Exposé XIV. Regular elements (continuation). Applications to algebraic groups

by A. Grothendieck, with an Appendix by J.-P. Serre

1. Construction of Cartan subgroups and of maximal tori for a smooth algebraic group

Theorem 1.1. Let $G$ be a smooth algebraic group over a field $k$. Then $G$ admits a maximal torus $T$, hence a Cartan subgroup $C=Cent{r}_G(T)$.

By virtue of (XII 3.2), it amounts to the same to find a maximal torus $T$ of $G$ or a Cartan subgroup $C$ of $G$. Moreover, since the maximal tori of $G$ are those of $G^0$, we may assume $G$ connected. We distinguish two cases:

1°) The field $k$ is finite. Let $T$ be the scheme of maximal tori of $G$ (Exp XII 1.10), which is a smooth scheme over $k$. Note that $G$ operates on $T$ via inner automorphisms, and by virtue of the conjugation theorem (XII 6.6 a)), two points of $T_{\bar{k}}$ rational over $\bar{k}$ are congruent under $G_{\bar{k}}(\bar{k})$. Taking into account that $T$ is smooth over $k$, hence $T_{\bar{k}}$ smooth over $\bar{k}$, it follows that $T_{\bar{k}}$ is isomorphic to $G_{\bar{k}}/N$, where $N$ is the stabilizer of an element of $T_{\bar{k}}(\bar{k})$, i.e. the normalizer of a maximal torus $T$ of $G_{\bar{k}}$. Consequently, $T$ is a "homogeneous space" under the action of the group $G$. A well-known theorem of Lang (Amer. J. Math. 78, 1956, pp. 555–563) tells us that every homogeneous space under a smooth connected algebraic group over a finite field $k$ admits a rational point. In particular, $T$ admits a rational point, i.e. $G$ admits a maximal torus $T$. QED.

2°) The field $k$ is infinite. We shall use the

Lemma 1.2. Let $G$ be a smooth algebraic group over a field $k$. Then $G$ admits a subgroup of type (C) (XIII 6.2).

By virtue of (XIII 6.3), this amounts to saying that $\mathfrak{g}$ contains a Cartan subalgebra $\mathfrak{d}$. This is trivial if $k$ is infinite, for then $\mathfrak{g}$ contains a regular element $a$, and one takes $\mathfrak{d}=Nil(a,\mathfrak{g})$. The case of finite $k$ is handled exactly as in the proof of 1°) above, but requires the prior construction of the scheme $\mathcal{D}$ of Cartan subalgebras of $\mathfrak{g}$ and the fact that this scheme is smooth over $k$, which will be carried out below (2.16).

To establish 1.1 in case 2°), where we have placed ourselves, it suffices in any event to know 1.2 for $k$ infinite. Let us also record for the record:

Lemma 1.3. Let $G$ be a smooth, connected, affine algebraic group whose reductive center (XII 4.1 and 4.4) is reduced to the unit group. Then $G$ is nilpotent if (and only if) its Lie algebra $\mathfrak{g}$ is nilpotent.

This is contained in (XII 4.9).

We can now give a procedure for constructing Cartan subgroups of $G$ (also valid when $k$ is finite, granting 1.2 in that case). Suppose first that $G$ is affine. We proceed by induction on $n=\dim G$, the assertion being trivial if $n=0$. So suppose $n>0$ and the assertion proved for dimensions $n^{\prime }<n$. Let $Z$ be the reductive center of $G$, and let

u : G ⟶ G′ = G/Z

be the canonical homomorphism. By virtue of (XII 4.7 c)), one has a bijective correspondence $C^{\prime }\mapsto C=u^{-1}(C^{\prime })$ between Cartan subgroups of $G^{\prime }$ and Cartan subgroups of $G$. So, replacing $G$ by $G^{\prime }$ if necessary, we may suppose that the reductive center $Z$ of $G$ is reduced to the unit element (since this is the case for that of $G^{\prime }$ by XII 4.7 b)).

closure $\bar{k}$ of $k$, $D_{\bar{k}}$ contains a Cartan subgroup of $G_{\bar{k}}$ (XIII 6.6 b)), hence every Cartan subgroup of $D$ is a Cartan subgroup of $G$ (XIII 2.8 a)). So we are reduced to finding a Cartan subgroup of $D$. If $\dim D=\dim G$, i.e. $D=G$, then the Lie algebra of $G$ is a Cartan subalgebra of itself, hence is nilpotent, hence by 1.3 $G$ is nilpotent, hence is a Cartan subgroup of itself. If $\dim D<\dim G$, then by the induction hypothesis there exists a Cartan subgroup of $D$, which is therefore a Cartan subgroup of $G$, which completes the proof in the case where $G$ is affine. In the general case, let $Z$ be the center of $G$; then $G/Z=G^{\prime }$ is affine (XII 6.1) and for every Cartan subgroup $C^{\prime }$ of $G^{\prime }$, its inverse image $C$ in $G$ is a Cartan subgroup of $G$ (XII 6.6 e)). One is reduced to finding a Cartan subgroup of the smooth affine algebraic group $G^{\prime }$, a case already treated.

Corollary 1.4. Let $G$ be a smooth group scheme of finite type over an artinian scheme $S$. Then $G$ admits a maximal torus $T$, hence a Cartan subgroup $C=Cent{r}_G(T)$. Every torus $S$ in $G$ is contained in a maximal torus.

One may indeed suppose $S$ local of residue field $k$; then by virtue of 1.1, $G_0=G{\times }_S\mathrm{Spec}(k)$ admits a maximal torus $T_0$, and by virtue of (IX 3.6 bis) and (X 2.3), $T_0$ comes from a torus $T$ of $G$, which is evidently a maximal torus. The last statement follows from this, applying the preceding result to the centralizer of $S$, which is indeed smooth over $k$ (XI 2.4).

Remarks 1.5. a) We shall give below (3.20, 3.21 and XV) variants of 1.4 in the case where $S$ is not assumed artinian.

b) I do not know whether every algebraic group $G$ (not necessarily smooth) over a field $k$ admits a maximal torus. The question only arises in characteristic $p>0$, and using 1.1 for a smooth quotient group of the form $G^{\prime }=G/I$, where $I$ is a suitable radicial subgroup of $G$ (for example, the kernel of a sufficiently high power of the Frobenius homomorphism), one is reduced, by taking the inverse image in $G$ of a maximal torus of $G^{\prime }$, to the case where $(G_{\bar{k}}{)}_{red}$ is a torus ($\bar{k}$ always denoting the algebraic closure of $k$). It is easy to see that the answer is affirmative when $G$ is commutative (or more generally nilpotent): then $G$ admits a unique maximal torus, which one may construct for example by descent from the maximal torus of $G_{\bar{k}}$ ¹.

c) In the case where $G$ is affine, and $k$ is perfect or $G$ solvable, 1.1 is known and due to Rosenlicht; his proof is very different from the one given here.

d) When $k$ is infinite, 1.1 is a consequence of the much more precise result that the scheme $T$ of maximal tori of $G$ is a rational variety, proved below (6.1). The method is essentially a combination of the proof of 1.1 and of the explicit description of the structure of the scheme $\mathcal{D}$ of Cartan subalgebras of $\mathfrak{g}$. To reach the desired result, we must first generalize to the case of an arbitrary base prescheme certain results from (XIII 4 to 6) (this is the goal of the next two sections), and refine the previous construction proving 1.1, by using the fact that every Cartan subgroup of $G$ is contained in one and only one subgroup of type (C) of $G$ (Nos 4, 5).

2. Lie algebras over an arbitrary prescheme: regular sections and Cartan subalgebras

Lemma 2.1. Let $A$ be a ring, $\mathfrak{d}$ a Lie algebra over $A$, and for every $s\in \mathrm{Spec}(A)$, let $\mathfrak{d}(s)=\mathfrak{d}{\otimes }_{A}k(s)$ be the corresponding Lie algebra over the residue field $k(s)$. Suppose the $A$-module $\mathfrak{d}$ is of finite presentation. The following conditions are equivalent:

• (i) For every $s\in \mathrm{Spec}(A)$, $\mathfrak{d}(s)$ is nilpotent.
• (ii) For every $x\in \mathfrak{d}$, $ad(x)$ is nilpotent.
• (iii) There exists an integer $N⩾0$ such that for every sequence $x_1,\cdots ,x_N$ of elements of $\mathfrak{d}$, one has

$$ad(x_1)ad(x_2)\cdots ad(x_N)=0.$$

When $A$ is a field, the equivalence of (i) and (iii) is the definition of "nilpotent", that of (ii) and (iii) is a well-known consequence of Engel's theorem (Bourbaki, Groupes et algèbres de Lie, Chap. I, § 4, N° 2). In the general case, one has trivially (iii) ⇒ (ii), and (ii) ⇒ (i) thanks to the preceding result and to the fact that (ii) is stable by passage to quotients and by localization. It remains to prove (i) ⇒ (iii). When $A$ is local artinian with maximal ideal $\mathfrak{m}$, let $n>0$ be an integer such that ${\mathfrak{m}}^n=0$, let $N$ be an integer such that condition (iii) is satisfied for $\mathfrak{d}(s)=\mathfrak{d}{\otimes }_{A}A/\mathfrak{m}$, and take $N^{\prime }=nN$; one sees at once that this integer satisfies (iii). When $A$ is noetherian, there exist finitely many elements $s_i\in \mathrm{Spec}(A)$ and ideals of definition ${\mathfrak{q}}_i\subset A_{s_i}$ such that the natural map

𝔡 ⟶ ∏_i 𝔡 ⊗_A A_i,    where A_i = A_{s_i}/𝔮_i,

is injective; then by what precedes there exists for every $i$ an integer $N_i$ satisfying (iii) for the Lie algebra $\mathfrak{d}{\otimes }_A{A}_i$, and taking $N$ to be the largest of the $N_i$, one satisfies (iii) for $\mathfrak{d}$. Finally, the general case reduces to the noetherian case by the usual procedure explained in (EGA IV 8).

Definition 2.2. Let $S$ be a prescheme, $\mathfrak{d}$ a quasi-coherent Lie algebra over $S$ that is a module of finite presentation. One says that $\mathfrak{d}$ is locally nilpotent if for every $s\in S$, the Lie algebra $\mathfrak{d}(s)$ over $k(s)$ is nilpotent. One says that $\mathfrak{d}$ is strictly locally nilpotent if it is locally free, and if on every open $U$ of $S$ on which it is of constant rank $r$, its Killing polynomial reduces to $P_{\mathfrak{d}}(t)=t^r$.

Of course, if $\mathfrak{d}$ is a locally free Lie algebra (as a module) over $S$, one defines its Killing polynomial as a polynomial

P_𝔡 ∈ A[t],    where A = Γ(Sym_{𝒪_S}(𝔡)) ≃ Γ(W(𝔡))

is the ring of sections of the structure sheaf of the vector bundle $W(\mathfrak{d})$ defined by $\mathfrak{d}$.

It is evident that the two notions just introduced in 2.2 are stable by base change, and of local nature for the fpqc topology. Let us note:

Proposition 2.3. If $\mathfrak{d}$ is strictly locally nilpotent, it is locally nilpotent. The converse is true when $\mathfrak{d}$ is locally free and $S$ is reduced.

The proof is immediate.

Definition 2.4. Let $S$ be a prescheme, $\mathfrak{g}$ a Lie algebra over $S$ that is a locally free module of finite type. Let $\mathfrak{d}$ be a Lie subalgebra of $\mathfrak{g}$; one says that it is a Cartan subalgebra if it satisfies the following conditions:

• (i) $\mathfrak{d}$ is a locally direct factor submodule (hence also locally free of finite type).
• (ii) For every $s\in S$, $\mathfrak{d}(s)$ is a Cartan subalgebra of $\mathfrak{g}(s)$.

