Exposé VIII. Diagonalizable groups

by A. Grothendieck

Version 1.1 of 8 November 2009: additions in 1.2, 1.4, 1.7, 3.1, 3.4, 4.5.1, 6.4, 6.8 — 1.5.1 and section 7 to be revised.¹

1. Biduality

Let $C$ be a category, which we identify, as is usual, with a full subcategory of $\hat{C}=\mathrm{Hom}(C^{\circ },(Ens))$ (cf. Exp. I). Let $I$ be a commutative group functor, i.e. an object of Ĉ endowed with a commutative group structure (cf. I, 2.1).² For every $X\in Ob(\hat{C})$, the object $\mathrm{Hom}(X,I)$ carries a commutative group structure induced by that of $I$. For every group $G$ in Ĉ, let

$$D(G)={\mathrm{Hom}}_{gr}.(G,I)$$

be the subobject of $\mathrm{Hom}(G,I)$ defined, for every $S\in Ob(C)$, by

$$(\ast )D(G)(S)={\mathrm{Hom}}_{S-gr.}(G_S,I_S),$$

where $G_S=G\times S$ and $I_S=I\times S$ are considered as $S$-groups, i.e. groups in $\hat{C}/S$. Then $D(G)$ is a Ĉ-subgroup of $\mathrm{Hom}(G,I)$. In this way one obtains a contravariant functor $D$ from the category of Ĉ-groups to the category of commutative Ĉ-groups.

The right-hand side of (*) can also be interpreted as the subset of $\mathrm{Hom}(G\times S,I)$ consisting of morphisms $G\times S\to I$ which are "multiplicative with respect to the first argument $G$". Moreover, the preceding formulas remain valid more generally when $S$ is an arbitrary object of Ĉ, not necessarily coming from $C$.

If we now take for $S$ a group in Ĉ, which we shall denote $G^{\prime }$, then in the left-hand side $\mathrm{Hom}(G^{\prime },D(G))$ of (*) we can single out the subset ${\mathrm{Hom}}_{gr}.(G^{\prime },D(G))$ consisting of morphisms that respect the group structures of $G^{\prime }$ and $D(G)$.

It then corresponds to the subset of $\mathrm{Hom}(G\times G^{\prime },I)$ consisting of morphisms that are multiplicative with respect to the first and with respect to the second argument — which one may call bilinear morphisms from $G\times G^{\prime }$ to $I$, or pairings of $G$ and $G^{\prime }$ with values in $I$. One thus finds

(**)   Hom_gr.(G′, D(G)) ⥲ Hom_bil.(G × G′, I),

which is an isomorphism functorial in the pair $(G,G^{\prime })$. Since the right-hand side is symmetric in $G$ and $G^{\prime }$, one deduces a functorial bijection

(***)  Hom_gr.(G′, D(G)) ⥲ Hom_gr.(G, D(G′)).

In other words, "it amounts to the same thing" to give a group homomorphism $G^{\prime }\to D(G)$ or a group homomorphism $G\to D(G^{\prime })$, both reducing in effect to the datum of a pairing $G\times G^{\prime }\to I$.

Applying this to the case $G^{\prime }=D(G)$ and to the identity homomorphism $G^{\prime }\to D(G)$, one finds a canonical homomorphism

$$(\ast \ast \ast \ast )G\to D(D(G)).$$

Definition 1.0.³ We say that $G$ is reflexive (relative to $I$) if the preceding homomorphism is an isomorphism. We note that this implies that $G$ is commutative.

One thus sees that:

Proposition 1.0.1. The functor $D$ induces an anti-equivalence of the category of reflexive Ĉ-groups with itself.

In particular, if $G$, $H$ are two reflexive groups, $D$ induces an isomorphism

Hom_gr.(G, H) ⥲ Hom_gr.(D(H), D(G))

(it even suffices that $H$ be reflexive, as one sees from formula (***)).

Definition 1.0.2. As usual, we shall then say that a $C$-group $G$ is reflexive if it is reflexive as a Ĉ-group (without worrying whether $D(G)$ is representable or not).

One thus obtains, by $D$, an anti-equivalence of the category of reflexive $C$-groups $G$ such that $D(G)$ is representable, with itself.

Remark 1.0.3. To conclude these generalities, let us point out that the formation of duals $D(G)$ commutes with base extension, which therefore transforms reflexive groups into reflexive groups.

