Exposé VI_A. Generalities on algebraic groups

by P. Gabriel

¹ Version of 13/10/2024.

Throughout this entire chapter, $A$ will denote an Artinian local ring with residue field $k$. A group scheme $G$ over $\mathrm{Spec}A$ will be called simply an $A$-group. This $A$-group is said to be locally of finite type if the underlying scheme is locally of finite type over $A$; it is said to be algebraic if the underlying scheme is of finite type over $A$.

0. Preliminary remarks

0.1.

Let us first consider a group scheme $G$ over an arbitrary scheme $S$. We call multiplication the structural morphism µ : G ×_S G → G, and inversion the morphism $c:G\to G$ defined by the equalities $c(T)(x)=x^{-1}$ (with $T$ a scheme over $S$ and $x$ an element of $G(T)$). If $U$ and $V$ are subsets of the underlying set of $G$, we write $U\cdot V$ for the image under the multiplication morphism of the part of $G{\times }_{S}G$ consisting of points whose first projection lies in $U$ and second projection in $V$. Likewise, the notations $U^{-1}$ and $c(U)$ are equivalent.

Let $p{r}_1$ denote the projection of $G{\times }_{S}G$ onto the first factor and $\sigma :G{\times }_{S}G\to G{\times }_{S}G$ the morphism with components $p{r}_1$ and µ. For every $S$-scheme $T$, $\sigma (T)$ is the map $(x,y)\mapsto (x,xy)$; it follows that $\sigma$ is an automorphism. The composition of this automorphism with the projection $p{r}_2$ of $G{\times }_{S}G$ onto the second factor is the multiplication morphism. When $G$ is flat over $S$, $p{r}_2$ and hence µ are flat morphisms; when $G$ is smooth over $S$, $p{r}_2$ and hence µ are smooth morphisms, etc.

0.2.

We assume from now on that $S$ is the spectrum of an Artinian local ring $A$ with residue field $k$. We denote by $(Sch/k{)}_{red}$ the category of reduced schemes over $k$. For every scheme $X$ over $A$, the reduced scheme $X_{red}$ is an object of $(Sch/k{)}_{red}$, and the functor $X\mapsto X_{red}$ is right adjoint to the inclusion of $(Sch/k{)}_{red}$ into $(Sch/A)$. It follows that, for every $A$-group $G$, $G_{red}$ is a group in the category $(Sch/k{)}_{red}$,² i.e., for every reduced $k$-scheme $T$, $G_{red}(T)$ is equipped with a group structure, functorial in $T$. One should beware that $G_{red}$ is not necessarily a $k$-group, since the "multiplication" is only a morphism from (G_red ×_k G_red)_red into $G_{red}$.³

⁴ However, if $k$ is a perfect field, the inclusion of $(Sch/k{)}_{red}$ into $(Sch/k)$ commutes with products, so that groups in the category $(Sch/k{)}_{red}$ are identified with group schemes over $k$ whose underlying scheme is reduced. In this case, if $G$ is a $k$-group, $G_{red}$ is a group subscheme of $G$; this subgroup is not in general normal in $G$.

For example, if $k$ is a field of characteristic 3, the constant group $(\mathbb{Z}/2\mathbb{Z}{)}_k$ acts non-trivially on the diagonalizable group $D_k(\mathbb{Z}/3\mathbb{Z})$ (cf. Exp. I, 4.1 and 4.4); if $G$ denotes the semidirect product of $D_k(\mathbb{Z}/3\mathbb{Z})$ by $(\mathbb{Z}/2\mathbb{Z}{)}_k$ defined by this action, $G_{red}$ is identified with $(\mathbb{Z}/2\mathbb{Z}{)}_k$ and is not normal in $G$.⁵

Let $k$ be an arbitrary field, $k^{p^{-\infty }}$ its perfect closure, and $H$ a group in the category $(Sch/k{)}_{red}$. Then $(H{\otimes }_k{k}^{p^{-\infty }}{)}_{red}$ is a group scheme over the perfect field $k^{p^{-\infty }}$. Since $H{\otimes }_k{k}^{p^{-\infty }}$ and $(H{\otimes }_k{k}^{p^{-\infty }}{)}_{red}$ have the same underlying topological space, one sees that the groups of $(Sch/k{)}_{red}$ share with $k$-groups certain topological properties invariant under extension of the base field: for example, it will follow from 0.3 and the remarks just made that every group of $(Sch/k{)}_{red}$ is separated.

We shall encounter groups of $(Sch/k{)}_{red}$ in what follows whenever we deal with a non-empty, locally closed subset $U$ of an $A$-group $G$ such that $U\cdot U=U$ and $U^{-1}=U$: indeed, the reduced subscheme of $G$ defined by $U$ is then a group of $(Sch/k{)}_{red}$.

0.3.

An $A$-group $G$ is always separated, since the unit section $e:\mathrm{Spec}A\to G$ is a closed immersion. Indeed, let $x$ be the unique point of $\mathrm{Spec}A$ and $\eta$ the structural morphism $G\to \mathrm{Spec}A$. Since $\eta \circ e=i{d}_{\mathrm{Spec}A}$, for every affine open $U=\mathrm{Spec}B$ of $G$ containing $e(x)$, the morphism $B\to A$ has a section, and so is surjective. It follows that $e$ is a closed immersion.⁶ Now the diagonal of $G{\times }_{A}G$ is identified with the functor from $(Sch/A{)}^{\circ }$ with values in (Ens) that associates to every scheme $S$ over $A$ the inverse image of the unit element of $G(S)$ under the map $\varphi (S):(x,y)\mapsto x\cdot y^{-1}$ from $G(S)\times G(S)$ to $G(S)$. We therefore have the cartesian square below, so that the diagonal morphism, being obtained from a closed immersion by base change, is itself a closed immersion:

            φ
G ×_S G ─────────→ G
   │                ▲
   │ diagonal       │ unit section
   ▼                │
   G ─────────→ Spec A.

0.4.

⁷ Let $G$ be an $A$-scheme. We shall say that a point $g$ of $G$ is strictly rational over $A$ if there exists an $A$-morphism $s:\mathrm{Spec}A\to G$ sending the unique point of $\mathrm{Spec}A$ to $g$, i.e. if the morphism $A\to O_{G,g}$ admits a retraction. Note that one then has $\kappa (g)=k$, and so $g$ is a closed point of $G$ (if $B$ is the ring of an affine open neighborhood of $g$ and $\mathfrak{p}$ the prime ideal of $B$ corresponding to $g$, then $B/\mathfrak{p}\subset k$ is a finite $A$-algebra, hence an integral Artinian ring, hence a field).

Suppose henceforth that $G$ is an $A$-group; then such a section $s:\mathrm{Spec}A\to G$ defines an automorphism $r_s$ of the scheme $G$ over $A$, which we call right translation by $s$: for every morphism $\pi :S\to \mathrm{Spec}A$, $r_s(\pi )$ is the automorphism of $G(S)$ defined by $x\mapsto x\cdot G(\pi )(s)$, for every $x\in G(S)$. Similarly, we write ${\ell }_s$ for the left translation by $s$, i.e. the automorphism of $G$ defined by the equalities ${\ell }_s(\pi )(x)=G(\pi )(s)\cdot x$, for every $x\in G(S)$.