Definition 2.5. Let $S$, $\mathfrak{g}$ be as in 2.4. A section $a$ of $\mathfrak{g}$ is called quasi-regular if for every $s\in S$, $a(s)\in \mathfrak{g}(s)$ is a regular element of the Lie algebra $\mathfrak{g}(s)$ over $k(s)$. One says that $a$ is a regular section if it is quasi-regular and contained in a Cartan subalgebra of $\mathfrak{g}$.

Notions 2.4 and 2.5 are again stable by base change, and of local nature for the fpqc topology. Only the last assertion, and only for the notion "regular section", requires a proof, and follows from the fact that the Cartan subalgebra containing a given regular section is uniquely determined. More precisely:

Proposition 2.6. Let $S$, $\mathfrak{g}$ be as in 2.4 and let $a$ be a quasi-regular section of $\mathfrak{g}$. Then there exists at most one Cartan subalgebra of $\mathfrak{g}$ containing $a$. For one to exist, i.e. for $a$ to be a regular section, it is necessary and sufficient that $a$ satisfy the following condition: $\mathfrak{d}=Nil(a,\mathfrak{g})$ is a locally direct factor submodule of $\mathfrak{g}$, and $ad(a)$ induces an automorphism of $\mathfrak{g}/\mathfrak{d}$. In this case, $\mathfrak{d}$ is the unique Cartan subalgebra of $\mathfrak{g}$ containing $a$.

Suppose indeed that $a$ is contained in the Cartan subalgebra $\mathfrak{d}$ of $\mathfrak{g}$. Then $ad(a{)}_{\mathfrak{g}/\mathfrak{d}}$ is bijective on each fiber, hence (since $\mathfrak{g}/\mathfrak{d}$ is locally free of finite type) it is an automorphism of $\mathfrak{g}/\mathfrak{d}$; on the other hand by virtue of 2.1 $ad(a{)}_{\mathfrak{d}}$ is locally nilpotent, hence $\mathfrak{d}=Nil(a,\mathfrak{g})$, which proves the uniqueness of $\mathfrak{d}$, and the necessity of the stated regularity criterion. For sufficiency, note that the hypothesis made on $a$ implies that the formation of $Nil(a,\mathfrak{g})$ commutes with base extension and in particular with passage to fibers, which shows in particular that the fibers $\mathfrak{d}(s)$ of $\mathfrak{d}=Nil(a,\mathfrak{g})$ are Cartan subalgebras of the $\mathfrak{g}(s)$; moreover $\mathfrak{d}$ is a Lie subalgebra of $\mathfrak{g}$ by virtue of (XIII 4.1), hence it is a Cartan subalgebra.

Corollary 2.7. Under the conditions of 2.4, let $\mathfrak{d}$ be a Cartan subalgebra of $\mathfrak{g}$, $a$ a section of $\mathfrak{d}$ that is regular in $\mathfrak{g}$, $u$ an automorphism of $\mathfrak{g}$. In order for $u$ to preserve $\mathfrak{d}$, it is necessary and sufficient that $u(a)$ be a section of $\mathfrak{d}$.

Indeed, by transport of structure $u(a)$ is then a regular section of $\mathfrak{g}$, contained in the two Cartan subalgebras $\mathfrak{d}$ and $u(\mathfrak{d})$, which are therefore identical.

Corollary 2.8. Under the conditions of 2.4, let $\mathfrak{d}$ be a Cartan subalgebra of $\mathfrak{g}$. Then for every $s\in S$ such that $k(s)$ is infinite, there exists an open neighborhood $V$ of $s$ and a regular section $a$ of $\mathfrak{g}$ over $V$ such that $\mathfrak{d}=Nil(a,\mathfrak{g}|V)$ (i.e. such that $a$ is a section of $\mathfrak{d}|V$).

Indeed, the fact that $k(s)$ is infinite ensures the existence of a regular element $a_0$ of $\mathfrak{g}(s)$ contained in $\mathfrak{d}(s)$; one can extend it to a section $a$ of $\mathfrak{d}$ on an open neighborhood $U$ of $s$, and since $ad(a{)}_{\mathfrak{g}/\mathfrak{d}}$ induces an automorphism of $\mathfrak{g}(s)/\mathfrak{d}(s)$, it follows that, on restricting $V$, $ad(a{)}_{\mathfrak{g}/\mathfrak{d}}$ is an automorphism, which implies that $a$ is a quasi-regular section of $\mathfrak{g}|V$ satisfying the desired condition.

Let $\mathfrak{g}$ be as in 2.4; then examination of its Killing polynomial implies at once that the function

s ↦ nilpotent rank of 𝔤(s)

on $S$ is upper semi-continuous. We are above all interested in the case when this function is in fact continuous, i.e. locally constant. Here are some variants of this property:

Proposition 2.9. Let $S$, $\mathfrak{g}$ be as in 2.4. Consider the following conditions:

• (C₀) The nilpotent rank of the $\mathfrak{g}(s)$ ($s\in S$) is a locally constant function of $s$.
• (C₁) There locally exists, for the fpqc topology, a Cartan subalgebra of $\mathfrak{g}$.
• (C₁′) Like (C₁), with "fpqc topology" replaced by "étale topology".
• (C₂) Condition (C₀) is satisfied, and for every $S^{\prime }$ over $S$, every quasi-regular section of ${\mathfrak{g}}_{S^{\prime }}=\mathfrak{g}{\otimes }_S{S}^{\prime }$ is regular.
• (C₃) Every $s\in S$ has an open neighborhood $V$ on which the Killing polynomial of $\mathfrak{g}$ is of the form

tʳ(tⁿ⁻ʳ + c₁ tⁿ⁻ʳ⁻¹ + ⋯ + c_{n−r})

where for every $s\in V$, $c_{n-r}(s)\in Sym(\mathfrak{g}(s{)}^{\vee })$ is non-zero.

With these notations, one has the implications

(C₃) ⇒ (C₂) ⇒ (C₁′) ⇒ (C₁) ⇒ (C₀),

and when $S$ is reduced, the five conditions considered are equivalent.

Let us also note that the conditions considered are manifestly stable by base change, and of local nature for the fpqc topology.

The implications (C₁′) ⇒ (C₁) ⇒ (C₀) are trivial, the implication (C₀) ⇒ (C₃) when $S$ is reduced is immediate. Note moreover:

Corollary 2.10. Suppose condition (C₀) satisfied. Let $U$ be the set of elements of $W(\mathfrak{g})$ that are regular in their fiber. Then $U$ is open; in particular, for every section $a$ of $\mathfrak{g}$ over $S$, the set $V$ of $s\in S$ such that $a(s)\in \mathfrak{g}(s)$ is regular, is open.

Indeed, the first assertion follows from the second (applied to ${\mathfrak{g}}_{S^{\prime }}$ for every base change $S^{\prime }$ over $S$). For the second, since one may suppose here $S$ reduced, hence (C₃) verified, it suffices to examine the Killing polynomial of $a$ in $\mathfrak{g}$.

The implication (C₂) ⇒ (C₁′) now follows easily from b) in the more precise statement that follows:

Corollary 2.11. Suppose condition (C₂) satisfied. Then:

a) For every $s\in S$ and every Cartan subalgebra ${\mathfrak{d}}_0$ of $\mathfrak{g}(s)$ such that ${\mathfrak{d}}_0$ contains a regular element of $\mathfrak{g}(s)$ (condition automatically satisfied if $k(s)$ is infinite), there exists an open neighborhood $V$ of $s$ and a subalgebra $\mathfrak{d}$ of $\mathfrak{g}|V$ whose fiber at $s$ is ${\mathfrak{d}}_0$. If $S_1$ is a subprescheme of $S$ containing $s$, and if one has already extended ${\mathfrak{d}}_0$ to a Cartan subalgebra ${\mathfrak{d}}_1$ of $\mathfrak{g}{\otimes }_S{S}_1$, then one can find an open neighborhood $V$ of $s$ in $S$ and a Cartan subalgebra $\mathfrak{d}$ of $\mathfrak{g}|V$ such that $\mathfrak{d}{\otimes }_V(S_1\cap V)$ is equal to ${\mathfrak{d}}_1|(S_1\cap V)$.

b) For every $s\in S$ such that $\mathfrak{g}(s)$ contains a regular element (condition automatically satisfied if $k(s)$ is infinite), there exists an open neighborhood $V$ of $s$ and a Cartan subalgebra $\mathfrak{d}$ of $\mathfrak{g}|V$.

Statement b) follows from a) by taking ${\mathfrak{d}}_0=Nil(a_0,\mathfrak{g}(s))$, $a_0$ being a regular element of $\mathfrak{g}(s)$. To prove a), let us say the second formulation, one considers a regular element $a_0$ of $\mathfrak{g}(s)$ contained in ${\mathfrak{d}}_0$, one extends it to a neighborhood of $s$ in $S_1$ as a section of ${\mathfrak{d}}_1$, and one extends this last to a section of $\mathfrak{g}$ in a neighborhood of $s$. By virtue of 2.10, this section is quasi-regular in an open neighborhood $V$ of $s$, hence regular by virtue of (C₂), hence by virtue of 2.6 $\mathfrak{d}=Nil(a,\mathfrak{g}|V)$ satisfies the required conditions.

Remark 2.12. I do not know if statement 2.11 a) and b) is valid without the hypothesis of existence of regular points (when $k(s)$ is finite).

It remains to prove the implication (C₃) ⇒ (C₂). Let us also note the following equivalent form of (C₃):

• (C₃′) One has (C₀), i.e. the nilpotent rank of the $\mathfrak{g}(s)$ ($s\in S$) is locally constant, and on every open $V$ of $S$ where this rank has value $r$, the Killing polynomial of $\mathfrak{g}|V$ is divisible by tʳ.

It is necessary to show that this condition implies that every quasi-regular section $a$ of $\mathfrak{g}$ is regular. Taking 2.6 into account, this is contained in the following lemma (applied to the endomorphism $ad(a)$ of $\mathfrak{g}$), (iv) ⇒ (iii):

Lemma 2.13. Let $A$ be a ring, $M$ a projective $A$-module of finite type, $u$ an endomorphism of $M$. The following conditions are equivalent:

• (i) $M$ is the direct sum of two stable submodules $M^{\prime }$, $M^{\prime \prime }$ such that $u|M^{\prime }$ is nilpotent and $u|M^{\prime \prime }$ is an automorphism of $M^{\prime \prime }$.
• (ii) There exists an integer $n>0$ such that $Im{u}^n+Ker{u}^n=M$.
• (iii) The nil-space $Nil(u)={\bigcup }_{n>0}Ker{u}^n$ is a direct factor in $M$, and $M=Nil(u)+u(M)$.

These conditions are implied by the following (and are equivalent to it when $A$ is reduced):

• (iv) Locally on $\mathrm{Spec}(A)$ (for the Zariski topology) the characteristic polynomial $P_u(t)$ of $u$ can be put in the form

tʳ(tⁿ⁻ʳ + c₁ tⁿ⁻ʳ⁻¹ + ⋯ + c_{n−r}),

where $c_{n-r}$ is invertible.

The equivalence of (i) (ii) (iii) is immediate and is recorded for the record. The fact that (i) implies (iv) when $A$ is reduced follows from the fact that in this case a nilpotent endomorphism of a projective module of rank $r$ has characteristic polynomial tʳ, while in any case the characteristic polynomial of an automorphism of a projective module of finite type has as constant term the determinant of $u$ up to sign (locally on $\mathrm{Spec}(A)$), hence an invertible element of $A$. Finally, to prove (iv) ⇒ (i), one notes that $M$ is a module over the polynomial ring A[t], by letting $t$ act as $u$, and the well-known identity

P(u) = 0

shows that $M$ is annihilated by P A[t], hence can be considered as a module over $A[t]/PA[t]$. Now writing $P=t^{r}Q$, where the constant term of $Q$ is invertible, one sees at once that

P A[t] = tʳ A[t] ∩ Q A[t],

hence $A[t]/PA[t]$ decomposes into the product of the rings $A[t]/t^{r}A[t]$ and $A[t]/QA[t]$, whence a corresponding decomposition of $M$ as a sum of two A[t]-modules, i.e. as a sum of two $A$-submodules $M^{\prime }$ and $M^{\prime \prime }$ stable under $u$; this is the decomposition envisaged in (i).