We shall be interested henceforth in the case where $C=(Sch)/S$, the category of preschemes over $S$, and $I=G_{m,S}$, the "multiplicative group over $S$" (cf. Exp. I). For every ordinary group $M$, we consider the $S$-group M_S. One sees at once that for every prescheme in groups $J$ over $S$, there is a canonical isomorphism (functorial in $M$ and $J$, and compatible with base extension):

Hom_{S-gr.}(M_S, J) = Hom_gr.(M, J(S)).

Applying this to $J=I=G_{m,S}$ and to a variable $S^{\prime }$ over $S$, one finds a functorial isomorphism:

$$(\mathrm{1.0.4})D(M_S)(S^{\prime })\stackrel{\sim }{\to }{\mathrm{Hom}}_{gr}.(M,G_m(S^{\prime })).$$

One thus recovers the functor already considered in I, 4.4, also denoted $D_S(M)$, which is representable for $M$ commutative, since

D_S(M) = D(M_S) = Spec O_S(M),

where $O_S(M)$ denotes the algebra of the group $M$ with coefficients in O_S. (Let us note moreover, in the general case, that $D(M)$ does not change if one replaces $M$ by its abelianization, so that no information is lost by assuming $M$ commutative.)

Definition 1.1. A prescheme in groups $G$ over $S$ is said to be diagonalizable if it is isomorphic to a scheme of the form $D_S(M)=D(M_S)={\mathrm{Hom}}_{S-gr.}(M_S,G_m)$ for some suitable commutative group $M$.

We say that $G$ is locally diagonalizable if every point of $S$ admits an open neighborhood $U$ such that $G|U$ is diagonalizable.

Theorem 1.2. Let $\Gamma$ be a constant commutative group scheme over $S$, i.e. isomorphic to a group scheme of the form M_S, where $M$ is an ordinary commutative group. Then $\Gamma$ is reflexive, i.e. the canonical homomorphism

$$\Gamma \to D(D(\Gamma ))$$

is an isomorphism.⁴ The diagonalizable group $D(M_S)$ is therefore also reflexive.

Taking the definitions into account, this follows from the following statement (which one will apply to a prescheme $S^{\prime }$ over $S$):

Corollary 1.3. Let $G=D(M_S)$. Then every homomorphism of $S$-groups

$$u:G\to G_{m,S}$$

is defined by a uniquely determined section of M_S over $S$, i.e. by a uniquely determined locally constant map from $S$ to $M$.

Proof. Since by definition

$$G_{m,S}=GL(1{)}_S={\mathrm{Aut}}_{O_S-mod.}(O_S),$$

one sees that giving a group homomorphism $G\to G_{m,S}$ is equivalent to giving on O_S a structure of $G$-O_S-module compatible with the natural O_S-module structure on O_S (cf. I, 4.7). By I, 4.7.3, this also amounts to giving an $M$-grading on O_S, i.e. a decomposition of O_S as a direct sum of modules $L_m$ ($m\in M$).

Now it is well known that a direct factor of a locally free module of finite type is locally free of finite type; therefore each $L_m$ is, in a neighborhood of each point of $S$, either zero or free of rank 1, and in the latter case identical to O_S in that neighborhood. Let $S_m$ be the open subset of $S$ consisting of the points where this second alternative occurs. Expressing that O_S is the direct sum of the $L_m$, one sees that the union of the $S_m$ is $S$, and that the $S_m$ are pairwise disjoint. Hence giving a group homomorphism $G\to G_{m,S}$ is equivalent to giving a decomposition of $S$ as a union of pairwise disjoint open subsets $S_m$ ($m\in M$), i.e. to giving a locally constant map from $S$ to $M$. This establishes 1.3, hence 1.2.

Corollary 1.4. A diagonalizable group is reflexive; the same therefore holds for a locally diagonalizable group. If $M$, $N$ are two ordinary commutative groups, the natural homomorphism

Hom_{S-gr.}(M_S, N_S) → Hom_{S-gr.}(D(N_S), D(M_S))

is bijective.

⁵ The preceding isomorphism being compatible with base extension, one deduces an isomorphism of $S$-functors in groups:

(1)    Hom_{S-gr.}(M_S, N_S) ⥲ Hom_{S-gr.}(D(N_S), D(M_S)).

