equal to the union of the $G_s^0$ as $s$ runs through $S$. Then $G^0$ is a part of $G$ stable under the group law of $G$ (cf. 3.0), and for any morphism $S^{\prime }\to S$ one has:

G⁰(S′) = {u ∈ G(S′) | u(S′) ⊂ G⁰}.

When $G^0$ is an open part of $G$, it is clear that $G^0$ is representable by the subscheme of $G$ induced on the open $G^0$.

Proposition 3.5. Let $S$ be a quasi-compact and quasi-separated scheme, and $G$ an $S$-group with fibers locally of finite type. Then there exists a quasi-compact open subset $U$ of $G$ containing $G^0$.

The unit section $ϵ$ being an immersion, $ϵ(S)$ is a quasi-compact subspace of $G$, so there exists a quasi-compact open subset $V$ of $G$ containing $ϵ(S)$. Since $S$ is quasi-separated and $V$ quasi-compact, $V$ is quasi-compact over $S$ (EGA IV₁, 1.2.4), so $V{\times }_{S}V$ is quasi-compact over $S$, hence quasi-compact. Then V · V = µ(V ×_S V) is quasi-compact. Set $V_s=V\cap G_s$ and $V_s^0=V\cap G_s^0$. Then $V_s^0$ is an open subset of $G_s^0$, dense in $G_s^0$ since $G_s^0$ is irreducible (VI_A, 2.4), so $V_s^0\cdot V_s^0=G_s^0$ (VI_A, 0.5), which shows that $V_s\cdot V_s\supset G_s^0$, hence $V\cdot V\supset G^0$. Finally, since $V\cdot V$ is quasi-compact, there exists a quasi-compact open subset $U$ of $G$ containing $V\cdot V$ and a fortiori $G^0$. ∎

Corollary 3.6. Let $G$ be an $S$-group with fibers locally of finite type and connected. Then $G$ is quasi-compact over $S$.

Our assertion being local on $S$ (EGA I, 6.6.1), one reduces to the case where $S$ is affine. By 3.5, there then exists a quasi-compact open $U$ of $G$ containing $G^0=G$, hence $G$ is quasi-compact, hence quasi-compact over the affine scheme $S$ (EGA I, 6.6.4 (v)). ∎

Proposition 3.7. ²⁷ Let $G$ be an $S$-group locally of finite presentation. Then:

(i) $G^0$ is ind-constructible in $G$.

(ii) If moreover $G$ is quasi-separated over $S$, and $S$ quasi-compact and quasi-separated, then $G^0$ is constructible.

(iii) Consequently, if $G$ is quasi-separated over $S$, $G^0$ is locally constructible.

Proof. Let us first show the first assertion. Since $\pi :G\to S$ is locally of finite presentation, given $s\in S$, there exists an open subset $U$ of $G$ containing $ϵ(s)$ and an open subset $V$ of $S$ containing $s$ such that $\pi (U)\subset V$ and such that the morphism ${\pi }^{\prime }:U\to V$ deduced from $\pi$ is of finite presentation. Then $T={ϵ}^{-1}(U)$ is an open subset of $S$ and if we let $W={\pi }^{\prime -1}(T)$ and ${\pi }^{\prime \prime }={\pi }^{\prime }{|}_W$, then ${\pi }^{\prime \prime }:W\to T$ is of finite presentation, and admits as section the morphism ${ϵ}^{\prime \prime }:T\to W$ deduced from $ϵ$.