This completes the proof of 2.13, hence of 2.9.

Corollary 2.14. When condition (C₃) of 2.9 is satisfied, the Cartan subalgebras of $\mathfrak{g}$ are strictly nilpotent (2.2).

The proof is immediate.

Remark 2.15. a) Let us point out that one can prove a converse of 2.14: (C₃) is equivalent to the fact that for every $S^{\prime }$ over $S$, every quasi-regular section of ${\mathfrak{g}}_{S^{\prime }}$ is regular and every Cartan subalgebra of ${\mathfrak{g}}_{S^{\prime }}$ is strictly nilpotent, or again, every quasi-regular section of ${\mathfrak{g}}_{S^{\prime }}$ is contained in a strictly nilpotent Cartan subalgebra of ${\mathfrak{g}}_{S^{\prime }}$.

b) Contrary to the other conditions (C₀) to (C₂), condition (C₃) is of "infinitesimal nature". More precisely, when $S$ is locally noetherian, $\mathfrak{g}$ satisfies condition (C₃) if and only if for every base change $S^{\prime }\to S$ with $S^{\prime }$ local artinian (if one wishes, $S^{\prime }$ the spectrum of an artinian quotient of a local ring of $S$), ${\mathfrak{g}}_{S^{\prime }}$ satisfies the same condition. Similarly, when (C₀) is satisfied, the condition for a section of $\mathfrak{g}$ to be regular is of infinitesimal nature.

c) When $\mathfrak{g}$ is the Lie algebra of a smooth prescheme in groups of finite presentation over $S$, then we shall see that conditions (C₀), (C₁), (C₁′), (C₂) on $\mathfrak{g}$ are equivalent (5.2 a)); I do not know what is the case in general (except that, even for $S$ local artinian, (C₀) does not imply (C₁)). However, even in the case where $\mathfrak{g}$ comes from a $G$, and $S$ being local artinian, it is not true in general that (C₂) implies (C₃), since $\mathfrak{g}$ can be nilpotent without being strictly nilpotent. One obtains an example of this fact starting from a smooth affine group scheme $G$ over the spectrum $S$ of a discrete valuation ring such that the Lie algebra of the generic fiber is non-nilpotent (for example the generic fiber is an adjoint semisimple group), and that of the special fiber is nilpotent (for example, the special fiber being a vector group): then for $n$ large enough, the Lie algebra of $G_n=G{\times }_S{S}_n$ is not strictly nilpotent, however it is nilpotent.

Let us still place ourselves under the conditions of 2.4, and let

$$\mathcal{D}:(Sch{)}^{\circ }/S⟶(Ens)$$

be the functor defined by

𝒟(S′) = set of Cartan subalgebras of 𝔤_{S′}.

Let us also introduce the functor

X(S′) = set of pairs (𝔡, a), where 𝔡 is a Cartan subalgebra of 𝔤_{S′} and a a section of 𝔡.

One thus has two projections $(\mathfrak{d},a)\mapsto \mathfrak{d}$ and $(\mathfrak{d},a)\mapsto a$:

p : X ⟶ 𝒟    and    ψ : X ⟶ W(𝔤).

When (C₀) is satisfied, we also consider the open $U$ of regular points of $W(\mathfrak{g})$ (cf. 2.10) and condition (C₂) is then expressed by the fact that the morphism

$${\psi }^{-1}(U)⟶U$$

induced by $\psi$ is an isomorphism (a priori, it is a monomorphism thanks to 2.6). Note that it is trivial that the morphism $p:X\to \mathcal{D}$ is representable by a projection of vector bundles (i.e. for every $S$-morphism $S^{\prime }\to \mathcal{D}$, corresponding to a Cartan subalgebra $\mathfrak{d}$ of ${\mathfrak{g}}_{S^{\prime }}$, $X{\times }_{\mathcal{D}}{S}^{\prime }$ is representable by a vector bundle over $S^{\prime }$, namely $W(\mathfrak{d})$); so if $\mathcal{D}$ is representable, the same is true of $X$. Now one has:

Theorem 2.16. Let $S$ be a prescheme, $\mathfrak{g}$ a Lie algebra over $S$ that is a locally free ${\mathcal{O}}_S$-module of finite type; suppose condition (C₀) of 2.9 is satisfied.

a) The functor $\mathcal{D}$ of Cartan subalgebras of $\mathfrak{g}$ defined above is representable by a quasi-projective prescheme of finite presentation over $S$. The same is true of the functor $X$ defined above.

b) When condition (C₂) of 2.9 is satisfied, $\mathcal{D}$ and $X$ are smooth over $S$, and the morphism ${\psi }^{-1}(U)\to U$ induced by $\psi$ is an isomorphism.

c) Still assuming condition (C₂) satisfied, let $s\in S$, ${\mathfrak{d}}_0$ a Cartan subalgebra of $\mathfrak{g}(s)$, corresponding to a point $d$ of $\mathcal{D}$ rational over $k(s)$. Suppose that ${\mathfrak{d}}_0$ contains a regular point of $\mathfrak{g}(s)$ (condition automatically satisfied if $k(s)$ is infinite). Let $r$ be the infinitesimal rank of $\mathfrak{g}(s)$, $n$ its rank over $k(s)$; then there exists an open neighborhood $V$ of $d$ in $\mathcal{D}$ that is $S$-isomorphic to an open $V^{\prime }$ of $S[t_1,\cdots ,t_{n-r}]$.

Proof. One may suppose $\mathfrak{g}$ of constant rank $n$ and of constant nilpotent rank $r$. The assertions made about $X$ in a) and b) follow at once from the assertions made about $\mathcal{D}$ and from the fact that $X$ is a vector bundle over $\mathcal{D}$ defined by a locally free module, and are stated only for the record.

a) The functor $\mathcal{D}$ is a subfunctor of the functor $Gras{s}_{n-r}(\mathfrak{g})$ whose value at $S^{\prime }$ is the set of locally free quotient modules of rank $n-r$ of ${\mathfrak{g}}_{S^{\prime }}$, and it is well known that this latter functor is representable by a projective prescheme smooth over $S$ (cf. for example Séminaire Cartan 1960/61, Exp. 12, Nos 2 and 3, whose constructions transpose as they stand to the case of preschemes) ^{2 3}. So one is reduced to a relative problem, namely the following: given a locally free quotient module of rank $n-r$ of $\mathfrak{g}$, or equivalently, a locally free submodule $\mathfrak{d}$ of rank $r$ that is locally a direct factor, represent the following functor: $F(S^{\prime })=\varnothing$ if $\mathfrak{d}$ is not a Cartan subalgebra of ${\mathfrak{g}}_{S^{\prime }}$, $F(S^{\prime })=\varnothing$ otherwise. In fact, we shall see that $F$ is representable by a subprescheme of finite presentation of $S$ (which will show that $\mathcal{D}\to Grass$ is representable by an immersion of finite presentation, and will finish proving a)). One begins by expressing the condition that ${\mathfrak{d}}_{S^{\prime }}$ is a Lie subalgebra of ${\mathfrak{g}}_{S^{\prime }}$; one sees at once that this is expressed by the fact that $S^{\prime }\to S$ factors through a certain closed subprescheme $S_1$ of $S$, of finite presentation over $S$ (whose local equations on $S$ can be written immediately using a basis of $\mathfrak{g}$ adapted to the submodule $\mathfrak{d}$). One may therefore suppose that one has already $S=S_1$. One must then express that ${\mathfrak{d}}_{S^{\prime }}$ contains locally for fpqc a quasi-regular section of ${\mathfrak{g}}_{S^{\prime }}$, and for this one considers $V=W(\mathfrak{d})\cap U$, where $U$ is the open of regular points of $W(\mathfrak{g})$ (2.10); then the structure morphism $V\to S$ being smooth and quasi-compact, its image $S_1$ is an open part of $S$ and the immersion morphism $S_1\to S$ is quasi-compact i.e. of finite presentation. The condition envisaged on $S^{\prime }$ is then expressed by saying that $S^{\prime }\to S$ factors through $S_1$. So one is reduced to the case where $S=S_1$, and using the theory of descent, to the case where $\mathfrak{d}$ admits a section $a$ that is a quasi-regular section of $\mathfrak{g}$. It remains finally to express that the section $a_{S^{\prime }}$ of ${\mathfrak{g}}_{S^{\prime }}$ deduced from $a$ satisfies $ad(a_{S^{\prime }}{)}_{{\mathfrak{g}}_{S^{\prime }}/{\mathfrak{d}}_{S^{\prime }}}$ bijective, which amounts again to saying that $S^{\prime }\to S$ factors through a certain open subprescheme of finite presentation of $S$, namely S_D, where $D$ is the determinant of $ad(a{)}_{\mathfrak{g}/\mathfrak{d}}$. But then one sees at once that $\mathfrak{d}|S_D$ is a Cartan subalgebra of $\mathfrak{g}|S_D$, hence S_D represents the functor $F$, which proves a).

b) Is immediate thanks to 2.11 a) and (XI 1.5). Of course b) is also a consequence of the more precise statement c).

c) Let $a_0$ be a regular point of $\mathfrak{g}(s)$ contained in ${\mathfrak{d}}_0$; extend it to a section $a$ of $\mathfrak{g}$ on an open neighborhood $V$ of $s$; one may evidently suppose $V=S$. On the other hand let $M_0$ be a complement of the vector space ${\mathfrak{d}}_0$ in $\mathfrak{g}(s)$; then in an open neighborhood $V$ of $S$ there exists a submodule $M$ of $\mathfrak{g}$, a direct factor of $\mathfrak{g}|V$, such that $M(s)=M_0$, and one may again suppose $V=S$. Let now $\mathcal{V}$ be the subfunctor of $\mathcal{D}$ such that $\mathcal{V}(S^{\prime })$ is the set of Cartan subalgebras ${\mathfrak{d}}^{\prime }$ of ${\mathfrak{g}}_{S^{\prime }}$ satisfying the two following conditions:

• 1°) ${\mathfrak{d}}^{\prime }$ is a complement of $M_{S^{\prime }}$, and
• 2°) the unique section of $(a_{S^{\prime }}+M_{S^{\prime }})\cap {\mathfrak{d}}^{\prime }$ is a regular section of ${\mathfrak{g}}_{S^{\prime }}$.

Condition 1°) corresponds to an open $V_1$ of $\mathcal{D}$ (induced by the open of $Gras{s}_{n-r}(\mathfrak{g})$ defined by the same condition 1°)); the conjunction of 1°) and 2°) corresponds to an open of $V_1$ by virtue of 2.10 and (C₂). Hence $\mathcal{V}$ is represented by an open subprescheme $V$ of $\mathcal{D}$, evidently containing $d$.

On the other hand let ${\mathcal{V}}^{\prime }$ be the subfunctor of $W(M)$ defined by

${\mathcal{V}}^{\prime }(S^{\prime })=setofsections{u}^{\prime }of{M}_{S^{\prime }}$ such that:

• (i) $a_{S^{\prime }}+u^{\prime }$ is a regular section of ${\mathfrak{g}}_{S^{\prime }}$, and
• (ii) the unique Cartan subalgebra ${\mathfrak{d}}^{\prime }$ of ${\mathfrak{g}}_{S^{\prime }}$ containing $a_{S^{\prime }}+u^{\prime }$ is a complement of $M_{S^{\prime }}$.