Since $G{\otimes }_{A}k$ and $G$ have the same underlying topological space $G$, since $G{\otimes }_{A}k$ is a $k$-group, and since $s{\otimes }_{A}k$ depends only on $g$ and not on $s$, one sees that the automorphisms of $G$ induced by $r_s$ and ${\ell }_s$ (or by $r_{s\otimes k}$ and ${\ell }_{s\otimes k}$) depend only on $g$ and not on $s$; when $P$ is a subset of $G$, we may therefore write $r_g(P)$ or $P\cdot g$ (resp. ${\ell }_g(P)$ or $g\cdot P$) instead of $r_s(P)$ (resp. ${\ell }_s(P)$), in agreement with 0.1.

Remark 0.4.1. If $g$ is a strictly rational point of $G$ and if $A\to A^{\prime }$ is a morphism of Artinian local rings, then $G^{\prime }=G{\otimes }_A{A}^{\prime }$ has a unique point $g^{\prime }$ above $g$, and $g^{\prime }$ is strictly rational over $A^{\prime }$; moreover, if one writes $P^{\prime }$ for the inverse image of $P$ in $G^{\prime }$, then $P^{\prime }\cdot g^{\prime }$ is the inverse image of $P\cdot g$ (cf. EGA I, 3.4.8).

Proposition 0.5. ⁸ Let $G$ be an $A$-group and U, V two dense open subsets of $G$. Then $U\cdot V$ (i.e. the image of $U{\times }_{A}V$ under multiplication) is equal to the whole underlying space of $G$.

Proof. Since $G$ and $G{\otimes }_{A}k$ have the same underlying space, we may, possibly replacing $A$ by $k$ and $G$ by $G{\otimes }_{A}k$, assume that $A=k$. Let $g\in G$. Set $K=\kappa (g)$; then left translation ${\ell }_g$ is an automorphism of G_K. Since the projection $G_K\to G$ is open, U_K and V_K are dense open subsets of G_K, as is the image of V_K under ${\lambda }_g$. There exists therefore $v\in V_K$ such that $u={\ell }_g(v)$ belongs to U_K. Let $L$ be an extension of $K$ containing $\kappa (v)$ (and so $\kappa (u)$), and let $g_L$ and $v_L$ be the $L$-points of G_L deduced from $g$ and $v$. Then $g_L\cdot v_L=u^{\prime }$ is a point of G_L above $u$, and so $g_L=u^{\prime }\cdot v_L^{-1}$ lies above $U\cdot V$, whence $g\in U\cdot V$, which proves the proposition.

Corollary 0.5.1. ⁹ If $G$ is an irreducible $A$-group, then $G$ is quasi-compact.

Proof. Let $U$ be a non-empty affine open subset of $G$; then $U$ is dense in $G$, so by 0.5 the morphism µ : U ×_A U → G is surjective, and therefore $G$ is quasi-compact since $U{\times }_{A}U$ is.

Corollary 0.5.2. ¹⁰ Let $G$ be an $A$-group scheme, and $H$ a group subscheme of $G$ over $A$. Then $H$ is closed.

Proof. Let $k^{\prime }$ be the perfect closure of the residue field $k$ of $A$. Since the underlying topological spaces of $G$ and $H$ are unchanged under the base change $A\to k\to k^{\prime }$, we may suppose that $A=k$ is a perfect field. We may then suppose that $G$ and $H$ are reduced, hence geometrically reduced.

Let $\bar{H}$ be the closure of $H$; then µ⁻¹(H̄) is a closed subset of $G\times G$ containing $H\times H$. Since the morphism $H\to \mathrm{Spec}k$ (resp. $\bar{H}\to \mathrm{Spec}k$) is universally open, and since $H$ is dense in $\bar{H}$, then $H\times H$ is dense in $H\times \bar{H}$ and $H\times \bar{H}$ is dense in $\bar{H}\times \bar{H}$, so $H\times H$ is dense in $\bar{H}\times \bar{H}$. Therefore µ(H̄ × H̄) ⊂ H̄, and so, since $\bar{H}\times \bar{H}$ is reduced, µ induces a morphism µ′ : H̄ × H̄ → H̄.

Let then $g\in \bar{H}$, and set $K=\kappa (g)$. Since the projection ${\bar{H}}_K\to \bar{H}$ is open, H_K and ${\ell }_g(H_K)$ are two dense open subsets of ${\bar{H}}_K$, so there exist $u,v\in H_K$ such that $\ell (g)(v)=u$. One concludes, as in the proof of 0.5, that $g$ belongs to $H\cdot H=H$, whence $\bar{H}=H$.

1. Local properties of an $A$-group locally of finite type

We shall first see that, if $G$ is locally of finite type and flat over $A$, one can "make strictly rational any closed point of $G$" by means of a finite, flat extension of the base.

1.1.

Unless explicit mention is made to the contrary, we assume from now on that $G$ is an $A$-group locally of finite type. When $A$ is a field $k$, we shall obtain in Exposé VII_B very precise results on the local rings of $G$.¹¹ We content ourselves here with a few elementary results:

Proposition 1.1.1. Let $x$ be a point of an $A$-group $G$ locally of finite type and flat over $A$. Then the local ring $O_{G,x}$ is Cohen–Macaulay, and there exists a system $a_1,\cdots ,a_n$ of parameters of $O_{G,x}$ such that $O_{G,x}/(a_1,\cdots ,a_n)$ is a finite and flat (hence finite and free) $A$-module.

We first assume $A$ equal to its residue field $k$; it then suffices to prove that $O_{G,x}$ is Cohen–Macaulay and one may limit oneself to the case where $x$ is a closed point (cf. EGA 0_IV, 16.5.13). By Lemma 1.1.2 below, $G$ contains a closed point $y$ such that $O_{G,y}$ is Cohen–Macaulay. By SGA 1, I § 9, this amounts to saying that, for every finite extension $K$ of the base field $k$ and every point ȳ of $\bar{G}=G{\otimes }_{k}K$ above $y$, $O_{\bar{G},\bar{y}}$ is Cohen–Macaulay. If the extension $K$ has been chosen large enough — i.e. if $K$ contains a normal extension of $k$ containing the residue fields $\kappa (y)$ and $\kappa (x)$ — then ȳ is (strictly) rational over $K$, as is every point $\bar{x}$ of Ḡ above $x$.¹² Since the automorphism $r_{\bar{x}}\circ (r_{\bar{y}}{)}^{-1}$ sends ȳ to $\bar{x}$, it follows that $O_{\bar{G},\bar{x}}$, and hence $O_{G,x}$ (SGA 1, I § 9), are Cohen–Macaulay.

When $A$ is again assumed arbitrary, the preceding argument applies to $k{\otimes }_{A}G$, so that $k{\otimes }_A{O}_{G,x}$ is Cohen–Macaulay. If $a_1,\cdots ,a_n$ is a sequence of elements of $O_{G,x}$ whose image in $k{\otimes }_A{O}_{G,x}$ is a system of parameters, it follows from SGA 1, IV 5.7 or from EGA 0_IV, 15.1.16, that $a_1,\cdots ,a_n$ is an $O_{G,x}$-regular sequence and that $O_{G,x}/(a_1,\cdots ,a_n)$ is finite and flat (hence finite and free) over $A$.

Lemma 1.1.2. Every non-empty scheme $X$, locally of finite type over an Artinian ring $A$, contains a closed point $x$ whose local ring is Cohen–Macaulay.

We may of course assume $X$ affine with algebra $B$, and argue by induction on $\dim X$ (the assertion is clear if $X$ is discrete, since all local rings are then Artinian). Since $B$ is of finite type over $A$, if $\dim B>0$, $B$ contains an element $a$ that is non-invertible and not a zero-divisor.¹³ The closed subscheme $X^{\prime }=\mathrm{Spec}B/(a)$ of $X$ is then of dimension strictly less than $\dim X$, and by induction contains a closed point $x$ such that $O_{X^{\prime },x}$ is Cohen–Macaulay. Since $O_{X^{\prime },x}=O_{X,x}/(a)$ and $a$ is non-invertible and not a zero-divisor in $O_{X,x}$, then $O_{X,x}$ is Cohen–Macaulay (see also EGA IV_2, 6.11.3).