For every $S$-prescheme $T$, one has

Hom_{S-gr.}(M_S, N_S)(T) = Hom_gr.(M, Γ(N_T/T)),

and, by I 1.8, $\Gamma (N_T/T)$ is the abelian group of locally constant maps $T\to N$. On the other hand, let ${\mathrm{Hom}}_{gr}.(M,N{)}_S$ be the constant $S$-group associated with the ordinary abelian group ${\mathrm{Hom}}_{gr}.(M,N)$. One has an evident homomorphism of $S$-functors in commutative groups:

(2)    Hom_gr.(M, N)_S  -θ→  Hom_{S-gr.}(M_S, N_S),

which is always a monomorphism. Moreover, it is an isomorphism if $M$ is of finite type.⁶

From the foregoing one deduces point (a) of the following corollary; point (b) follows from it by the descent results "recalled" in 1.7.⁷

Corollary 1.5. a) Let $M$, $N$ be two ordinary commutative groups, with $M$ of finite type. Then one has an isomorphism

Hom_gr.(M, N)_S ⥲ Hom_{S-gr.}(D(N_S), D(M_S));

consequently ${\mathrm{Hom}}_{S-gr.}(D(N_S),D(M_S))$ is representable.

b) More generally, if $G$, $H$ are locally diagonalizable, with $H$ of finite type, then ${\mathrm{Hom}}_{S-gr.}(G,H)$ is representable.

⁸ From 1.5 one concludes:

Corollary 1.6. Under the conditions of 1.5, if $S$ is connected, one has

Hom_{S-gr.}(D_S(N), D_S(M)) ⥲ Hom_gr.(M, N)

and

Isom_{S-gr.}(D_S(N), D_S(M)) ⥲ Isom_gr.(M, N).

1.7. Descent of representability

⁹ In this paragraph we "recall" some descent results that will be used frequently in what follows.

Scholium 1.7.1. Let $S$ be a prescheme and $X$, $Y$, $T$ $S$-preschemes. If $(T_i)$ is an open covering of $T$, and if we set $T_{ij}=T_i\cap T_j=T_i{\times }_T{T}_j$, then, since giving a morphism of $T$-preschemes $X_T\to Y_T$ is local on $T$, one has an exact sequence of sets:

(1)    Hom_T(X_T, Y_T) → ∏_i Hom_{T_i}(X_{T_i}, Y_{T_i}) ⇒ ∏_{i,j} Hom_{T_{ij}}(X_{T_{ij}}, Y_{T_{ij}})

i.e. ${\mathrm{Hom}}_S(X,Y)$ is a local $S$-functor, that is, a sheaf on $(Sch)/S$ endowed with the Zariski topology.

More generally, by IV 4.5.13, ${\mathrm{Hom}}_S(X,Y)$ is a sheaf on $(Sch)/S$ for every topology coarser than the canonical topology — for example, for the (fpqc) topology.

If $G$, $H$ are $S$-preschemes in groups, one deduces that the subfunctor ${\mathrm{Hom}}_{S-gr.}(X,Y)$ is a sheaf for the (fpqc) topology (hence a fortiori a local functor).

Lemma 1.7.2.¹⁰ Let $F$ be a local $S$-functor.

(i) Suppose there exists an open covering $(S_i)$ of $S$ such that the restriction $F_i=F{\times }_S{S}_i$ of $F$ to each $S_i$ is representable by an $S_i$-prescheme $X_i$. Then $F$ is representable by an $S$-prescheme $X$.

(ii) Suppose $F$ is an (fpqc) sheaf and that there exists a faithfully flat quasi-compact morphism $S^{\prime }\to S$ such that the restriction $F^{\prime }=F{\times }_S{S}^{\prime }$ of $F$ is representable by an $S^{\prime }$-prescheme $X^{\prime }$. Then $X^{\prime }$ is endowed with a descent datum (cf. IV 2.1) relative to $S^{\prime }\to S$.

If moreover this descent datum is effective (which is the case if $X^{\prime }$ is affine over $S^{\prime }$), then $F$ is representable by an $S$-prescheme $X$.