Then condition 1°) corresponds to an open subprescheme $V_1^{\prime }$ of $W(M)$, namely the inverse image of the open $U$ of regular points of $W(\mathfrak{g})$ (cf. 2.10) by the translation morphism $m\mapsto a+m$. The conjunction of (i) and (ii) corresponds to an open $V^{\prime }$ of $V_1^{\prime }$, namely the inverse image of $V$ by the obvious morphism $V_1^{\prime }\to \mathcal{D}$ (associating to $u^{\prime }$ the unique Cartan subalgebra ${\mathfrak{d}}^{\prime }$ of ${\mathfrak{g}}_{S^{\prime }}$ containing $a_{S^{\prime }}+u^{\prime }$). The restriction of this last morphism to $V^{\prime }$ is a morphism

$$V^{\prime }⟶V,$$

which is evidently an isomorphism. This proves c).

Corollary 2.17. Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$. Then the scheme $\mathcal{D}$ of Cartan subalgebras of $\mathfrak{g}$ (2.16 a)) is quasi-projective, smooth and irreducible. When $\mathfrak{g}$ contains a regular element (for example when $k$ is infinite), $\mathcal{D}$ is a rational variety, i.e. its function field is a pure extension of $k$.

The fact that $\mathcal{D}$ is irreducible follows from the fact that one has a surjective morphism ${\psi }^{-1}(U)\to \mathcal{D}$, and ${\psi }^{-1}(U)$ is irreducible, being isomorphic to the open $U$ of $W(\mathfrak{g})$. The assertion on the function field is an immediate consequence of c).

Remark 2.18. I do not know if this conclusion remains valid if $k$ is finite, without supposing that $\mathfrak{g}$ contains a regular point; compare 2.12. One can prove that this is the case when $\mathfrak{g}$ is the Lie algebra of an algebraic group $G$ smooth over $k$, at least when $G/radical$ is an "adjoint" semisimple group, by using a result of Chevalley pointed out below (cf. Appendix). It is plausible that this result remains valid without restriction on $G$; it would suffice for this that the cited result of Chevalley be proved for every (not necessarily adjoint) semisimple algebraic group.

3. Subgroups of type (C) of group preschemes over an arbitrary prescheme

Theorem 3.1. Let $S$ be a prescheme, $G$ a smooth $S$-prescheme in groups, $\mathfrak{g}$ its Lie algebra (which is a locally free module of finite type over $S$), $\mathfrak{h}$ a Lie subalgebra of $\mathfrak{g}$ that is (as a module) a locally direct factor in $\mathfrak{g}$, and such that for every $s\in S$, the geometric fiber ${\mathfrak{h}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$. Let $a$ be a quasi-regular section of $\mathfrak{g}$ (2.5). Then

$$M_a=Trans{p}_G(a,\mathfrak{h})$$

(subfunctor of $G$ whose $S^{\prime }$-valued points are the $g\in G(S^{\prime })$ such that $ad(g)\cdot a_{S^{\prime }}\in \Gamma (S^{\prime },{\mathfrak{h}}_{S^{\prime }})$) is representable by a closed subprescheme ⁴ of $G$ smooth over $S$, whose structure morphism to $S$ is surjective.

Consider the canonical morphism

ϕ : G ×_S W(𝔥) ⟶ W(𝔤)

given by $(g,x)\mapsto ad(g)\cdot x$; then $M_a$ is $S$-isomorphic to ${\phi }^{-1}(a)$, the inverse image of $a$ (considered as a section of $W(\mathfrak{g})$ over $S$) by $\phi$. It will suffice for the smoothness of $M_a$ to show that $\phi$ is smooth at points of $G{\times }_{S}W(\mathfrak{h})$ lying over $Im(a)$; more generally $\phi$ is smooth at every point lying over a point of $W(\mathfrak{g})$ that is regular in its fiber $W(\mathfrak{g}(s))$ over $S$. To see this, since the source and target of $\phi$ are smooth, hence flat locally of finite presentation over $S$, one is reduced to making the verification fiber by fiber, which reduces us to the case where $S$ is the spectrum of an algebraically closed field, $G$ thus being a locally algebraic group over $k$, $\mathfrak{h}$ a Lie subalgebra of its Lie algebra $\mathfrak{g}$, containing a Cartan subalgebra of $\mathfrak{g}$, and $a$ a regular point of $\mathfrak{g}$. One may evidently suppose (taking into account that $\phi$ is a $G$-morphism) that the point of $G\times W(\mathfrak{h})$ envisaged is of the form $(e,a)$. One may evidently suppose $G$ connected, hence of finite type over $k$, but then our assertion is none other than (XIII 5.4). Moreover, the fact that $M_a$ is a closed subprescheme of $G$ (of finite presentation over $S$) is trivial, since $M_a$ is the inverse image of $W(\mathfrak{h})$ by the morphism $g\mapsto ad(g)\cdot a$ of $G$ into $W(\mathfrak{g})$. The surjectivity of the structure morphism $M_a\to S$ also reduces to the case of an algebraically closed base field, but then $\mathfrak{h}$ contains a Cartan subalgebra $\mathfrak{d}$ by hypothesis, which is therefore conjugate to the Cartan subalgebra $\mathfrak{d}=Nil(a,\mathfrak{g})$ by the conjugation theorem (XIII 6.1 a)), hence $\mathfrak{h}$ contains a conjugate of $a$. This completes the proof of 3.1.

Corollary 3.2. Let $G$, $\mathfrak{g}$ be as in 3.1, with $G$ of finite type over $S$; let $\mathfrak{k}$ and $\mathfrak{h}$ be two Lie subalgebras of $\mathfrak{g}$, locally direct factors (as modules); suppose that one is under one of the two following hypotheses:

a) For every $s\in S$, the geometric fiber ${\mathfrak{k}}_s$ is nilpotent and contains a regular element of ${\mathfrak{g}}_s$; the geometric fiber ${\mathfrak{h}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$.

b) $\mathfrak{k}$ is a Cartan subalgebra of $\mathfrak{g}$.

Then $Trans{p}_G(\mathfrak{k},\mathfrak{h})$ is a closed subprescheme of $G$ smooth over $S$; moreover, in case a), its structure morphism to $S$ is surjective.

The fact that the transporter is representable by a closed subprescheme of finite presentation of $G$ is immediate, and left to the reader. Let us first prove smoothness in case a). Suppose first that there exists a section $a$ of $\mathfrak{k}$ that is quasi-regular in $\mathfrak{g}$. Then it suffices to apply 3.1 and the

Lemma 3.3. Under the conditions of 3.2 a), if $a$ is a section of $\mathfrak{k}$ quasi-regular in $\mathfrak{g}$, then one has

Transp_G(𝔨, 𝔥) = Transp_G(a, 𝔥).

Indeed, taking the definitions into account, this amounts to showing that if $a$ is moreover a section of $\mathfrak{h}$, then one has $\mathfrak{k}\subset \mathfrak{h}$. Now since by hypothesis $\mathfrak{k}$ is locally nilpotent, it follows from 2.1 that $\mathfrak{k}\subset Nil(a,\mathfrak{g})$; on the other hand $Nil(a,\mathfrak{g})\subset \mathfrak{h}$ because $ad(a{)}_{\mathfrak{g}/\mathfrak{h}}$ is injective (being so fiber by fiber by virtue of (XIII 4.8 b))). Whence the conclusion. In the general case, one reduces to the case where $S$ is affine noetherian by the standard procedure, then to the case where $S$ is local artinian (smoothness being a property of infinitesimal nature), and by flat descent to the case where its residue field is infinite, hence the fiber $K_0$ admits an element that is regular in ${\mathfrak{g}}_0$. One lifts this element to an element of $K=\Gamma (\mathfrak{k})$, which reduces us to the preceding case. Thus, in case a) we have proved the smoothness of the transporter; as for the fact that its structure morphism is surjective, it reduces to the case where $S$ is the spectrum of an algebraically closed field, hence where $K$ contains a regular point of $\mathfrak{g}$, and one applies 3.3 and 3.1.

To prove b), one is reduced by the definition of smoothness (XI 1.1) to proving that if $S$ is affine, $S_0$ a subscheme defined by a quasi-coherent nilpotent ideal $J$, $g_0$ an element of $G(S_0)$ that transports ${\mathfrak{k}}_0$ into ${\mathfrak{g}}_0$, then $g_0$ lifts to an element $g$ of $G(S)$ that transports $\mathfrak{k}$ into $\mathfrak{g}$. Now the hypothesis on $g_0$ implies that one is under the conditions of a), already treated. This completes the proof.

Of course, when in 3.2 b) $\mathfrak{h}$ satisfies the stronger hypothesis of 3.1, then (and only then) the structure morphism $Trans{p}_G(\mathfrak{d},\mathfrak{h})\to S$ is surjective. Using Hensel's lemma (XI 1.10), one concludes from 3.1 and 3.2:

Corollary 3.4. Under the conditions of 3.1 for $G$ and $\mathfrak{h}$, and supposing $G$ of finite type over $S$:

a) For every quasi-regular section $a$ of $\mathfrak{g}$, there locally exists for the étale topology a conjugate of $a$ that is a section of $\mathfrak{h}$.

b) For every Cartan subalgebra $\mathfrak{d}$ of $\mathfrak{g}$, $\mathfrak{d}$ is locally for the étale topology conjugate to a subalgebra of $\mathfrak{h}$.

In particular, when $\mathfrak{h}$ is itself a Cartan subalgebra of $\mathfrak{g}$, one finds:

Corollary 3.5. Let $G$ be a smooth $S$-prescheme in groups of finite type, $\mathfrak{g}$ its Lie algebra, $\mathfrak{d}$ and ${\mathfrak{d}}^{\prime }$ two Cartan subalgebras of $\mathfrak{g}$. Then $Trans{p}_G(\mathfrak{d},{\mathfrak{d}}^{\prime })$ is identical to the strict transporter of $\mathfrak{d}$ into ${\mathfrak{d}}^{\prime }$, and is a closed subprescheme of $G$ smooth over $S$, with surjective structure morphism. Locally for the étale topology, $\mathfrak{d}$ and ${\mathfrak{d}}^{\prime }$ are conjugate.

The fact that the transporter is identical here to the strict transporter follows trivially from the fact that $\mathfrak{d}$ and ${\mathfrak{d}}^{\prime }$ are locally direct factors in $\mathfrak{g}$ and have the same rank at every point. Hence 3.5 is a particular case of 3.4. In particular, if $\mathfrak{d}={\mathfrak{d}}^{\prime }$:

Corollary 3.6. Let $G$ be as in 3.5, and let $\mathfrak{d}$ be a Cartan subalgebra of $\mathfrak{g}$. Then $Nor{m}_G(\mathfrak{d})$ is a closed subprescheme in groups of $G$ smooth over $S$, whose Lie algebra is identical to $\mathfrak{d}$.

Indeed, this last assertion amounts to saying that $\mathfrak{d}$ is its own normalizer in $\mathfrak{d}$, which follows at once from the fact that this is true fiber by fiber.

Corollary 3.7. Let $G$, $\mathfrak{g}$ be as in 3.5. Then conditions (C₂), (C₁′), (C₁) of 2.9 are equivalent; in other words, if $\mathfrak{g}$ admits locally for the fpqc topology a Cartan subalgebra, then every quasi-regular section of $\mathfrak{g}$ is regular.

Indeed, let $a$ be a quasi-regular section; let us prove that it is regular. The question being local for the fpqc topology, one may suppose that $\mathfrak{g}$ admits a Cartan subalgebra $\mathfrak{d}$. By virtue of 3.4 a), $a$ is then locally for the étale topology conjugate to a section of $\mathfrak{d}$, which reduces us to the case where $a$ is a section of $\mathfrak{d}$, where the conclusion is trivial from the definition.

Definition 3.8. Let $G$ be a smooth prescheme in groups over a prescheme $S$. One calls a subgroup of type (C) of $G$ a subprescheme in groups $D$ of $G$, smooth over $S$, with connected fibers, such that $\mathfrak{d}=Lie(D)$ is a Cartan subalgebra (2.4) of $\mathfrak{g}=Lie(G)$, i.e. such that for every $s\in S$, $D_s$ is a subgroup of type (C) of the algebraic group $G_s$ (XIII 6.2).