Proposition 1.2. Let $A$ be an Artinian local ring, $G$ an $A$-group locally of finite type and flat over $A$, and $x$ a closed point of $G$. There exists an $A$-algebra $A^{\prime }$ that is local, finite and free over $A$, such that every point of $G{\otimes }_A{A}^{\prime }$ above $x$ is strictly rational over $A^{\prime }$.

¹⁴ Indeed, let $k_1$ be a normal extension of finite degree of $k$ containing the residue field $\kappa (x)$ of $x$. By Lemma V 4.1.2, there exists an $A$-algebra A_1 that is local, finite and free over $A$, with residue field $k_1$. In this case (cf. N.D.E. (11)) all the points $g_1,\cdots ,g_n$ of $G{\otimes }_A{A}_1$ above $x\in G$ have $k_1$ as residue field (i.e. $g_1,\cdots ,g_n$ are rational over A_1 in the sense of Exp. V, § 4.e)).

Let then $B_1,\cdots ,B_n$ be the local rings of $g_1,\cdots ,g_n$. By 1.1.1, $B_1,\cdots ,B_n$ have quotients $B_1^{\prime },\cdots ,B_n^{\prime }$ that are Artinian and finite and free over A_1. Set $A^{\prime }=B_1^{\prime }{\otimes }_{A_1}\cdots {\otimes }_{A_1}{B}_n^{\prime }$. Then $A^{\prime }$ is local, finite and free over A_1 and, for each $i=1,\cdots ,n$, we have surjective homomorphisms

B_i ⊗_{A_1} A′ ↠ B′_i ⊗_{A_1} A′ ↠ A′,

the second induced by the multiplication map $B_i^{\prime }{\otimes }_{A_1}{B}_i^{\prime }\twoheadrightarrow B_i^{\prime }$. Consequently, $A^{\prime }$ answers the question.

1.3.

Let $e$ denote the unit element (or origin) of $G$, i.e. the image of the unique point of $\mathrm{Spec}A$ by the unit section $\mathrm{Spec}A\to G$. By definition itself, $e$ is strictly rational over $A$.

Proposition 1.3.1. ¹⁵ Let $G$ be a group locally of finite type and flat over an Artinian ring $A$, and $K$ (resp. $\bar{k}$) the perfect closure (resp. an algebraic closure) of the residue field $k$ of $A$.

(1) For every closed point $x$ of $\bar{G}=G{\otimes }_k\bar{k}$, the local rings $O_{\bar{G},\bar{e}}$ and $O_{\bar{G},x}$ are isomorphic. In particular, the tangent spaces $T_{e}G$ and $T_{x}G$ have the same dimension.

(2) The following assertions are equivalent:

(i) $G{\otimes }_{A}K$ is reduced. (i bis) $O_{G,e}{\otimes }_{A}K$ is reduced. (ii) $G$ is smooth over $A$. (ii bis) $G$ is smooth over $A$ at the origin.

Proof. (1) Let $x$ be a closed point of Ḡ; there is exactly one $\bar{k}$-morphism $s:\mathrm{Spec}\bar{k}\to \bar{G}$ whose image is $x$; right translation $r_s$ then induces an isomorphism from $O_{\bar{G},\bar{e}}=O_{G,e}{\otimes }_k\bar{k}$ onto $O_{\bar{G},x}$, whence assertion (1).

Let us prove assertion (2). By SGA 1, II.2.1, one reduces immediately to the case where $A$ is a field ($A=k$). The implications (i) ⇒ (i bis), (ii) ⇒ (ii bis), (ii) ⇒ (i) and (ii bis) ⇒ (i bis) are obvious, so it suffices to prove that (i bis) implies (ii).

Now, it follows from (i bis) that $O_{G,e}{\otimes }_k\bar{k}$ is reduced. Hence, by (1), $O_{\bar{G},x}$ is reduced for every closed point $x$ of Ḡ, so that Ḡ is reduced. Then, since Ḡ is locally of finite type over $\bar{k}$, there exists at least one closed point $y$ such that $O_{\bar{G},y}$ is regular. Since, by (1), the local rings of the closed points of Ḡ are all isomorphic to $O_{\bar{G},\bar{e}}$, one sees that all these local rings are regular, so that Ḡ is smooth over $\bar{k}$, and hence $G$ is smooth over $k$.

¹⁶ One may now give the examples below, signalled by M. Raynaud, of group schemes $G$ over a non-perfect field $k$ such that $G_{red}$ is not a group scheme over $k$.

Examples 1.3.2. Let $k$ be a non-perfect field of characteristic $p>0$, $t\in k-k^p$, $\bar{k}$ an algebraic closure of $k$, and $\alpha \in \bar{k}$ such that ${\alpha }^p=t$.

(1) Consider the additive group $G_{a,k}=\mathrm{Spec}k[X]$, and let $G$ be the group subscheme, finite over $k$, defined by the additive polynomial $X^{p^2}-t{X}^p$. Then

G_red = Spec k[X]/X(X^{p(p−1)} − t)

is étale at the origin. If it were a group scheme over $k$, it would be smooth (by 1.3.1); but $G_{red}$ is not geometrically reduced, so it is not a group scheme over $k$.

(2) Consider $G_{a,k}^4=\mathrm{Spec}k[X,Y,U,V]$ and let $G$ be the group subscheme defined by the ideal $I$ generated by the additive polynomials $P=X^p-t{Y}^p$ and $Q=U^p-t{V}^p$. Then $G$ is of dimension 2 and irreducible, since $(G_{\bar{k}}{)}_{red}\cong \mathrm{Spec}\bar{k}[Y,V]$ is.

Let $A=k[X,Y,U,V]$ and $\mathfrak{m}$ its augmentation ideal. Denote by x, y, u, v the images of dX, dY, dU, dV in ${\Omega }_{A/k}^1{\otimes }_A(A/\mathfrak{m})$, viewed as linear forms on the tangent space $k^4=T_0{G}_{a,k}^4$. Let us show that the subspace $E=T_0{G}_{red}$ equals $k^4$. Otherwise, there would exist a linear form $f=ax+by+a^{\prime }u+b^{\prime }v$, with $a,b,a^{\prime },b^{\prime }\in k$ not all zero, vanishing on $E$. Recall that the formation of ${\Omega }_{A/k}^1$ (and hence of tangent spaces) commutes with base change (cf. EGA IV_4, 16.4.5), and identify $f$ with its image in $({\bar{k}}^4)\ast$. Since $(G_{\bar{k}}{)}_{red}\subset (G_{red}{)}_{\bar{k}}$, then $f$ vanishes on the subspace $T_0(G_{\bar{k}}{)}_{red}$ of ${\bar{k}}^4$, which is defined by the equations $g_1=x-\alpha y$ and $g_2=u-\alpha v$, and so f = λg_1 + µg_2, with λ, µ ∈ k̄. Now λg_1 + µg_2 belongs to $k^4$ only if λ = µ = 0! This contradiction shows that $E=k^4$, and so $T_0(G_{red}{)}_{\bar{k}}={\bar{k}}^4$.