Proof. (i) It follows from the hypothesis that $X_i{\times }_S{S}_j$ and $X_j{\times }_S{S}_i$ both represent the restriction of $F$ to $S_{ij}=S_i{\times }_S{S}_j$, hence, by the Yoneda lemma, there is a unique isomorphism of $S_{ij}$-preschemes

c_{ji} : X_i ×_S S_j ⥲ X_j ×_S S_i;

one then has isomorphisms of preschemes over $S_{ijk}=S_i{\times }_S{S}_j{\times }_S{S}_k$:

       c_{ji} × id_{S_k}                       c_{kj} × id_{S_i}
X_i ×_S S_j ×_S S_k ─────→ X_j ×_S S_i ×_S S_k ≅ X_j ×_S S_k ×_S S_i ─────→ X_k ×_S S_j ×_S S_i

       c_{ki} × id_{S_j}
X_i ×_S S_k ×_S S_j ─────→ X_k ×_S S_i ×_S S_j

and since all these objects represent the restriction of $F$ to $S_{ijk}$, this diagram is commutative, i.e. the $c_{ji}$ satisfy the usual cocycle relation $c_{kj}\circ c_{ji}=c_{ki}$.

It follows that the $X_i$ glue together into an $S$-prescheme $X$ such that $X{\times }_S{S}_i=X_i$ for every $i$. For every $Y$ over $S_i$, one therefore has

(*)    F(Y) = F_i(Y) = Hom_{S_i}(Y, X ×_S S_i) = Hom_S(Y, X) = h_X(Y).

Then, for $Y\to S$ arbitrary, the $Y_i=Y{\times }_S{S}_i$ form an open covering of $Y$; set $Y_{ij}=Y_i{\times }_Y{Y}_j=Y{\times }_S{S}_{ij}$. Since $F$ (resp. $h_X$) is a local functor by hypothesis (resp. because the Zariski topology is coarser than the canonical topology), $F(Y)$ and $h_X(Y)$ both identify, in view of (*), with the kernel of the double arrow:

∏_i F(Y_i)  ⇒  ∏_{i,j} F(Y_{ij})

∏_i h_X(Y_i)  ⇒  ∏_{i,j} h_X(Y_{ij}).

This proves (i).

(ii) It follows from the hypothesis that $F_1^{\prime \prime }=F^{\prime }{\times }_{S^{\prime }}{S}_1^{\prime \prime }$ (where $S_1^{\prime \prime }=S^{\prime \prime }=S^{\prime }{\times }_S{S}^{\prime }$ is considered as $S^{\prime }$-prescheme via the first projection) is represented by $X_1^{\prime \prime }=X^{\prime }{\times }_{S^{\prime }}{S}_1^{\prime \prime }$; similarly, $F_2^{\prime \prime }=F^{\prime }{\times }_{S^{\prime }}{S}_2^{\prime \prime }$ is represented by $X_2^{\prime \prime }=X^{\prime }{\times }_{S^{\prime }}{S}_2^{\prime \prime }$. Now $F_1^{\prime \prime }=F{\times }_S{S}^{\prime \prime }=F_2^{\prime \prime }$, hence there exists a (unique) $S^{\prime \prime }$-isomorphism $c:X_1^{\prime \prime }\stackrel{\sim }{\to }{X}_2^{\prime \prime }$. Then, denoting by $q_i$ (resp. $p_{ji}$) the projection of $S^{\prime \prime \prime }=S^{\prime }{\times }_S{S}^{\prime }{\times }_S{S}^{\prime }$ onto the $i$-th factor (resp. onto the factors $i$ and $j$), $X_i^{\prime \prime \prime }=X^{\prime }{\times }_{S^{\prime }}{S}_i^{\prime \prime \prime }$ (where $S_i^{\prime \prime \prime }=S^{\prime \prime \prime }$ considered as $S^{\prime }$-prescheme via $q_i$), and $p_{ji}^{\ast }(c):X_i^{\prime \prime \prime }\stackrel{\sim }{\to }{X}_j^{\prime \prime \prime }$ the isomorphism of $S^{\prime \prime \prime }$-preschemes deduced from $c$ by base change, one obtains a diagram of isomorphisms of $S^{\prime \prime \prime }$-preschemes:

              p_{21}^*(c)
        X‴_1 ─────────→ X‴_2
            ╲              │
  p_{31}^*(c) ╲             │ p_{32}^*(c)
              ╲             ↓
                X‴_3

and since all these objects represent the restriction of $F$ to $S^{\prime \prime \prime }$, this diagram is commutative, i.e. the usual cocycle relation $p_{32}^{\ast }(c)\circ p_{21}^{\ast }(c)=p_{31}^{\ast }(c)$ holds, i.e. $c$ is a descent datum on $X^{\prime }$ relative to $S^{\prime }\to S$ (cf. IV 2.1).