Theorem 3.9. Let $G$ be a smooth $S$-prescheme in groups of finite presentation, $\mathfrak{g}$ its Lie algebra. Then:

a) The map

$$D\mapsto \mathfrak{d}=Lie(D)$$

establishes a bijective correspondence between subgroups of type (C) of $G$ and Cartan subalgebras of $\mathfrak{g}$.

b) If $D$ and $\mathfrak{d}$ correspond, one has

$$Nor{m}_G(D)=Nor{m}_G(\mathfrak{d}),$$

it is a closed subprescheme of $G$ smooth over $S$, and one has

$$D=Nor{m}_G(D{)}^0=Nor{m}_G(\mathfrak{d}{)}^0.$$

c) Two subgroups of type (C) $D$ and $D^{\prime }$ of $G$ are conjugate locally for the étale topology.

Proof. Let $D$ be a subgroup of type (C) of $G$, and $\mathfrak{d}$ its Lie algebra; then $D\subset Nor{m}_G(\mathfrak{d})$, and by virtue of Definition 3.8 and 3.6 this is an inclusion of preschemes in groups smooth over $S$, inducing an isomorphism on the Lie algebras. Since $D$ has connected fibers, one therefore has $D=Nor{m}_G(\mathfrak{d}{)}^0$. Hence the map envisaged in a) is injective; let us prove that it is surjective. So let $\mathfrak{d}$ be a Cartan subalgebra of $\mathfrak{g}$; then by virtue of 3.6 $Nor{m}_G(\mathfrak{d})=N$ is a closed subprescheme in groups of $G$ smooth over $S$, admitting $\mathfrak{d}$ as Lie algebra. Since $G$ is of finite presentation over $S$, the same is true of $N$, hence (as was pointed out in XII after 7.3) the union of the connected components of the fibers of $N$ is the underlying set of an open subprescheme in groups $N^{\circ }$ of $N$, which is evidently a subgroup of type (C) of $G$ having Lie algebra $\mathfrak{d}$. This proves a); the first assertion of b) follows from it at once; and the formula $D=Nor{m}_G(\mathfrak{d}{)}^0$ has already been proved. Finally, c) follows from a) and 3.5.

Corollary 3.10. Suppose that $\mathfrak{g}$ admits locally for the fpqc topology a Cartan subalgebra (or equivalently, by virtue of 3.9 a), that $G$ admits locally for the fpqc topology a subgroup of type (C)). Consider the functor $\mathcal{D}:(Sch{)}^{\circ }/S\to (Ens)$ defined by $\mathcal{D}(S^{\prime })=$ set of subgroups of type (C) of $G_{S^{\prime }}$. Then this functor is representable by a quasi-projective and smooth prescheme over $S$, with connected geometric fibers. When $S$ is the spectrum of a field $k$, hence $G$ a smooth algebraic group over $k$, and $\mathfrak{g}$ admits a regular point (condition automatically satisfied if $k$ is infinite), then $\mathcal{D}$ is a rational variety over $k$.

Indeed, by virtue of 3.9 a), the functor $\mathcal{D}$ is canonically isomorphic to the functor envisaged in 2.16; on the other hand by virtue of 3.7 condition (C₂) is satisfied. Hence 3.10 follows from 2.16 and 2.17.

Corollary 3.11. Let $D$ be a subgroup of type (C) of $G$, and let $N$ be its normalizer in $G$. Then the quotient sheaf $G/N$ is canonically isomorphic to the functor $\mathcal{D}$ of 3.10, hence representable by a quasi-projective and smooth prescheme over $S$, with connected geometric fibers.

Proposition 3.12. Let $G$ be a smooth $S$-prescheme in groups of finite presentation over $S$, $H$, $K$ two subpreschemes in groups, smooth of finite presentation over $S$, with $K$ having connected fibers; $\mathfrak{g}$, $\mathfrak{k}$, $\mathfrak{h}$ the corresponding Lie algebras. Suppose that one of the two following conditions is realized for the geometric fibers of these latter:

a) For every $s\in S$, ${\mathfrak{k}}_s$ contains a regular element of ${\mathfrak{g}}_s$, and ${\mathfrak{h}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$.

b) For every $s\in S$, ${\mathfrak{k}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$.

Under these conditions, in order to have $H\supset K$, it is necessary and sufficient that one have $\mathfrak{h}\supset \mathfrak{k}$.

Of course, we need only prove that if $\mathfrak{h}\supset \mathfrak{k}$, then $H\supset K$. In case b), the inclusion $\mathfrak{h}\supset \mathfrak{k}$ shows that one is in fact under the conditions of a), so it suffices to prove a). Proceeding as in 3.2 a) by reduction to the case $S$ local artinian, one is reduced to the case where there exists a section $a$ of $\mathfrak{k}$ that is quasi-regular in $\mathfrak{g}$. In this case, proceeding as in (XIII 5.5), one is reduced to the following statement:

Corollary 3.13. Let $G$ be a smooth $S$-prescheme in groups of finite presentation over $S$, $H$ a subprescheme in groups of $G$ smooth of finite presentation over $S$, $\mathfrak{g}$ and $\mathfrak{h}$ the Lie algebras, $a$ a section of $\mathfrak{h}$ that is quasi-regular in $\mathfrak{g}$. Suppose that for every $s\in S$, the geometric fiber ${\mathfrak{h}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$. Let

$$M_a=Trans{p}_G(a,\mathfrak{h}),$$

which is a closed subprescheme of $G$ smooth over $S$ (cf. 3.1), so that $M_a^0$ (union of the connected components of the identity element in the fibers of $M_a$) is an open part of $M_a$, which we shall endow with the structure induced by $M_a$. One then has $H^0=M_a^0$.

Evidently one has $H\subset M_a$, whence $H^0\subset M_a^0$. Since this is an inclusion of preschemes smooth over $S$, to prove that it is an equality, one is reduced to verifying it on the fibers, which reduces us to the case where $S$ is the spectrum of an algebraically closed field, a case treated in (XIII 5.4).

Corollary 3.14. Let $G$ be a smooth $S$-prescheme in groups of finite presentation, $D$ a subgroup of type (C) of $G$, $H$ a subprescheme in groups of $G$ smooth over $S$ and of finite presentation over $S$, $\mathfrak{d}$ and $\mathfrak{h}$ their Lie algebras. Then one has

Transp_G(D, H) = Transp_G(𝔡, 𝔥),

and this functor is representable by a closed subprescheme of $G$ smooth over $S$.

Indeed, the identity between the two transporters follows from 3.12 b), which allows one to apply 3.2.

Corollary 3.15. Let $G$, $H$ be as in 3.14 and suppose that for every $s\in S$, the geometric fiber ${\mathfrak{h}}_s$ contains a Cartan subalgebra of ${\mathfrak{g}}_s$. Suppose moreover that $\mathfrak{g}$ admits locally for the fpqc topology a Cartan subalgebra. Then locally for the étale topology, $H$ contains a subgroup of type (C) of $G$.

By virtue of 3.7 and 3.9 a), $G$ admits locally for the étale topology a subgroup of type (C), so one may suppose that $G$ admits such a subgroup, say $D$. Then the hypothesis on $\mathfrak{h}$ means also that the structure morphism of the transporter considered in 3.14 is surjective (taking into account the conjugation theorem XIII 6.1 a)). One concludes by Hensel's lemma (XI 1.10).

Corollary 3.16. Let $G$, $H$, $K$ be as in 3.12 a); suppose moreover that for every $s\in S$, the geometric fiber ${\mathfrak{k}}_s$ is nilpotent (i.e. $\mathfrak{k}$ is locally nilpotent). Then one has

Transp_G(K, H) = Transp_G(𝔨, 𝔥),

and this functor is representable by a closed subprescheme of $G$ smooth over $S$, whose structure morphism to $S$ is surjective. $H$ contains locally for the étale topology a subgroup conjugate to $K$.

The identity of the two transporters is again contained in 3.12 a), the assertion on its structure is then none other than 3.2 a), and the last assertion of 3.16 is then a consequence of Hensel's lemma.

Remarks 3.17. a) In 3.12 and 3.16, one may replace the hypothesis that $K$ is smooth over $S$ by the following weaker hypothesis: the sheaf $e_K\ast ({\Omega }_{K/S}^1)={\omega }_K^1$ of relative 1-differentials along the unit section is locally free. In this way, 3.12 contains (XIII 5.5).

b) Let $G$, $\mathfrak{g}$, $\mathfrak{h}$ be as in 3.1, with $G$ of finite presentation over $S$. Then $N=Nor{m}_G(\mathfrak{h})$ is not necessarily smooth over $S$ along the unit section, or equivalently, there does not necessarily exist a subprescheme in groups $H$ of $S$ smooth over $S$ whose Lie algebra is $\mathfrak{h}$, even if $S$ is the spectrum of a field. When such an $H$ exists, so that one has (taking $H$ with connected fibers) $N=Nor{m}_G(H)$, I do not know whether $N$ is smooth over $S$. In this question, one may evidently reduce to the case where $S$ is local artinian.

To conclude this section, let us examine the case where $G$ is "semisimple" over $S$:

Theorem 3.18. Let $S$ be a prescheme, $G$ a smooth $S$-prescheme in groups over $S$, whose geometric fibers are "adjoint" semisimple groups, i.e. semisimple with reductive center (XII 4.1 and 4.4) reduced to the unit group. Then the subgroups of type (C) of $G$ are identical with its maximal tori, hence also with its Cartan subgroups (XII 3.1).

Taking definitions into account, one is reduced to the case where $S$ is the spectrum of an algebraically closed field, and to proving then that for a maximal torus $T$ of $G$, the Lie algebra $\mathfrak{t}$ of $T$ is a Cartan subalgebra of $\mathfrak{g}$, that is to say (taking into account the inequality nilpotent rank of 𝔤 ⩾ nilpotent rank of G = dim T = rank 𝔱 = r) that there exists $x\in \mathfrak{t}$ with $\dim Nil(x,\mathfrak{g})=r$. Since $\mathfrak{t}$ is abelian and a fortiori nilpotent, it amounts to the same to say that there exists $a\in \mathfrak{t}$ such that $ad(a{)}_{\mathfrak{g}/\mathfrak{t}}$ is injective (XIII 5.7 a)). Now consider the characters $\alpha$ of $T$ that intervene in the representation of $T$ induced by the adjoint representation of $G$. The structure theory of the semisimple group $G$ (Bible, 13 th. 1 a) and th. 3, cor. 2), more precisely the "big cell" of $G$, semidirect product of $T$ and subgroups $P_{\alpha }$ isomorphic to the additive group $G_a$, preserved by $T$ and corresponding to the "root" characters of $G$ for the torus $T$, shows that the eigenspace of $\mathfrak{g}$ relative to the trivial character is none other than $\mathfrak{t}$, and the other eigenspaces are of dimension 1, the characters $\alpha$ associated being none other than the roots of $G$ for $T$. By virtue of the computation of the reductive center of $G$ as the intersection of the kernels of the characters of $T$ that intervene in the adjoint representation of $G$ (XII 4.8), one sees that the fact that $G$ is adjoint is interpreted as saying that the roots generate the lattice $M=\mathrm{Hom}(T,G_m)$. Now a well-known lemma of the theory of roots tells us that every root is part of a system of simple roots, hence of a basis of the group generated by the roots, and consequently of a basis of the dual $M$ of $T$ ⁵. We conclude:

Corollary 3.19. If $G$ is an adjoint semisimple algebraic group over an algebraically closed field $k$, $T$ a maximal torus of $G$, then for every root $\alpha$ of $G$ with respect to $T$, $\alpha :T\to G_m$, the corresponding homomorphism ${\alpha }^{\prime }:\mathfrak{t}\to k$ is non-zero.