On the other hand, $R=XV-YU$ belongs to $\sqrt{I}$, since $R^p=(X^p-t{Y}^p)V^p-Y^p(U^p-t{V}^p)$. Consequently, the tangent space $F$ at the point $(\alpha ,1,\alpha ,1)$ of $(G_{red}{)}_{\bar{k}}$ is contained in the hyperplane $H$ of ${\bar{k}}^4$ with equation $\alpha dV+dX-dU-\alpha dY=0$, hence is of dimension $\le 3$.¹⁷ Therefore, by point (1) of 1.3.1, $G_{red}$ is not a group scheme over $k$.

2. Connected components of an $A$-group locally of finite type

2.1.

Let us first consider an arbitrary $A$-group $G$, and let $G^{\prime }$ be the connected component of the origin $e$ of $G$. This connected component is evidently closed, so that we may identify it with the reduced closed subscheme of $G$ having $G^{\prime }$ as underlying space.¹⁸

Proposition 2.1.1. For every extension $K$ of the residue field $k$ of $A$, $G^{\prime }{\otimes }_{A}K$ has as underlying space the connected component of the origin in the $K$-group $G{\otimes }_{A}K$ (i.e. $G^{\prime }$ is geometrically connected).

Indeed, let $(G{\otimes }_{A}K{)}^{\prime }$ be the connected component of the origin in $G{\otimes }_{A}K$. Since the image of $(G{\otimes }_{A}K{)}^{\prime }$ in $G$ is connected and contains the unit element of $G$, this image is contained in $G^{\prime }$, so that $(G{\otimes }_{A}K{)}^{\prime }$ is contained in the inverse image

$G^{\prime }{\otimes }_{A}K$ of $G^{\prime }$ in $G{\otimes }_{A}K$. The proposition therefore follows from the connectedness of $G^{\prime }{\otimes }_{A}K$, which is proved in Lemma 2.1.2.

Lemma 2.1.2. Let $X$ and $Y$ be two connected schemes over a field $k$. If $X$ contains a rational point, then $X{\times }_{k}Y$ is connected.

We give below a direct proof of this result from EGA IV_2 (4.5.8 and 4.5.14).

Suppose first $Y$ non-empty, connected and affine, with algebra $B$. In this case, $X{\times }_{k}Y$ is the spectrum of the quasi-coherent O_X-algebra $\mathcal{B}=O_X{\otimes }_{k}B$. We want to show that every subset $U$ of $X{\times }_{k}Y$ that is open, closed, and non-empty coincides with $X{\times }_{k}Y$. Now $U$ is affine over $X$ and its affine O_X-algebra is a direct factor of $\mathcal{B}$. It therefore follows from Lemma 2.1.3 below that the image of $U$ in $X$ is open and closed,

i.e. coincides with all of $X$. This image contains in particular a rational point $x$ of $X$, so that $U$ meets the inverse image of $x$ in $X{\times }_{k}Y$. Since this inverse image is isomorphic to $Y$, hence connected, $U$ contains this inverse image. The same would hold for the complement of $U$ in $X{\times }_{k}Y$ if $U$ were distinct from $X{\times }_{k}Y$, which would be absurd.

If $Y$ is now an arbitrary $k$-scheme, what precedes shows that the fibers of the canonical projection $X{\times }_{k}Y\to Y$ are connected. If $x$ is a rational point of $X$, these fibers all meet the subscheme $x{\times }_{k}Y$, which is itself connected, whence the proposition.

Lemma 2.1.3. Let $X$ be a scheme and $\mathcal{A}$ a quasi-coherent O_X-algebra which is locally¹⁹ a direct factor of a free O_X-module. The image of $\mathrm{Spec}\mathcal{A}$ in $X$ is then open and closed.

Let $V$ be this image. It is clear that $V$ is contained in the support of $\mathcal{A}$. Conversely, if $x$ belongs to the support of $\mathcal{A}$, then ${\mathcal{A}}_x$ is non-zero and is a free $O_{X,x}$-module, since by Kaplansky every projective module over a local ring is free. Consequently the fiber of $\mathrm{Spec}\mathcal{A}$ at $x$, which is affine with algebra ${\mathcal{A}}_x{\otimes }_{O_{X,x}}\kappa (x)$, is non-empty. One thus sees that the image of $\mathrm{Spec}\mathcal{A}$ coincides with the support of $\mathcal{A}$.

If $s$ is the unit section of $\mathcal{A}$, the equality $s_x=0$ implies that $s$, and hence $\mathcal{A}$, are

zero in a neighborhood of $x$. So the support of $\mathcal{A}$ is closed. On the other hand, one may assume that $\mathcal{A}$ is a direct factor of the direct sum $L$ of a family $(O_X^{\alpha })$ of copies of O_X.²⁰ For every $\beta$, write ${\varphi }_{\beta }$ for the restriction to $\mathcal{A}$ of the canonical projection of $L={\oplus }_{\alpha }{O}_X^{\alpha }$ onto $O_X^{\beta }$. For $x\in X$, denote by ${\mathcal{A}}_x^{\ast }$ the dual module ${\mathrm{Hom}}_{O_{X,x}}({\mathcal{A}}_x,O_{X,x})$. Since ${\mathcal{A}}_x$ is a direct factor of $L_x$, the restriction map

Hom_{O_{X,x}}(L_x, O_{X,x}) → Hom_{O_{X,x}}(𝓐_x, O_{X,x}) = 𝓐_x^*

is surjective. If ${\mathcal{A}}_x\ne 0$ then, since ${\mathcal{A}}_x$ is free and non-zero, there exist $a\in {\mathcal{A}}_x$ and a linear form $\varphi \in {\mathcal{A}}_x^{\ast }$ such that $\varphi (a)=1$. There exists therefore a family $({\xi }^{\alpha })$ of elements of $O_{X,x}$ such that, for every $u\in {\mathcal{A}}_{X,x}$, one has $\varphi (u)={\sum }_{\alpha }{\xi }^{\alpha }{\varphi }^{\alpha }(u)$ (this sum being finite since ${\varphi }^{\alpha }(u)=0$ except for finitely many $\alpha$). Applying this to $u=a$, one obtains that there exist ${\alpha }_1,\cdots ,{\alpha }_n$ such that

1 = φ(a) = ξ^{α_1} φ^{α_1}(a) + ⋯ + ξ^{α_n} φ^{α_n}(a).

There exists then an open neighborhood $U$ of $x$ such that $a$ and the ${\xi }^{{\alpha }_i}$ come from sections $a\in \mathcal{A}(U)$ and ${\xi }^{{\alpha }_i}\in O_X(U)$, and the equality ${\sum }_i{\xi }^{{\alpha }_i}{\varphi }^{\alpha }(a)=1$ on $U$ shows that ${\tilde{a}}_y\ne 0$ for every $y\in U$. ("The support of a projective module is open.")

2.2.

The notations being still those of 2.1, it is clear that $G^{\prime }$ is a reduced $k$-scheme. Lemma 2.1.2 shows that $G^{\prime }{\times }_k{G}^{\prime }$ is connected, so that $(G^{\prime }{\times }_k{G}^{\prime }{)}_{red}$ is the reduced subscheme of $G{\times }_{A}G$ having as underlying space the connected component of the origin. In particular, the multiplication morphism µ : G ×_A G → G induces a morphism µ′ : (G′ ×_k G′)_red → G′ that makes $G^{\prime }$ a group in $(Sch/k{)}_{red}$.

2.2.bis.

²¹ One recalls (cf. Exp. V) that, if $P$ is a scheme, one writes $P$ for the underlying topological space of $P$. Now one defines a sub-$A$-functor $G^0$ of $G$ by setting, for every $A$-scheme $S$,

G⁰(S) = {u ∈ G(S) | u(S) ⊂ G′}.