Suppose moreover that this descent datum is effective, i.e. that there exists an $S$-prescheme $X$ such that $X^{\prime }\simeq X{\times }_S{S}^{\prime }$ (by SGA 1, VIII 2.1, this is the case if $X^{\prime }$ is affine over $S^{\prime }$¹¹). Then, for every $Y\to S^{\prime }$, one has

(**)    F(Y) = F′(Y) = Hom_{S′}(Y, X ×_S S′) = Hom_S(Y, X) = h_X(Y).

Then, for $Y\to S$ arbitrary, set $Y^{\prime }=Y{\times }_S{S}^{\prime }$ and $Y^{\prime \prime }=Y^{\prime }{\times }_Y{Y}^{\prime }\simeq Y{\times }_S{S}^{\prime \prime }$. Then $Y^{\prime }\to Y$, like $S^{\prime }\to S$, is faithfully flat and quasi-compact, hence an $M$-effective epimorphism (where $M$ = family of faithfully flat quasi-compact morphisms), i.e. the equivalence relation

Y′ ×_Y Y′ ⇒ Y′

is $M$-effective and has quotient $Y$. Since $F$ (resp. $h_X$) is an (fpqc) sheaf by hypothesis (resp. because the (fpqc) topology is coarser than the canonical topology), $F(Y)$ and $h_X(Y)$ both identify, in view of (**), with the kernel of the double arrow

F(Y′) ⇒ F(Y′ ×_Y Y′)

h_X(Y′) ⇒ h_X(Y′ ×_Y Y′).

This proves (ii).

Corollary 1.7.3. Let $F$ be an (fpqc) sheaf on $(Sch)/S$. Suppose there exists an open covering $(S_i)$ of $S$ and, for each $i$, a faithfully flat quasi-compact morphism $S_i^{\prime }\to S_i$ such that the restriction $F_i^{\prime }=F{\times }_S{S}_i^{\prime }$ is representable by an $S_i^{\prime }$-prescheme $X_i^{\prime }$ affine over $S_i^{\prime }$. Then $F$ is representable by an $S$-prescheme $X$ affine over $S$ (such that $X{\times }_S{S}_i^{\prime }=X_i^{\prime }$ for every $i$).

If moreover each $X_i^{\prime }\to S_i^{\prime }$ is a closed immersion (resp. a finite étale morphism), the same holds for $X\to S$.

The first assertion follows from 1.7.2. For the second, it suffices to verify that each morphism $X{\times }_S{S}_i\to S_i$ is a closed immersion (resp. finite and étale), which follows from EGA IV₂, 2.7.1 (resp. and IV₄, 17.7.3).

Remark 1.7.4. Assertion 1.5 (b) follows, as announced, from 1.7.1 and 1.7.2 (i).

2. Schematic properties of diagonalizable groups

They are summarized in the following.

Proposition 2.1. Let $S$ be a non-empty prescheme, $M$ an ordinary commutative group, $G=D(M_S)$ the diagonalizable $S$-group defined by $M$. Then:

a) $G$ is faithfully flat over $S$, and affine over $S$ (a fortiori quasi-compact over $S$).

b) $M$ of finite type ⇔ $G$ of finite type over $S$ ⇔ $G$ of finite presentation over $S$.

c) $M$ finite ⇔ $G$ finite over $S$ ⇔ $G$ of finite type over $S$ and annihilated by an integer $n>0$. Then $deg(G/S)=Card(M)$.

c′) $M$ a torsion group ⇔ $G$ integral over $S$.

d) $M=0$ ⇔ $G$ = unit $S$-group.

e) $M$ of finite type, and the order of its torsion subgroup is prime to the residue characteristics of $S$ ⇔ $G$ is smooth over $S$.