This result is essentially equivalent to Theorem 3.18, for given $t\in \mathfrak{t}$, $ad(t)$ is semisimple and its eigenvalues in $\mathfrak{g}/\mathfrak{t}$ are none other than the ${\alpha }^{\prime }(t)$, hence $ad(t{)}_{\mathfrak{g}/\mathfrak{t}}$ is injective if and only if the ${\alpha }^{\prime }(t)$ are $\ne 0$, and there exists $t\in \mathfrak{t}$ having this property if and only if all the ${\alpha }^{\prime }$ are $\ne 0$.

Corollary 3.20. Let $S$ be a prescheme, $G$ an $S$-prescheme in groups, smooth, of finite presentation over $S$, whose geometric fibers are connected reductive algebraic groups (i.e. extensions of a semisimple group by a torus). Then for every $s\in S$ there exists an open neighborhood $U$ of $s$ such that $G|U$ admits a maximal torus. ⁶

We shall see indeed in XVI that the hypothesis just made on $G$ implies that $G$ is affine over $S$ and of locally constant reductive rank; hence (XII 4.7 c)) $G$ admits a reductive center $Z$, $G^{\prime }=G/Z$ is a smooth and affine group over $S$ whose reductive center is reduced to the unit group, and finally the maximal tori of $G$ and of $G^{\prime }$ are in bijective correspondence. Moreover, one sees at once that the geometric fibers of $G^{\prime }$ are connected semisimple groups, and moreover adjoint by definition (their reductive center being trivial). So it suffices to limit oneself to the case where $G$ is semisimple and adjoint, and by virtue of 3.18 one is reduced to finding an open neighborhood $U$ of $s$ and a subgroup of type (C) of $G|U$, or equivalently (3.9 a)) a Cartan subalgebra of $\mathfrak{g}|U$. Now this is possible if $k(s)$ is infinite, since by virtue of 3.7 $\mathfrak{g}$ satisfies condition (C₂) of 2.9, hence one may apply 2.11 b). In fact, statement 3.20 remains valid without supposing $k(s)$ infinite. Indeed, by the preceding argument, it suffices to know that for every adjoint semisimple group $G$ over a finite field $k$, the Lie algebra $\mathfrak{g}$ of $G$ contains a regular element. Now this statement has been proved by Chevalley (using the properties of the Coxeter element of the Weyl group...), cf. the Appendix below by J.-P. Serre.

Remarks 3.21. a) Statement 3.20 remains valid, with essentially the same proof, replacing $G$ by a closed subprescheme in groups $H$ smooth over $S$, having everywhere the same rank as $G$ (for example a "parabolic subgroup" of $G$), provided that $k(s)$ is infinite. I do not know if here again the hypothesis that $k(s)$ is infinite is superfluous. One can show that this is the case at least if $H$ is parabolic, thanks to the construction of the radical $U$ of $H$ and of the quotient $H/U$, which reduces us to the semisimple case. Unfortunately, the method of regular elements seems here powerless, since one easily constructs examples (for example with the projective group and its "standard" Borel subgroup) where the Lie algebra $\mathfrak{h}$ of $H$ does not contain any regular element.

b) The proof of 3.20 in fact shows a more precise result (by invoking 3.9 b)) in the case where $k(s)$ is infinite, namely that every maximal torus $T_0$ of $G_s$ comes from a maximal torus $T$ on an open neighborhood of $s$. I do not know if this statement remains valid when $k(s)$ is no longer supposed infinite; the difficulty obviously coming from the fact that the Lie algebra ${\mathfrak{t}}_0$ of $T_0$ does not in general contain a regular element of the Lie algebra ${\mathfrak{g}}_0$ of the fiber $G_s$. An affirmative answer to this problem would imply the following statement (which is proved only in the case of an infinite residue field or when $A$ is separated and complete): Let $A$ be a local ring, with residue field $k$, $M$ an "Azumaya algebra" over $A$, i.e. an algebra such that $M$ is a free module of finite type over $A$, and $M_0=M{\otimes }_{A}k$ a central simple algebra over $k$, $D_0$ a commutative subalgebra of $M_0$ separable over $k$, such that $[M_0:k]=([D_0:k]{)}^2$; then there exists a commutative subalgebra $D$ of $M$, which is a direct factor module in $M$ and such that $D{\otimes }_{A}k=D_0$ (?). (Note that the datum of $M$ is equivalent to the datum of a principal homogeneous bundle under the projective group $PGL(n{)}_A$, whence an "inner" twisted form $G$ of $PGL(n)$, and the maximal tori of $G$ correspond bijectively to the commutative subalgebras $D$ of $M$ étale over $A$ of rank $n$ over $A$.)

c) Applying 3.20 to the centralizer of a subtorus $Q$ of $G$ ($G$ a reductive $S$-prescheme in groups), one deduces that every such $Q$ is contained, locally for the Zariski topology, in a maximal torus of $G$.

4. A digression on Borel subgroups

Definition 4.1. Let $G$ be a smooth algebraic group over an algebraically closed field. One calls a Borel subgroup of $G$ a smooth, solvable, connected algebraic subgroup that is maximal for these properties.

When $G$ is affine, one thus recovers the terminology of (Bible 6 def. 1). Let us note at once that if $Z$ is a connected and smooth subgroup of $G$ contained in the center (or more generally, solvable and invariant), then for every Borel subgroup $B$ of $G$, the image BZ of $B\times Z$ by the morphism $(b,z)\mapsto bz$ of $B\times Z$ into $G$ is a smooth, solvable, connected subgroup of $G$ containing $B$, hence identical to $B$, hence $B$ contains $Z$, hence $B$ is the inverse image of an algebraic subgroup $B^{\prime }$ of $G^{\prime }=G/Z$, and it is immediate that $B^{\prime }$ is a Borel subgroup of $G^{\prime }$. Taking $Z=Centr(G^0{)}_{red}^{\circ }$, $G^{\prime }=G/Z$ is affine (XII 6.1), hence the Borel subgroups of $G^{\prime }$ are conjugate and for such a $B^{\prime }$, $G^{\prime }/B^{\prime }$ is a projective variety (Bible 6 th. 4). Consequently:

Proposition 4.2. Let $G$ be as in 4.1. Then the Borel subgroups of $G$ are conjugate. If $B$ is a Borel subgroup, then $G/B$ is a projective variety. The maximal tori of $B$ (resp. the Cartan subgroups of $B$, $G$ being connected) are maximal tori of $G$ (resp. Cartan subgroups of $G$).

It remains to prove the last assertion, and one may evidently suppose $G=G^0$. For the Cartan subgroups, it follows from the analogous assertion in $G^{\prime }$ (Bible 6 th. 4 cor. 4) and from (XII 6.6 e)). For the maximal tori, it follows from the preceding, since by (XII 6.6 c)) the maximal tori of a smooth algebraic group are the maximal tori of its Cartan subgroups.

Corollary 4.3. Suppose $G$ connected. Then every element of $G$ is contained in a Borel subgroup of $G$.

One is reduced to the same statement in $G^{\prime }$, which is well known (Bible 6 th. 5 d)).

Corollary 4.4. Let $B$ be a Borel subgroup of $G$, $C$ a Cartan subgroup of $B$, $N$ its normalizer in $G$; then $N\cap B=C$.

Indeed, $N\cap B=Nor{m}_B(C)$, so one is reduced to showing that when $G$ is connected and solvable, a Cartan subgroup $C$ is its own connected normalizer. Now with the preceding notations, $C$ is the inverse image of a Cartan subgroup $C^{\prime }$ of $G^{\prime }$, so one is reduced to the case where $G$ is affine. Since one knows that the normalizer of a Cartan subgroup is smooth ($C$ being its own connected normalizer, cf. for example XII 6.6 c)), it suffices to see that $C$ and $N$ have the same $k$-valued points, which is none other than (Bible th. 6 d)).

Definition 4.5. Let $G$ be a smooth prescheme in groups of finite presentation over a prescheme $S$. One calls a Borel subgroup of $G$ any subprescheme in groups $B$ of $G$, smooth of finite presentation, such that for every $s\in S$, the geometric fiber $B_s$ is a Borel subgroup of $G_s$.

This is therefore, as one verifies at once, a notion stable by base change and of local nature for the fpqc topology (for if $k^{\prime }$ is an algebraically closed extension of an algebraically closed field $k$, then an algebraic subgroup $B$ of the smooth algebraic group $G$ over $k$ is a Borel subgroup of $G$ if and only if $B_{k^{\prime }}$ is one of $G_{k^{\prime }}$). It follows from this definition that if $G$ is a smooth algebraic group over an arbitrary field $k$, $B$ a Borel subgroup of $G$, then $G/B$ is a projective variety, every maximal torus $T$ of $B$ is a maximal torus of $G$, its normalizer in $B$ is identical to its centralizer $C$, and is a Cartan subgroup of $G$ when $G$ is connected.

Remarks 4.6. Unfortunately, it is no longer true in general (even if $G$ is affine over $S$ and $S$ is the spectrum of the algebra of dual numbers over an algebraically closed field $k$) that two Borel subgroups of $G$ are conjugate locally for the fpqc topology. As a consequence of this regrettable fact, let us point out that if $G$ is a smooth, affine, connected algebraic group over a non-perfect field $k$, it is not possible in general to define in a natural way a homogeneous space $\mathcal{D}$ under $G$, playing the role of a flag variety, i.e. of the variety of Borel subgroups of $G$ (which, over the algebraic closure $\bar{k}$ of $k$, would therefore be isomorphic to $G_{\bar{k}}/B$, where $B$ is a Borel subgroup of $G_{\bar{k}}$). Indeed, when the quotient of $G_{\bar{k}}$ by its radical $R$ is an adjoint semisimple group, then the kernel of $G_{\bar{k}}\to {\mathrm{Aut}}_{\bar{k}}(\mathcal{D})$ is the radical of $G_{\bar{k}}$, hence if $\mathcal{D}$ came from a homogeneous space $D$ under $G$, the radical $R$ would come from a subgroup $R$ of $G$. Now one easily constructs examples where $G_{\bar{k}}/R$ is adjoint but $R$ is not "defined over $k$". It is easy to see that under these conditions, the functor $\mathfrak{B}:(Sch{)}^{\circ }/S\to (Ens)$ such that $\mathfrak{B}(S^{\prime })=$ set of Borel subgroups of $G_{S^{\prime }}$ is not representable by a smooth $S$-prescheme. From the infinitesimal point of view (III § 3), the non-validity of the conjugation theorem is expressed by the fact that if $B$ is a Borel subgroup of the smooth algebraic group $G$, the cohomology group $H^1(B,\mathfrak{g}/\mathfrak{b})$ ⁷ can be different from zero.

We shall see on the other hand in a later exposé that when $G$ is semisimple, or more generally reductive, such unpleasant phenomena do not occur. It is these phenomena no doubt, as well as the absence of good existence theorems, that explain why Borel subgroups play only a relatively effaced role in the study of general group schemes from the scheme-theoretic point of view, while they will dominate the theory of semisimple group schemes in the later exposés.

Proposition 4.8. ⁸ Let $G$ be a smooth $S$-prescheme in groups of finite presentation with connected fibers, $B$ a Borel subgroup of $G$; then $B$ is identical to its own normalizer, and is a closed subprescheme of $G$.

Indeed, by virtue of (XII 7.10), one is reduced to proving that over an algebraically closed field $k$, every element of $G(k)$ that normalizes $B$ is in $B(k)$, which for $G$ affine is a fundamental result of Chevalley (Bible 9 th. 1); the general case reduces to it at once by the reduction already used in 4.2.