Let $c:G\to G$ be the inversion morphism; since $c(G^{\prime })=G^{\prime }$, we have $c\circ u\in G^0(S)$ for every $u\in G^0(S)$. On the other hand, if $u,v\in G^0(S)$, then $u⊠v$ sends $S$ into the subspace of $G{\times }_{A}G$ consisting of points whose two projections belong to $G^{\prime }$; this subspace is identified with the underlying space of $G^{\prime }{\times }_A{G}^{\prime }$, which is connected by Lemma 2.1.2. Consequently, µ ∘ (u ⊠ v) sends $S$ into $G^{\prime }$. This shows that $G^0$ is a sub-$A$-functor in groups of $G$.

If the connected component of $e$ is open in $G$, then the subfunctor $G^0$ is representable by the subscheme induced by $G$ on this open set, which is therefore a group subscheme of $G$; we shall also write $G^0$ for it. In this case, with the notations of 2.1, one has $G^{\prime }=(G^0{)}_{red}$ and the topological spaces $G^{\prime }$ and $G^0$ coincide.²²

2.3.

In accordance with our conventions of 1.1, we again assume from now on that $G$ is locally of finite type over $A$. Then $G$ is locally noetherian, hence locally connected,²³ hence: every connected component of $G$ is open.

We then write $G^0$ for the subscheme induced by $G$ on the connected component $G^{\prime }$ of the unit element. By 2.2.bis, $G^0$ is a group subscheme of $G$, which we shall call the neutral component of $G$;

for every $A$-scheme $S$ we therefore have:

G⁰(S) = {u ∈ G(S) | u(S) ⊂ G⁰ = G′}.

Let $G^{\alpha }$ be an arbitrary connected component of $G$ and ${\nu }^{\alpha }:G^{\alpha }{\times }_A{G}^0\to G$ the morphism defined by the equalities

$${\nu }^{\alpha }(S)(g,\gamma )=g\gamma g^{-1},$$

for every $S\in (Sch/A)$, $g\in G^{\alpha }(S)$, $\gamma \in G^0(S)$.

If $e$ is the origin of $G$, the restriction of ${\nu }^{\alpha }$ to $G^{\alpha }{\times }_{A}e$ is the trivial morphism; since $G^{\alpha }{\times }_A{G}^0$ is connected by 2.1.2, one sees that ${\nu }^{\alpha }$ factors through $G^0$. Therefore, for every $A$-scheme $S$, $G^0(S)$ is a normal subgroup of $G(S)$. We have thus obtained the following proposition:²⁴

Proposition 2.3.1. Let $G$ be an $A$-group locally of finite type. Then the neutral component $G^0$ is an open normal group subscheme of $G$.

Proposition 2.4. Let $G$ be an $A$-group locally of finite type.

(i) $G^0$ is irreducible, and $G^0{\otimes }_{A}k$ is geometrically irreducible over $k$.

(ii) $G^0$ is quasi-compact, hence of finite type over $A$.

²⁵ Proof. (i) Since $G^0$ and $G^0{\otimes }_{A}k$ have the same underlying topological space, it suffices to show the second assertion. Let $\bar{k}$ be an algebraic closure of $k$. By 2.2, $(G{\otimes }_A\bar{k}{)}_{red}$ is a $\bar{k}$-group locally of finite type and reduced, hence smooth over $\bar{k}$ (1.3.1). A fortiori the local rings of $(G{\otimes }_A\bar{k}{)}_{red}$ are integral domains, so,²⁶ since $G{\otimes }_A\bar{k}$ is locally noetherian, the connected components of $G{\otimes }_A\bar{k}$ are irreducible (cf. EGA I, 6.1.10). In particular, the connected component $G^0{\otimes }_A\bar{k}$ (cf. 2.1.1) is irreducible.

(ii) Let us now show that $G^0$ is of finite type over $A$. Since $G^0$ is locally of finite type over $A$, it suffices to prove that $G^0$ is quasi-compact. As $G^0$ is irreducible, this follows from 0.5.1.

Corollary 2.4.1. Every connected component of $G$ is irreducible,²⁷ of finite type over $A$,²⁸ and of the same dimension as $G^0$.

One may indeed assume $A$ equal to its residue field $k$. Let then $C$ be a connected component of $G$, $x$ a closed point of $C$, $\kappa (x)$ the residue field of $x$, and $k^{\prime }$ a normal extension of $k$ containing $\kappa (x)$ and of finite degree over $k$. The canonical projection $\pi :C{\otimes }_k{k}^{\prime }\to C$ is open and closed;²⁹ consequently, if $C^{\prime }$ is a connected component of $C{\otimes }_k{k}^{\prime }$, the projection $C^{\prime }\to C$ is surjective, so $C^{\prime }$ contains a point $y\in {\pi }^{-1}(x)$, and such a point is rational over $k^{\prime }$ (cf. the proof of 1.2), so that $C^{\prime }$ is the image by translation $r_y$ of $G^0{\otimes }_k{k}^{\prime }$. Now $G^0{\otimes }_k{k}^{\prime }$ is of finite type over $k^{\prime }$ by 2.4, and ${\pi }^{-1}(x)$ is finite (of cardinality $\le [k^{\prime }:k]$), so $C{\otimes }_k{k}^{\prime }$ is of finite type over $k^{\prime }$, and hence $C$ is of finite type over $k$.

On the other hand, since $G^0{\otimes }_k{k}^{\prime }$ is irreducible by 2.4, the same holds for $C^{\prime }$,³⁰ and hence also for $C$, since the projection $C^{\prime }\to C$ is surjective.

Finally, we have seen above that $C{\otimes }_k{k}^{\prime }$ is a disjoint union of finitely many translates of $G^0{\otimes }_k{k}^{\prime }$. Since dimension is invariant under extension of the base field (cf. EGA IV_2, 4.1.4), it follows that $C$ has the same dimension as $G^0$. (Moreover, by EGA IV_2, 5.2.1, one has ${\dim }_{g}G=\dim G^0$ for every point $g\in G$.)

2.5.

This paragraph has been added. The results that follow appear in Exp. VI_B, but could (or should) have figured in VI_A, and it is useful to have them available from now on, in order to make precise Theorem 3.2 below.

Lemma 2.5.1. Let $(A,\mathfrak{m})$ be an Artinian local ring, and $k=A/\mathfrak{m}$ its residue field.

(i) If $X$ is an $A$-scheme such that $X{\otimes }_{A}k$ is locally of finite type (resp. of finite type) over $k$, then the same holds for $X$ over $A$.

(ii) Let $u:X\to Y$ be a morphism of $A$-schemes. If $u{\otimes }_{A}k$ is an immersion (resp. a closed immersion), then so is $u$.

Proof. (i) Suppose $X{\otimes }_{A}k$ locally of finite type over $k$. Let $U=\mathrm{Spec}B$ be an affine open of $X$. By hypothesis, there exist elements $x_1,\cdots ,x_n$ of $B$ whose images generate $B/\mathfrak{m}B$ as a $k$-algebra, and it follows from the "nilpotent Nakayama lemma" that the $x_i$ generate $B$ as an $A$-algebra. This proves that $X$ is locally of finite type over $A$. If, in addition, $X{\otimes }_{A}k$ is quasi-compact, then so is $X$ (which has the same underlying topological space), and hence $X$ is of finite type over $A$. This proves (i).