The verification of (a) to (d) is trivial, and is left to the reader. Let us prove (e). If $G$ is smooth over $S$, it is locally of finite presentation over $S$, hence of finite presentation over $S$ since it is affine over $S$, hence $M$ is of finite type. So we may already assume $M$ of finite type, hence $G$ of finite presentation over $S$. Then¹² $G$ is smooth over $S$ if and only if its geometric fibers are, which reduces us to the case where $S$ is the spectrum of an algebraically closed field $k$. Writing $M=T\times L$, with $T$ the torsion subgroup and $L$ free, $L\simeq {\mathbb{Z}}^r$, one has $D(M)=D(T)\times D(L)$, where $D(L)\simeq G_m^r$ is smooth over $k$. So $G=D(M)$ is smooth over $k$ if and only if $D(T)$ is, which means, since $D(T)$ is finite over $k$ of degree equal to the order $n$ of $T$, that $D(T)(k)$ has $n$ elements. Now $T$ is isomorphic to a sum of groups $\mathbb{Z}/n_i\mathbb{Z}$, $n$ being the product of the $n_i$, so $D(T)$ is the product of the $D(\mathbb{Z}/n_i\mathbb{Z})={\mu }_{n_i}$ (group scheme of $n_i$-th roots of unity), hence

card(D(T)(k)) = ∏_i card μ_{n_i}(k),

where $card{\mu }_{n_i}(k)$ = (number of $n_i$-th roots of unity in $k$) $⩽n_i$, equality being attained if and only if $n_i$ is prime to the characteristic $p$ of $k$. Hence one has $card(D(T)(k))=n$ (where $n={\prod }_i{n}_i$) if and only if all the $n_i$ are prime to $p$, i.e. if and only if $n$ is prime to $p$. QED.

3. Exactness properties of the functor D_S

Theorem 3.1. Let $S$ be a prescheme, and

0 → M′ -u→ M -v→ M″ → 0

an exact sequence of ordinary commutative groups. Consider the sequence of transposed homomorphisms:

0 → D_S(M″) -v^t→ D_S(M) -u^t→ D_S(M′) → 0.

(i) $v^t$ induces an isomorphism of $D_S(M^{\prime \prime })$ with the kernel of $u^t$, and $u^t$ is faithfully flat and quasi-compact.

(ii)¹³ $D_S(M^{\prime })$ represents the (fpqc) quotient sheaf $D_S(M)/D_S(M^{\prime \prime })$.

Let $M$ denote the family of faithfully flat quasi-compact morphisms. First, (ii) follows from (i) (cf. IV, 4.6.5.1). Indeed, the equivalence relation in $D_S(M)$ defined by $u^t$ is the same as that defined by the subgroup $Ker(u^t)=D_S(M^{\prime \prime })$; since $u^t\in M$, this equivalence relation is $M$-effective (cf. IV, 3.3.2.1), and therefore $D_S(M^{\prime })$ represents the quotient sheaf for the (fpqc) topology (cf. IV, 4.6.5).

The first assertion of (i) is a trivial consequence of the definition of the functors $D_S(-)$; more generally, for any exact sequence

$$M^{\prime }\to M\to M^{\prime \prime }\to 0$$

(without zero on the left), one will have a transposed exact sequence:

$$0\to D_S(M^{\prime \prime })\to D_S(M)\to D_S(M^{\prime }).$$

(This is valid more generally in the context of the beginning of §1.) On the other hand, since $D_S(M)$ and $D_S(M^{\prime })$ are affine over $S$, $u^t$ is necessarily an affine morphism, a fortiori quasi-compact (whatever the homomorphism $u:M^{\prime }\to M$). The second assertion of (i) will therefore follow from point (a) of the following:

Corollary 3.2. Let $S$ be a non-empty prescheme, $u:M^{\prime }\to M$ a homomorphism of ordinary commutative groups, $u^t:G\to G^{\prime }$ the transposed homomorphism. Then:

a) For $u$ to be a monomorphism, it is necessary and sufficient that $u^t$ be faithfully flat.

b) For $u$ to be an epimorphism, it is necessary and sufficient that $u^t$ be a monomorphism (and then $u^t$ is even a closed immersion).

To prove (a), one notes that if $u$ is a monomorphism, then $O_S(M)$ is a module over $O_S(M^{\prime })$ admitting a non-empty basis (namely, the system of sections defined by any system of representatives of $M$ modulo $M^{\prime }$), a fortiori it is faithfully flat. Conversely, if this is the case, then $u^t:O_S(M^{\prime })\to O_S(M)$ is injective, which (for $S\ne \varnothing$) implies that $u:M^{\prime }\to M$ is injective.