Remark 4.8.1. One can generalize Definition 4.5 by also introducing the notion of parabolic subgroup of $G$: one calls thus a subprescheme in groups $P$ of $G$, smooth and of finite presentation over $S$, such that for every $s\in S$, the geometric fiber $P_s$ is a parabolic subgroup of $G_s$, i.e. contains a Borel subgroup of $G_s$. Proposition 4.8 extends (with the same reduction proof to the "set-theoretic" statement, which is known) to the case of a parabolic subgroup $P$ of $G$. Let us note the following consequence of this result (cf. XVI). If $P$ is a parabolic subgroup of $G$, then $G/P$ is representable by a quasi-projective prescheme of finite presentation over $S$ (N.B. one assumes $G$ to have connected fibers). Moreover $G/P$ is evidently smooth over $S$, and moreover with connected and proper geometric fibers, whence one can conclude easily, using (EGA III 5.5.1), that $D=G/P$ is in fact proper, hence projective, over $S$. Moreover, if its relative dimension is $n$, it is known that the invertible sheaf ${\Omega }_{D/S}^n$ is such that its inverse induces on the geometric fibers of $D/S$ ample sheaves, hence (EGA III 4.7.1) $({\Omega }_{D/S}^n{)}^{-1}$ is ample on $D$ relative to $S$.

One sees easily, by reduction to the affine case and to the case of an algebraically closed base field, that if $u:G\to G^{\prime }$ is an epimorphism of smooth algebraic groups, then for every Borel subgroup $B$ of $G$, $u(B)=B^{\prime }$ is a Borel subgroup of $G^{\prime }$. We are interested in the case where one obtains in this way a bijective correspondence between Borel subgroups of $G$ and of $G^{\prime }$:

Proposition 4.9. Let $G$, $G^{\prime }$ be two smooth $S$-preschemes in groups of finite presentation with connected fibers, $u:G\to G^{\prime }$ a faithfully flat (i.e. surjective) homomorphism of groups ⁹. Suppose one is in one of the following two cases (where one has set $N=Keru$):

a) $N$ is central in $G$.

b) $S$ is the spectrum of a field $k$, and if $\bar{k}$ denotes an algebraic closure of it, $N_{\bar{k}}=Ker{u}_{\bar{k}}$ is contained in the radical of $G_{\bar{k}}$, i.e. in the largest smooth connected solvable invariant subgroup of $G_{\bar{k}}$.

Then the map $B^{\prime }\mapsto u^{-1}(B^{\prime })$ induces a bijection of the set of Borel subgroups of $G^{\prime }$ with the analogous set for $G$.

Case b) follows at once from the correspondence between algebraic subgroups of $G^{\prime }$ and algebraic subgroups of $G$ containing $N$, and from the fact that when $k$ is algebraically closed, the Borel subgroups of $G$ contain the radical of $G$ (which is immediate by the reasoning preceding 4.2).

Let us prove case a). By virtue of (XII 7.12), the map $H^{\prime }\mapsto H=u^{-1}(H^{\prime })$ establishes a bijective correspondence between subpreschemes in groups $H^{\prime }$ of $G^{\prime }$ that are smooth of finite presentation over $S$, with connected fibers, and that have at every $s\in S$ the same reductive rank and the same nilpotent rank as $G^{\prime }$, and the subpreschemes in groups $H$ of $G$ having the analogous properties. Now Borel subgroups (of $G^{\prime }$, or of $G$) have the properties in question. It remains to prove that if $H^{\prime }$, $H$ correspond, then $H^{\prime }$ is a Borel subgroup of $G^{\prime }$ if and only if $H$ is one of $G$. By definition, this question reduces to the case where $S$ is the spectrum of an algebraically closed field. Now since $N$ is central in $G$ hence in $H$, it follows that $H$ is solvable if and only if $H^{\prime }$ is. Finally, taking the correspondence between algebraic subgroups of $G^{\prime }$ and algebraic subgroups of $G$ containing $N$ into account, one sees at once that $H^{\prime }$ has the maximal character of Definition 4.1 if and only if $H$ does, which completes the proof.

Corollary 4.10. With the notations of 4.7, if $B^{\prime }$ and $B$ are Borel subgroups of $G^{\prime }$ and $G$ which correspond, one has

$$\mathfrak{b}=Lie(u{)}^{-1}({\mathfrak{b}}^{\prime })$$

where $\mathfrak{g}$, ${\mathfrak{g}}^{\prime }$, $\mathfrak{b}$, ${\mathfrak{b}}^{\prime }$ are the Lie algebras of $G$, $G^{\prime }$, $B$, $B^{\prime }$, and where $Lie(u):\mathfrak{g}\to {\mathfrak{g}}^{\prime }$ is the homomorphism deduced from $u$.

This statement follows trivially from the definitions and from the relation $u^{-1}(B^{\prime })=B$.

We can now prove the principal result of the present section:

Theorem 4.11. Let $G$ be a smooth algebraic group over an algebraically closed field $k$, $\mathfrak{g}$ its Lie algebra. Then $\mathfrak{g}$ is equal to the union of the Lie algebras $\mathfrak{b}$ of the Borel subgroups $B$ of $G$.

One may evidently suppose $G$ connected. Let $R$ be the radical of $G$ and let $G^{\prime }=G/R$. Then 4.9 b) and 4.10 reduce us to proving Theorem 4.11 for $G^{\prime }$ instead of $G$, i.e. one may suppose $G$ semisimple. Let then $Z$ be the center of $G$, identical to the reductive center, and let $G^{\prime }=G/Z$. The same reasoning (now using 4.9 a)) reduces us to proving the theorem for $G^{\prime }$, i.e. one may suppose $G$ semisimple adjoint. Let $B$ be a Borel of $G$, $T$ a maximal torus of $B$ hence of $G$, and let $\mathfrak{b}$ and $\mathfrak{t}$ be the Lie algebras. By virtue of 3.18, $T$ is a subgroup of type (C) of $G$, i.e. $\mathfrak{t}$ is a Cartan subalgebra of $\mathfrak{g}$, hence the union of the conjugates of $\mathfrak{t}$ is dense in $\mathfrak{g}$ (XIII 5.1 (i) ⇒ (vii)). A fortiori the union of the conjugates of $\mathfrak{b}$ is dense in $\mathfrak{g}$. Now let $X$ be the closed subscheme of $G/B\times W(\mathfrak{g})$ whose $k$-valued points are the $(g^{\prime },x)$ such that $x\in Ad(g)\cdot \mathfrak{b}$ (cf. XIII 1). Then the morphism $\psi :X\to W(\mathfrak{g})$ induced by the second projection is proper since $G/B$ is proper over $k$; on the other hand we have just seen that it is dominant, hence it is surjective, which proves 4.11.

The only result of the present section that we shall use in the rest of this exposé is the following corollary:

Corollary 4.12. Let $k$ be an infinite field, $G$ a smooth algebraic group over $k$, $T$ a maximal torus of $G$, $\mathfrak{g}$ and $\mathfrak{t}$ the Lie algebras, $u:G\to GL(V)$ a linear representation ($V$ a finite-dimensional vector space over $k$), whence a representation of Lie algebras $u^{\prime }:\mathfrak{g}\to \mathfrak{gl}(V)$, making $V$ into a $\mathfrak{g}$-module. Then the minimum of the nullity of $u^{\prime }(x)$ ($x\in \mathfrak{g}$) is attained for an element $x\in \mathfrak{t}$.

One is reduced at once to the case where $k$ is algebraically closed. One may evidently suppose $G$ connected, and replacing $G$ by $G/(Keru{)}_{red}$ if necessary, one may suppose $G$ affine. Using 4.11 and the conjugation theorem of maximal tori of $G$, one is reduced to the case where $G$ is moreover solvable. Then $G$ is a semidirect product $T\cdot V$, where $V$ is the "unipotent part" of $G$, which is a smooth connected unipotent group (Bible 6 th. 3). Hence $\mathfrak{g}$ is the direct sum (as a vector space) of the Lie subalgebras $\mathfrak{t}$ and $\mathfrak{v}=Lie(V)$. By virtue of the Lie-Kolchin theorem (Bible 6 th. 1), $\mathfrak{v}$ admits a composition series by stable subspaces ${\mathfrak{v}}_i$ such that ${\mathfrak{v}}_i/{\mathfrak{v}}_{i+1}=w_i$ is of dimension 1. Then for each $i$, one has an induced representation $u_i:G\to GL(w_i)=G_m$ and the corresponding homomorphism of Lie algebras $u_i^{\prime }:\mathfrak{g}\to k$, so that for every $x\in \mathfrak{g}$, the nullity of $u^{\prime }(x)$ is equal to the number of $i$ such that $u_i^{\prime }(x)=0$. Since $V$ is unipotent, the $u_i$ are trivial on $V$, hence the $u_i^{\prime }$ are trivial on $\mathfrak{v}$, which proves that if $x=t+v$ ($t\in \mathfrak{t}$, $v\in \mathfrak{v}$), then $u_i^{\prime }(x)=u_i^{\prime }(t)$ for all $i$, hence the nullity of $u^{\prime }(x)$ is equal to that of $u^{\prime }(t)$. Assertion 4.12 follows at once.

5. Relations between Cartan subgroups and Cartan subalgebras

Applying 4.12 to the adjoint representation of $G$, one finds:

Theorem 5.1. Let $G$ be a smooth algebraic group over an infinite field, $T$ a maximal torus of $G$, $\mathfrak{g}\supset \mathfrak{t}$ the Lie algebras; then $\mathfrak{t}$ contains a regular element of $\mathfrak{g}$.

Corollary 5.2. Let $G$ be a smooth $S$-prescheme in groups of finite presentation, $\mathfrak{g}$ its Lie algebra.

a) Conditions (C₀) to (C₂) of 2.9 on $\mathfrak{g}$ are equivalent; in particular, if the infinitesimal rank of the fibers of $\mathfrak{g}$ at the points of $S$ is locally constant, then $\mathfrak{g}$ admits locally for the étale topology a Cartan subalgebra, hence (by 3.9 a)) $G$ admits locally for the étale topology a subgroup of type (C).

b) Let $H$ be a subprescheme in groups of $G$ smooth of finite presentation over $S$, with connected fibers having the same reductive rank as $G$ at each $s\in S$ (for example, $H$ is a maximal torus or a Cartan subgroup of $G$); let $D$ be a subgroup of type (C) of $G$; then one has $H\subset D$ if and only if one has $\mathfrak{h}\subset \mathfrak{d}$.

c) Suppose condition (C₀) is satisfied, i.e. the infinitesimal rank of $G$ is locally constant. Let $H$ be a subprescheme in groups of $G$ smooth of finite presentation over $S$, with connected nilpotent fibers having the same reductive rank as $G$ at each point (for example, $H$ is a maximal torus or a Cartan subgroup of $G$). Then $H$ is contained in one and only one subgroup $D$ of type (C) of $G$.

d) Suppose that $G$ admits locally for fpqc a Cartan subgroup; then the same is true of every subgroup $D$ of type (C) of $G$.