Let us prove (ii). Suppose $u{\otimes }_{A}k$ is an immersion (resp. a closed immersion). Then $u$ is a homeomorphism of $X$ onto a locally closed (resp. closed) part of $Y$, and for every $x\in X$, the ring morphism ${\varphi }_x:O_{Y,u(x)}\to O_{X,x}$ is such that ${\varphi }_x{\otimes }_{A}k$ is surjective. By the nilpotent Nakayama lemma, it follows that ${\varphi }_x$ is surjective, so $u$ is an immersion (resp. a closed immersion).

Proposition 2.5.2. Let $A$ be an Artinian local ring with residue field $k$, and let $u:G\to H$ be a quasi-compact morphism between $A$-group schemes locally of finite type.

(a) The set $u(G)$ is closed in $H$, and its connected components are irreducible and all of the same dimension.

(b) One has dim G = dim u(G) + dim Ker(u).

(c) If $u$ is a monomorphism, it is a closed immersion.

Proof. By the preceding lemma, it suffices to prove the proposition in the case where $A=k$. Moreover, since the properties under consideration are stable by

(fpqc) descent, and since dimension is invariant under extension of the base field, one may assume $k$ algebraically closed.

Let us prove (a). Denote by $C$ the reduced subscheme of $H$ whose underlying topological space is $u(G)$. Since $u(G)$ is stable under the inversion morphism of $H$, so is $C$. On the other hand, $u:G\to C$ is quasi-compact and dominant, so by EGA IV_2, 2.3.7 the same holds for $u{\times }_{k}i{d}_G$ and $i{d}_H{\times }_{k}u$, and hence for their composition $u{\times }_{k}u:G{\times }_{k}G\to C{\times }_{k}C$. Consequently, the multiplication of $H$ sends $C{\times }_{k}C$ into $C$, and so $C$ is a group subscheme of $H$.

So, replacing $H$ by $C$, we reduce to the case where $u$ is dominant. Then $G(k)$ is dense in $H$, hence meets every connected component of $H$, and hence acts transitively on the set of these connected components. It therefore suffices to show that $u(G)$ contains $H^0$. Replacing $G$ by $u^{-1}(H^0)$, we may suppose $H=H^0$; in this case, by 2.4, $H$ is irreducible and of finite type over $k$, hence noetherian. On the other hand, $u$ is locally of finite type (cf. EGA I, 6.6.6) and quasi-compact, hence of finite type. Consequently, by Chevalley's constructibility theorem (cf. EGA IV_1, 1.8.5), $u(G)$ is a constructible (and dense) part of $H=u(G)$, hence contains a dense open $U$ of $H$ (cf. EGA 0_III, 9.2.2). Then, by 0.5, one has $H=U\cdot U\subset u(G)$, whence $u(G)=H$. Taking 2.4.1 into account, this proves assertion (a).

Let us prove (b). 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 $H$. Since $u$ is of finite type, $Ker(u)$ is of finite type over $k$. On the other hand, replacing $H$ by the reduced closed subscheme $u(G)$, we may assume $u$ surjective. Denote by $u^0$ the restriction of $u$ to $G^0$. Since $G$ and $Ker(u)$ are equidimensional, and since $Ker(u{)}^0\subset Ker(u^0)$, we reduce to the case where $G$, and hence also $H$, are irreducible.

Then, by EGA IV_3, 9.2.6.2 and 10.6.1 (ii), the set of $y\in H$ such that dim u⁻¹(y) = dim G − dim H contains a non-empty open set $V$. Since $u$ is surjective, $U=u^{-1}(V)$ is then a non-empty open subset of $G$, hence contains a closed point $x$ of $G$, since $G$ is a Jacobson scheme (cf. EGA IV_3, 10.4.8). Then right translation $r_x$ is an isomorphism of $Ker(u)$ onto $u^{-1}(u(x))$, whence:

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

Let us prove (c), following [DG70], I, § 3.4. (Another proof is given in Exp. VI_B, 1.4.2.) Assume $u$ a monomorphism. If $C$ is a connected component of $G$, there exists a closed point $x\in G$ such that $C=r_x(G^0)$, and if one denotes by $u_C$ (resp. $u^0$) the restriction of $u$ to $C$ (resp. to $G^0$), one has $u_C=r_{u(x)}\circ u^0\circ r_x^{-1}$, so it suffices to show that $u^0$ is a closed immersion. We may therefore suppose $G=G^0$, so that $G$ is irreducible and of finite type over $k$.

Let $\xi$ be the generic point of $G$; then $O_{G,\xi }$ is an Artinian local ring; denote by $\mathfrak{m}$ its maximal ideal. On the other hand, let $h=u(\xi )$, $\mathfrak{n}$ the maximal ideal of $O_{H,h}$, and $A=O_{G,\xi }/\mathfrak{n}{O}_{G,\xi }$. Since $u$ is a monomorphism, so is the morphism $u_h:\mathrm{Spec}(A)\to \mathrm{Spec}(\kappa (h))$ obtained by base change, so the multiplication morphism $A{\otimes }_{\kappa (h)}A\to A$ is an isomorphism (cf. EGA I, 5.3.8), whence $A=\kappa (h)$. By Nakayama's lemma (since $\mathfrak{n}{O}_{G,\xi }$ is contained in $\mathfrak{m}$, hence nilpotent), it follows that the morphism $O_{H,h}\to O_{G,\xi }$ is surjective.

Let then $V$ be an affine open of $H$ containing $h$, $U$ a non-empty affine open of $G$ contained in $u^{-1}(V)$, $\varphi :O_H(V)\to O_G(U)$ the morphism of $k$-algebras induced by $u$, $\mathfrak{p}$ the prime ideal of $O_G(U)$ corresponding to $\xi$, and $\mathfrak{q}={\varphi }^{-1}(\mathfrak{p})$. Since $G$ is of finite type over $k$, $O_G(U)$ is generated as a $k$-algebra by a finite number of elements $a_1,\cdots ,a_n$. By what precedes, there exist elements $b_1,\cdots ,b_n$ and $s$ in $O_H(V)$ such that $s\notin \mathfrak{q}$ and such that one has in $O_G(U{)}_{\mathfrak{p}}$ the equalities $a_i/1=\varphi (b_i)/\varphi (s)$. There exist therefore elements $t_1,\cdots ,t_n$ of $O_G(U)-\mathfrak{p}$ such that $t_i(a_i\varphi (s)-\varphi (b_i))=0$. Then, setting $t=t_1\cdots t_n\varphi (s)\in O_G(U)-\mathfrak{p}$, the equalities $a_i/1=\varphi (b_i)/\varphi (s)$ already hold in $O_G(U{)}_t$ and, since $t\in O_G(U)=k[a_1,\cdots ,a_n]$, there exists $b\in O_H(V)$ such that $t/1=\varphi (b)/\varphi (s{)}^r$, for some $r\in \mathbb{N}$. Consequently, $\varphi$ induces a surjection of $O_H(V{)}_{sb}$ onto $O_G(U{)}_t$, and so $u$ is a local immersion at the point $\xi$.

The open subset $W$ of $G$ consisting of the points at which $u$ is a local immersion is therefore non-empty. Since $G$ is a Jacobson scheme, $W$ contains a closed point $y$, and to show $W=G$, it suffices to show that every closed point of $G$ belongs to $W$. Now every closed point $x$ is the image of $y$ by translation $r_x\circ r_y^{-1}$, hence belongs to $W$, whence $W=G$. This proves that $u$ is a local immersion.