For every $t\in T$, since $G_t^0$ is irreducible (VI_A, 2.4), $W\cap G_t^0$ is dense in $G_t^0$, hence irreducible, hence connected: it is therefore the connected component of ${\pi }^{\prime \prime -1}(t)$ containing ${ϵ}^{\prime \prime }(t)$. It then follows from EGA IV₃, 9.7.12 that the union $W^0$ of the $W\cap G_t^0$, for $t\in T$, is locally constructible in $W$. On the other hand, it follows from (VI_A, 0.5) that $G^0\cap {\pi }^{-1}(T)=W^0\cdot W^0$, i.e. $G^0\cap {\pi }^{-1}(T)$ is the image of $W^0{\times }_T{W}^0$ under the morphism µ″ : W ×_T W → π⁻¹(T) deduced from µ.²⁸ Since $W{\times }_{T}W$ (resp. ${\pi }^{-1}(T)$) is of finite presentation (resp. locally of finite presentation) over $T$, µ″ is locally of finite presentation and quasi-separated, by EGA IV₁, 1.4.3 and 1.2.2; if moreover ${\pi }^{-1}(T)$ is quasi-separated over $T$, then µ″ is quasi-compact (loc. cit. 1.2.4), hence of finite presentation. Since $W^0{\times }_T{W}^0$ is locally constructible in $W{\times }_{T}W$ (since $W^0$ is in $W$), it follows from Chevalley's constructibility theorem (loc. cit., 1.8.4 and 1.9.5 (viii)) that $G^0\cap {\pi }^{-1}(T)$ is ind-constructible in ${\pi }^{-1}(T)$, and is locally constructible in ${\pi }^{-1}(T)$ if $G$ (and hence ${\pi }^{-1}(T)$) is quasi-separated over $T$. This proves assertions (i) and (iii).

Suppose now $G$ quasi-separated over $S$, and $S$ quasi-compact and quasi-separated. Then, by 3.5, there exists a quasi-compact open $U$ of $G$ containing $G^0$. Since $G$ is quasi-separated over $S$, $G$ is quasi-separated, so the open $U$ is retrocompact (EGA IV₁, 1.2.7), and it suffices to show that $G^0$ is constructible in $U$ (EGA 0_III, 9.1.8). Moreover, $U$ being quasi-compact, hence quasi-compact over $S$ (EGA IV₁, 1.2.4), and quasi-separated over $S$, the restriction of $\pi$ to $U$ is of finite presentation, so by EGA IV₃, 9.7.12, $G^0$ is locally constructible in $U$, hence constructible in $U$, since $U$ is quasi-compact and quasi-separated (EGA IV₁, 1.8.1). This proves (ii), and it follows that for every quasi-compact and quasi-separated open $T$ of $S$ (e.g., for every affine open of $S$), $G^0\cap {\pi }^{-1}(T)$ is constructible. ∎

Corollary 3.8. Let S_0 be a quasi-compact and quasi-separated scheme, $I$ a filtered increasing preordered set, $(A_i{)}_{i\in I}$ a direct system of commutative quasi-coherent $O_{S_0}$-algebras, $A=\lim A_i$ (direct limit), $S_i=\mathrm{Spec}{A}_i$ for $i\in I$, and $S=\mathrm{Spec}A$ (cf. EGA II, 1.3.1).

Let $G$ be an S_0-group scheme locally of finite presentation. Then the canonical map $\lim G^0(S_i)\to G^0(S)$ is bijective.

Since $G$ is locally of finite presentation over S_0, the canonical map $\lim G(S_i)\to G(S)$ is bijective, by EGA IV₂, 8.14.2 c). It follows immediately that the canonical map $\lim G^0(S_i)\to G^0(S)$ is injective. Let us show that it is surjective. Let $g\in G^0(S)\subset G(S)$. There exists $i\in I$ such that $g$ factors through $g_i\in G(S_i)$ via $S\to S_i$; by hypothesis, $g^{-1}(G^0)=S$. But, by 3.7, $G^0$ is ind-constructible in $G$, so $g_i^{-1}(G^0)$ is ind-constructible in $S_i$. It then follows from EGA IV₂, 8.3.4, that there exists an index $j\ge i$ such that $g_j^{-1}(G^0)=S_j$, where $g_j$ is the map deduced from $g_i$ by the base change $S_j\to S_i$. This shows that $g_j\in G^0(S_j)$, hence that $g$ comes from an element of $\lim G^0(S_i)$. ∎