Proof. a) Suppose condition (C₀) is satisfied, and let us prove that the same is true of (C₂), i.e. that every quasi-regular section $a$ of $\mathfrak{g}$ is regular. One reduces as usual to the case $S$ affine noetherian, then, the question being "infinitesimal" (2.15 b)), to the case where $S$ is local artinian (in which case (C₀) is moreover trivially satisfied). One may suppose moreover that the residue field $k$ of $S$ is infinite. Note that by virtue of 3.7 it suffices to establish that $\mathfrak{g}$ admits a Cartan subalgebra. Let $T$ be a maximal torus of $G$; by virtue of 5.1 there exists a quasi-regular element $t$ contained in the Lie algebra $\mathfrak{t}$ of $T$; let us prove that it is regular, which will complete the proof. Consider the linear representation of $T$ in $\mathfrak{g}$ induced by the adjoint representation of $G$; there exists therefore a finite set $(u_i{)}_{i\in I}$ of characters of $T$, such that $\mathfrak{g}$ decomposes as a direct sum of submodules ${\mathfrak{g}}_i$ stable under $T$, $T$ operating on ${\mathfrak{g}}_i$ by $u_i$ (cf. I § 4.7.3). Let $u_i^{\prime }:\mathfrak{t}\to A$ be the homomorphism of Lie algebras deduced from $u_i:T\to G_m$ (N.B. $A$ denotes the ring of $S$). Consider the homomorphisms $u_{i0}$ and $u_{i0}^{\prime }$ deduced from the preceding by passage to the fibers, i.e. by the base change $A\to k$. Let $I^{\prime }$ be the set of $i\in I$ such that $u_{i0}^{\prime }\ne 0$, and let $I^{\prime \prime }=I-I^{\prime }$. The fact that $t$ is regular is expressed by the condition $u_{i0}^{\prime }(t_0)\ne 0$ for all $i\in I^{\prime }$, hence $u_i^{\prime }(t)$ invertible for $i\in I^{\prime }$. The Cartan subalgebra of ${\mathfrak{g}}_0$ defined by $t$, i.e. the kernel subspace of the semisimple endomorphism $ad(t_0)$ of ${\mathfrak{g}}_0$, is ${\sum }_{i\in I^{\prime \prime }}({\mathfrak{g}}_i{)}_0$. Consider

𝔡 = ∑_{i ∈ I″} 𝔤_i;

then $ad(t)$ is nilpotent in $\mathfrak{d}$; on the other hand it is an automorphism of $\mathfrak{g}/\mathfrak{d}\simeq {\sum }_{i\in I^{\prime }}{\mathfrak{g}}_i$. By virtue of 2.6, $t$ is therefore regular.

b) Follows from 3.12 a), $H$ satisfying the hypothesis that every geometric fiber of $\mathfrak{h}$ contains a regular element of that of $\mathfrak{g}$, thanks to 5.1.

c) By virtue of b), one is reduced to proving that $\mathfrak{h}$ is contained in one and only one Cartan subalgebra of $\mathfrak{d}$. By virtue of a) we know moreover that $\mathfrak{g}$ satisfies (C₂). One is thus reduced to the:

Lemma 5.3. Let $\mathfrak{g}$ be a Lie algebra over $S$ that is a locally free module of finite type and satisfies condition (C₂) of 2.9. Let $\mathfrak{h}$ be a Lie subalgebra of $\mathfrak{g}$ satisfying the following conditions: it is a locally direct factor module, it is locally nilpotent, and for every $s\in S$, the geometric fiber ${\mathfrak{h}}_s$ contains a regular element of ${\mathfrak{g}}_s$. Then $\mathfrak{h}$ is contained in one and only one Cartan subalgebra of $\mathfrak{g}$.

(N.B. in the case of interest to us, $\mathfrak{h}$ satisfies the stated conditions: it is locally nilpotent because $H$ has nilpotent fibers, and the condition on regular elements follows from 5.1.) Since 5.3 is local for the fpqc topology, it suffices to prove that at a point $s\in S$ such that $\kappa (s)$ is infinite, there exists an open neighborhood $U$ of $s$ such that existence and uniqueness hold for every base change $S^{\prime }\to S$ factoring through $U$. Take a regular element of $\mathfrak{g}\otimes \kappa (s)$ contained in $\mathfrak{h}\otimes \kappa (s)$, extend it to a section $a$ of $\mathfrak{h}$ on an open neighborhood $U$ of $s$; thanks to (C₂) one may suppose this section is regular (2.10). One may suppose $U=S$. A Cartan subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$ contains $a$, hence is identical to $\mathfrak{d}=Nil(a,\mathfrak{g})$ (2.6), whence uniqueness. Moreover since $\mathfrak{h}$ is locally nilpotent, one indeed has $\mathfrak{h}\subset \mathfrak{d}$, which proves existence.

d) Is a particular case of (XII 7.9 d)).

Corollary 5.4. Let $G$ be a smooth and connected algebraic group over an algebraically closed field $k$, $H$ a connected algebraic subgroup such that $\mathfrak{h}$ contains a Cartan subalgebra of $\mathfrak{g}$, i.e. such that $H$ contains a subgroup of type (C) of $G$. Then the number of conjugates of $H$ containing a regular element of $G(k)$ is equal to the number of conjugates of $\mathfrak{h}$ containing a regular element of $\mathfrak{g}$.

Indeed, let $g$ be a regular element of $G(k)$, $C$ the unique Cartan subgroup of $G$ containing $g$ (XIII 2); then the conjugates of $H$ containing $g$ are those containing $C$ (XIII 2.8 b)). Similarly, let $x$ be a regular element of $\mathfrak{g}$; then if a conjugate ${\mathfrak{h}}^{\prime }$ of $\mathfrak{h}$ contains $x$, then $ad(x{)}_{\mathfrak{g}/{\mathfrak{h}}^{\prime }}$ is injective (XIII 5.4), hence ${\mathfrak{h}}^{\prime }$ contains $Nil(x,\mathfrak{g})=\mathfrak{d}$, hence the number of conjugates of $\mathfrak{h}$ containing $x$ is equal to the number of conjugates containing the Cartan subalgebra $\mathfrak{d}$. Moreover $\mathfrak{d}$ is the Lie algebra of a subgroup $D$ of type (C) of $G$. One may evidently suppose $C\subset D$, and assertion 5.4 will follow from this: in order for $H$ to contain $C$, it is necessary and sufficient that $\mathfrak{h}$ contain $\mathfrak{d}$. Indeed, by virtue of (XIII 5.5) the relation $\mathfrak{h}\supset \mathfrak{d}$ implies $H\supset D$ and a fortiori $H\supset C$. Conversely, by 5.1 $\mathfrak{t}$ contains a regular element $x$ of $\mathfrak{g}$; then $H\supset C$ implies $\mathfrak{h}\ni x$, hence as already pointed out, this implies $\mathfrak{h}\supset \mathfrak{d}$. This completes the proof.

Let $G$ be as in 5.2 and suppose that the infinitesimal rank of the fibers of $G$ remains locally constant (condition (C₀)). Then thanks to 5.2 c), one finds a homomorphism of functors on $(Sch{)}^{\circ }/S$:

$$\mathcal{C}⟶\mathcal{D}$$

where

𝒞(S′) = set of Cartan subgroups of G_{S′}
𝒟(S′) = set of subgroups of type (C) of G_{S′}.

By virtue of 3.10 and 5.2 a), $\mathcal{D}$ is representable by a smooth and quasi-projective prescheme over $S$. Consider the subgroup $D$ of type (C) of $G_{\mathcal{D}}$ playing the universal role with respect to $G/S$; one may then consider the functor ${\mathcal{C}}_D:(Sch{)}^{\circ }/\mathcal{D}\to (Ens)$ defined in terms of the $\mathcal{D}$-group $D$ as $\mathcal{C}$ in terms of the $S$-group $G$. One then has the following result:

Proposition 5.5. Under the preceding conditions, considering $\mathcal{C}$ as a functor over the prescheme $\mathcal{D}$, then $\mathcal{C}$ is $\mathcal{D}$-isomorphic to the functor ${\mathcal{C}}_D$ "of Cartan subgroups of $D$".

This follows at once from the:

Corollary 5.6. Let $G$ be a smooth $S$-prescheme in groups of finite presentation, $D$ a subgroup of type (C) of $G$. Then there is a bijective correspondence between Cartan subgroups of $G$ contained in $D$, and Cartan subgroups of $D$ (more precisely, for a subgroup $H$ of $G$, $H$ is a Cartan subgroup of $G$ if and only if it is a Cartan subgroup of $D$).

Indeed, this is a particular case of (XII 7.9 c)), taking into account that over an algebraically closed field, a subgroup of type (C) of $G$ contains a Cartan subgroup of $G$.

For the following section, the main result obtained here is 5.2 c) for $H$ a Cartan subgroup of $G$, which permits stating 5.5 and thus furnishes a useful "dévissage" of $\mathcal{C}$.

6. Applications to the structure of algebraic groups

Theorem 6.1. Let $G$ be a smooth algebraic group over a field $k$. Consider the scheme $\mathcal{T}$ of maximal tori of $G$, isomorphic to the scheme $\mathcal{C}$ of Cartan subgroups of $G$, which is a homogeneous space under $G^0$, and a smooth affine connected algebraic scheme (XII 7.1 d)). Then $\mathcal{C}$ is a rational variety, i.e. the field of rational functions of $\mathcal{C}$ is a pure extension of $k$.

We shall first give the proof in the case where $k$ is infinite. One may evidently suppose $G$ connected, since $\mathcal{T}$ hence $\mathcal{C}$ does not change on replacing $G$ by $G^0$. Moreover, by virtue of (XII 7.6), $\mathcal{C}$ does not change on dividing $G$ by a central subgroup. This allows us, first dividing by the center of $G$, to suppose $G$ affine (XII 6.1), then, dividing by its reductive center (XII 4.1 and 4.4), to suppose that the reductive center of $G$ is trivial (XII 4.7 b)). Moreover we proceed by induction on $n=\dim G$, supposing the theorem proved for dimensions $n^{\prime }<n$. If $G$ is nilpotent, then $\mathcal{C}$ is reduced to a single point rational over $k$, and 6.1 is trivial. In the contrary case, the Lie algebra of $G$ is non-nilpotent (1.3), hence the Cartan subalgebras of $\mathfrak{g}$ are of dimension $n^{\prime }<n$, hence the subgroups of type (C) of $G$ are of dimension $n^{\prime }<n$. Consider then the morphism

$$\mathcal{C}⟶\mathcal{D}$$

envisaged in 5.5. We know by 3.10 ($k$ being an infinite field, hence $\mathfrak{g}$ containing a regular element) that $\mathcal{D}$ is a rational variety, i.e. the field $K$ of rational functions on $\mathcal{D}$ is a pure extension of $k$. Consider the fiber of $\mathcal{C}$ over the generic point $x$ of $\mathcal{D}$; by virtue of 5.5 this is the scheme of Cartan subgroups of a certain smooth and connected algebraic group $D_x$ over $K=\kappa (x)$ (namely $D_x=$ "the generic subgroup of type (C) of $G$"). The field $L$ of rational functions on $\mathcal{C}$ is therefore isomorphic to the field of rational functions on ${\mathcal{C}}_{D_x}$, which by the induction hypothesis (since $\dim D_x=n^{\prime }<n$) is a pure extension of $K$. So by transitivity $L$ is a pure extension of $k$.

When $k$ is finite, a different proof is needed. One may still suppose $G$ is affine and connected. Note that $k$ is perfect; it follows at once that the radical $R$ of $G_{\bar{k}}$ is "defined over $k$", i.e. comes from a subgroup $R$ of $G$. Suppose first $R\ne G$, i.e. $G$ non-solvable, and let

u : G ⟶ G′ = G/R

be the canonical morphism. Consider the corresponding morphism $C\mapsto u(C)$

$$v:{\mathcal{C}}_G⟶{\mathcal{C}}_{G^{\prime }}$$

(whose definition is immediate by virtue of (XII 7.1 e))). Let $x$ be the generic point of ${\mathcal{C}}_{G^{\prime }}$; then the fiber $v^{-1}(x)$ is identified with the scheme of Cartan subgroups of G_K (where $K=\kappa (x)$) whose image in $G_K^{\prime }$ is a certain Cartan subgroup $C_x^{\prime }$ (namely, "the generic Cartan subgroup of $G^{\prime }$"). It is therefore also the scheme of Cartan subgroups of $H=u_K^{-1}(C^{\prime })$ (XII 7.9 c)), and since $K$ is here an infinite extension of $k$, it follows from the already proved part of 7.1 that the field $L$ of rational functions of ${\mathcal{C}}_G$, equal to that of ${\mathcal{C}}_H$, is a pure transcendental extension of $K$. To prove that it is a pure transcendental extension of $k$, it therefore suffices to prove that this is the case of $K$, i.e. one is reduced to the case where $G$ is semisimple. One may moreover suppose that $G$ is adjoint (dividing $G$ by its reductive center if necessary). But then by virtue of 3.18 one has ${\mathcal{C}}_G\simeq {\mathcal{D}}_G$, and by virtue of 3.10 it suffices to prove that $\mathfrak{g}$ admits a regular point, which (as was pointed out in 3.20) is an unpublished result of Chevalley ¹⁰.