Since $G$ is irreducible, it follows that $u$ is an immersion. Indeed, for every $x\in G$, let $U_x$ and $V_x$ be open subsets of $G$ and $H$ such that $x\in U_x$ and such that $u$ induces a closed immersion of $U_x$ into $V_x$. Since $U_x$ is dense in $G$, $u(U_x)$ is dense in $u(G)\cap V_x$, and since $u(U_x)$ is closed in $V_x$, one has $u(U_x)=V_x$. Moreover, since $u$ is injective, one has $U_x=u^{-1}(V_x)$, and it follows that $u$ induces a closed immersion of $G$ into the open subscheme of $H$ covered by the $V_x$. So $u:G\to H$ is an immersion. But we have already seen that $u(G)$ is closed in $H$, hence $u$ is a closed immersion.

Lemma 2.5.3. ³¹ Let $A$ be an Artinian local ring, $k$ its residue field, $G$ a flat $A$-group, $X$ an $A$-scheme equipped with a left action µ : G ×_A X → X of $G$ and with a section $s_0:\mathrm{Spec}A\to X$. (This is the case, for instance, if $G^{\prime }$ is a second $A$-group and one is given an $A$-group morphism $G\to G^{\prime }$.)

Let $\varphi$ be the morphism µ ∘ (id_G × s_0) from $G=G{\times }_{A}A$ to $X$. If $\varphi$ is flat at a point $g$ of $G$, then $\varphi$ is flat.

Proof. Since $G$ is flat over $A$, the flatness-by-fibers criterion (EGA IV_3, 11.3.10.2) shows that it suffices to show that $\varphi {\otimes }_{A}k$ is flat; hence we may suppose $A=k$. In this case, the datum of $s_0$ is equivalent to that of a $k$-point $x_0\in X(k)$, and $\varphi$ is the morphism $h\mapsto h{x}_0$.

Let then $h\in G$; let us show that $\varphi$ is flat at the point $h$. Let $K$ be an extension of $k$ containing a copy of $\kappa (g)$ and of $\kappa (h)$; one has a cartesian square

            φ_K
G_K ────────────→ X_K
 │                 │
 │                 │
 ▼      φ          ▼
 G ────────────→ X

in which the two vertical arrows are faithfully flat. So, by V 7.4 (i), ${\varphi }_K$ is flat at every point $g^{\prime }\in G_K$ above $g$, and to show that $\varphi$ is flat at $h$, it suffices to show that ${\varphi }_K$ is flat at a point $h^{\prime }$ above $h$. We are thus reduced to the case where $g$ and $h$ are rational. Let then $u=h{g}^{-1}$ and ${\ell }_u$ (resp. µ_u) the left translation of $G$ (resp. of $X$) defined by $u$; since φ ∘ ℓ_u = µ_u ∘ φ, one obtains a commutative square

                ~
O_{G,h} ──────────→ O_{G,g}
    ▲                  ▲
    │                  │
    │       ~          │
O_{X, hx_0} ────→ O_{X, gx_0}

in which the horizontal arrows are isomorphisms. Since the morphism $O_{X,g{x}_0}\to O_{G,g}$ is flat, the morphism $O_{X,h{x}_0}\to O_{G,h}$ is also flat.

Proposition 2.5.4. Let $A$ be an Artinian local ring, $k$ its residue field, $G$ an $A$-group locally of finite type, $X\ne \varnothing$ an $A$-scheme locally of finite type equipped with a left action of $G$. Assume that the morphism $\varphi :G{\times }_{A}X\to X{\times }_{A}X$ defined set-theoretically by $(g,x)\mapsto (gx,x)$ is surjective. Then:

(i) The connected components of $X$ are of finite type, irreducible, and all of the same dimension.

(ii) More precisely, let $\bar{k}$ be an algebraic closure of $k$ and $x$ a closed point of $X{\otimes }_A\bar{k}$; its stabilizer $F$ is a closed group subscheme of $G{\otimes }_A\bar{k}$, and the dimension of the irreducible components of $X$ is $\dim G-\dim F$.

Taking Lemma 2.5.1 into account, we may suppose $A=k$. Suppose first $k$ algebraically closed. Then $G_{red}$ is a $k$-group locally of finite type, and hence, replacing $G$ by $G_{red}$ and $X$ by $X_{red}$, we may suppose $G$ and $X$ reduced.

Since $G{\times }_{k}X$ is locally of finite type over $k$, $\varphi$ is locally of finite type (cf. EGA I, 6.6.6 (v)), hence locally of finite presentation since $X{\times }_{k}X$ is locally noetherian. Let $x$ be a rational point of $X$; then the morphism ${\varphi }_x:G\to X$ obtained from $\varphi$ by base change is surjective and locally of finite presentation. If $\eta$ is a maximal point of $X$, then $O_{X,\eta }$ is a field (since $X$ is reduced), so ${\varphi }_x$ is flat at every point of $G$ above $\eta$. So, by Lemma 2.5.3, ${\varphi }_x$ is flat. Consequently, ${\varphi }_x:G\to X$ is faithfully flat and locally of finite presentation, hence open (cf. EGA IV_2, 2.4.6). Since $G^0$ is open in $G$, irreducible and quasi-compact (by 2.4), each orbit $G^{0}x={\varphi }_x(G^0)$, for $x$ running through the rational points of $X$, is an open subset of $X$, irreducible and quasi-compact, hence of finite type over $k$ (since $X$ is locally of finite type over $k$).

Since every non-empty open subset of $X$ contains a rational point, it follows that $X$ is covered by these open sets. Moreover, two such open sets are either disjoint or equal. Indeed, if ${\varphi }_x(G^0)\cap {\varphi }_y(G^0)$ is non-empty, it contains a rational point $z$, and there exist therefore two rational points $g,h\in G^0$ such that $gx=z=hy$, whence $x=g^{-1}z$ and $y=h^{-1}z$, and so ${\varphi }_x(G^0)={\varphi }_z(G^0)={\varphi }_y(G^0)$. It follows that the orbits ${\varphi }_x(G^0)$ are also closed, and so are at once the connected components and the irreducible components of $X$.

Finally, let x, y be two rational points of $X$. Since ${\varphi }_x$ is surjective, there exists a rational point $g\in G$ such that $y=gx$, and since $G^0$ is a normal subgroup of $G$, the orbit $G^{0}y$ is the image of $G^0$ under translation ${\ell }_g$ of $X$, so that $G^{0}y$ and $G^{0}x$ have the same dimension.

Moreover, by I, 2.3.3.1, the stabilizer of $x$ is represented by the closed subscheme $F$ of $G$ defined by the cartesian square below:

F ─────────→ G
│            │
│            │ φ_x
▼            ▼
Spec k ────→ X.

Then $F$ is a $k$-group locally of finite type, $F\cap G^0$ is a $k$-group of finite type containing $F^0$, and by 2.4.1, $F$ and $F\cap G^0$ are equidimensional, of the same dimension as $F^0$. Let $C={\varphi }_x(G^0)$ be the irreducible component of $X$ containing $x$. Proceeding as in the proof of point (b) of 2.5.2, one obtains that dim C = dim G⁰ − dim F⁰ = dim G − dim F.

In the general case (i.e. for $k$ an arbitrary field), let $\bar{k}$ be an algebraic closure of $k$. Let $C$ be a connected component of $X$ and $C^{\prime }$ a connected component of $C{\otimes }_k\bar{k}$; then $C^{\prime }$ is a connected component of $X^{\prime }=X{\otimes }_k\bar{k}$. The morphism $\pi :X^{\prime }\to X$ is open (cf. EGA IV_2, 2.4.10), and since it is integral, it is also closed; consequently $\pi (C^{\prime })=C$. Since $C^{\prime }$ is irreducible and quasi-compact, $C$ is irreducible and quasi-compact, hence of finite type over $k$ (since $X$ is locally of finite type over $k$).

Finally, since dimension is invariant under extension of the base field (cf. EGA IV_2, 4.1.4), $\dim C=\dim C^{\prime }$, and since all the irreducible components of $X^{\prime }$ have the same dimension, the same holds for those of $X$.

2.6. Complements.

This paragraph has been added, drawn from [Per75] II, §§ 1–2, with complements due to O. Gabber.³² This shows that the preceding results are valid for any group scheme $G$ over a field $k$. (This will be used in sections 5, 6 and 7 of Exp. VI_B.)

We fix a field $k$. Let us begin with the following lemma (loc. cit., II 2.1.1), which does not appear explicitly in EGA IV_2, § 4.4 (although it can perhaps be read between the lines at the beginning of loc. cit., § 4.4.1).

Lemma 2.6.0. Let $X$ be an irreducible $k$-scheme, $K$ an extension of $k$, $X^{\prime }$ an irreducible component of X_K. The projection $X^{\prime }\to X$ is surjective.

Indeed, let $B$ be a transcendence basis of $K$ over $k$ and $L=k(B)\subset K$. By EGA IV_2, 4.3.2 and 4.4.1, X_L is irreducible and $X^{\prime }$ dominates X_L. The morphism $X^{\prime }\to X_L$ is therefore integral and dominant, hence surjective. Since $X_L\to X$ is surjective (loc. cit., 4.4.1), so is $X^{\prime }\to X$.

For the rest of 2.6, we fix a $k$-group scheme $G$ and an action µ : G × X → X of $G$ on a $k$-scheme $X$ satisfying the following condition:

(⋆)    the morphism Φ : G × X → X × X, (g, x) ↦ (gx, x) is surjective

(this is the case in particular for $G$ acting on itself by left translations). We shall then say, for short: "Let $X$ be a $G$-scheme satisfying (⋆)". Finally, we write $k^{\prime }$ for the perfect closure of $k$.

Proposition 2.6.1. Let $X$ be a $G$-scheme satisfying $(\star )$.

(i) $X$ is geometrically pointwise irreducible over $k$, i.e. for every extension $K$ of $k$, each $x\in X_K$ belongs to a unique irreducible component of X_K.

(ii) Each local ring of $(X_{k^{\prime }}{)}_{red}$ is normal.

(iii) Let $\eta$ be a maximal point of $(X_{k^{\prime }}{)}_{red}$, $C={\eta }^{-}$, and $L$ the algebraic closure of $k$ in $\kappa (\eta )$. Then $C$ is an $L$-scheme, and is geometrically irreducible over $L$ (i.e. $C{\otimes }_{L}K$ is irreducible for every extension $K$ of $L$).

(iv) In particular, if $x$ is a rational point of $X$, then the irreducible component of $X$ containing $x$ is geometrically irreducible over $k$.

Proof. (i) Since hypothesis $(\star )$ is preserved by every base change $k\to K$, it suffices to show that each $x\in X$ belongs to a unique irreducible component of $X$. Since the morphism $\mathrm{Spec}(k^{\prime })\to \mathrm{Spec}(k)$ is a universal homeomorphism, we may moreover assume $k$ perfect. We may then assume $G$ and $X$ reduced. Let $\eta$ be a maximal point of $X$ and $z$ an arbitrary point of $Z={\eta }^{-}$. Since $X$ is reduced, the local ring $O_{X,\eta }$ equals $\kappa (\eta )$, and since $k$ is perfect, for every extension $K$ of $k$, $\kappa (\eta ){\otimes }_{k}K$ is normal (cf. EGA IV_2, 6.14.2), so every point of X_K above $\eta$ is normal.

Since $\Phi$ is surjective, there exists a point $\gamma$ of $G\times X$ such that $\Phi (\gamma )$ has projections $z$ and $\eta$. Let $K=\kappa (\gamma )$; there exist then rational points $g$ and ${\eta }^{\prime }$ of G_K and X_K such that ${\eta }^{\prime }$ lies above $\eta$ and $z^{\prime }=g{\eta }^{\prime }$ lies above $z$. By what precedes, ${\eta }^{\prime }$ is a normal point of X_K, hence so is $z^{\prime }$. Since $\pi :X_K\to X$ is flat, it follows that $z=\pi (z^{\prime })$ is a normal point of $X$ (cf. EGA IV_2, 2.1.13). This proves (ii) and (i).

The first assertion of (iii) then follows from (ii). Then, since $L$ is algebraically closed in $\kappa (\eta )$, $C$ is geometrically irreducible over $L$, by EGA IV_2, 4.5.9.

Finally, if $X$ has a rational point $x$, it follows from (iii) that $L=k$, and so $X$ is geometrically irreducible over $k$. This can also be seen directly as follows (cf. [Per75], II 2.1): let $C$ be the irreducible component of $X$ containing $x$ and let $K$ be an extension of $k$; X_K has a unique point $e_K$ above $e$, and, by (i), $e_K$ belongs to a unique irreducible component $C^{\prime }$ of X_K; on the other hand, by 2.6.0, every irreducible component of C_K contains $e_K$, hence equals $C^{\prime }$.

Notation. Let us write provisionally $C^0$ for the reduced closed subscheme of $G$ whose underlying space is the unique irreducible component of $G$ containing the unit element $e$.

Corollary 2.6.2. $C^0$ is geometrically irreducible over $k$ and is set-theoretically stable under the group law, i.e. $(C_{k^{\prime }}^0{)}_{red}$ is a subgroup of $(G_{k^{\prime }}{)}_{red}$. Consequently, $C^0$ is quasi-compact.

Indeed, by 2.6.1, $C^0$ is geometrically irreducible over $k$, so $C^0\times C^0$ is irreducible, so $\nu (C^0\times C^0)\subset C^0$, where $\nu$ denotes the morphism $(g,h)\mapsto g{h}^{-1}$. Since $\mathrm{Spec}{k}^{\prime }\to \mathrm{Spec}k$ is a universal homeomorphism, the same conclusion holds for $C_{k^{\prime }}^0$, and then for $H=(C_{k^{\prime }}^0{)}_{red}$, and so, since $H{\times }_{k^{\prime }}H$ is reduced, $\nu$ induces a morphism $H{\times }_{k^{\prime }}H\to H$, i.e. $H$ is a subgroup of $(G_{k^{\prime }}{)}_{red}$. Consequently, by 0.5.1, $H$ (and hence also $C^0$) is quasi-compact.

Recollection 2.6.3. Let $Y$ be a scheme. Recall (EGA 0_III, 9.1.1) that a subset $E$ of $Y$ is said to be retrocompact if the inclusion $E\hookrightarrow Y$ is quasi-compact, and that, by EGA IV_1, 1.9.5 (v) and 1.10.1, if $U$ is a retrocompact open subset of $Y$, then the closure $U^{-}$ of $U$ is the union of the closures $y^{-}$ of the points $y\in U$, and of course it is enough to take $y$ running through the maximal points of $U$, i.e. the maximal points of $Y$ contained in $U$.

Consequently, if $Y$ is quasi-separated and if $\eta$ is a maximal point of $Y$, then the intersection of the closed neighborhoods of $\eta$ equals ${\eta }^{-}$: indeed, if $y\in Y-{\eta }^{-}$, then $y$ is contained in an affine open $V$ not containing $\eta$; since $Y$ is quasi-separated, $V$ is retrocompact, so $V^{-}$ is the union of the ${\xi }^{-}$, for $\xi$ running through the maximal points of $Y$ belonging to $V$, and hence $\eta \notin V^{-}$, i.e. $Y-V^{-}$ is a closed neighborhood of $\eta$ not containing $y$.