To prove (b), one notes that if $u$ is an epimorphism, then $O_S(M^{\prime })\to O_S(M)$ is surjective, hence $u^t$ is a closed immersion and a fortiori a monomorphism. Conversely, if this is the case, then $Ker{u}^t$ = unit group; now setting $M^{\prime \prime }=Cokeru$, we have seen that $Ker{u}^t\simeq D_S(M^{\prime \prime })$, hence by 2.1 (d) one has $M^{\prime \prime }=0$, hence $u$ is an epimorphism.

One concludes from 3.1 in the usual way:

Corollary 3.3. Let $M^{\prime }-u\to M-v\to M^{\prime \prime }$ be an exact sequence of ordinary commutative groups; consider the transposed sequence

G″ -v^t→ G -u^t→ G′.

Then $v^t$ induces a faithfully flat quasi-compact morphism from $G^{\prime \prime }$ to $Ker{u}^t$, and the latter is a diagonalizable group isomorphic to $D_S(v(M))=D_S(Cokeru)$.

Corollary 3.4. Let $S$ be a prescheme, $u:G\to H$ a homomorphism of $S$-preschemes in locally diagonalizable groups, with $H$ of finite type over $S$. Set $G^{\prime }=Keru$. Then:

a) $G^{\prime }$ is locally diagonalizable; it is of finite type over $S$ if $G$ is.

b) The quotient $G/G^{\prime }$ "exists"; more precisely, the equivalence relation defined by $G^{\prime }$ in $G$ is $M$-effective (where $M$ = set of faithfully flat quasi-compact morphisms, cf. IV, 3.4). Moreover, $G/G^{\prime }$ is locally diagonalizable, of finite type over $S$.

c) The homomorphism $u:G\to H$ factors uniquely as

G -v→ G/G′ -w→ H,

where $v$ is the canonical homomorphism (so $v$ is faithfully flat and quasi-compact). Moreover $w$ is a closed immersion, and a fortiori a monomorphism.

Finally, the quotient $H^{\prime }=H/Imw=Cokerw=Cokeru$ exists; more precisely, the equivalence relation defined by $G/G^{\prime }$ in $H$ is $M$-effective, and $H^{\prime }$ is of finite type over $S$.

The first assertion of (c) is a consequence of (b), by definition of the quotient $G/G^{\prime }$ (cf. IV, 3.2.3).¹⁴ Let us show that the (fpqc) quotient sheaf $G/G^{\prime }$ is representable. This may be verified locally on $S$, as may all the other assertions; we may therefore suppose $G$ and $H$ diagonalizable, of the form $D_S(M)$ and $D_S(N)$.

Since $H$ is of finite type over $S$, $N$ is of finite type by 2.1 (b), hence by 1.5 $u$ is defined by a homomorphism $u_0:N\to M$. Then, in virtue of 3.1 and 3.2, $G^{\prime }$ is isomorphic to $D_S(Coker{u}_0)$ and $G/G^{\prime }$ is representable by $D_S(Im{u}_0)$; moreover, considering the exact sequence

0 → Ker u_0 → N -w_0→ Im u_0 → 0,

one obtains that $w$ is a closed immersion, and that the quotient $H^{\prime }=H/Imw$ is $D_S(Ker{u}_0)$; the latter is of finite type over $S$ since $N$, and hence $Ker{u}_0$, is of finite type.

Remarks. The existence-of-quotients result 3.4 will be substantially generalized in §5.

On the other hand, one will note that in the present § and the preceding one, the hypothesis $S\ne \varnothing$ was used only to ensure the validity of certain converses, enabling one to deduce from certain hypotheses on diagonalizable $S$-groups properties of the corresponding ordinary groups. The "direct"-sense results are valid without restriction on $S$, and the proofs given here apply in the general case.

Corollary 3.5. Let $G$ be a diagonalizable group prescheme over $S$, and let $n$ be an integer $\ne 0$. Then the subgroup ${}_{nG}$ of $G$, kernel of the homomorphism $n\cdot i{d}_G:G\to G$, is integral over $S$, and finite over $S$ if $G$ is of finite type over $S$.

Indeed, if $G=D_S(M)$, then ${}_{nG}=D_S(M/nM)$ by 3.1, and one concludes by 2.1 (b), (c), (c′).

4. Torsors under a diagonalizable group

Let $S$ be a prescheme, and $G=D_S(M)$ a diagonalizable group over $S$. We propose to determine the $G$-torsors (or principal homogeneous $G$-bundles) on $S$, in the sense of the "faithfully flat quasi-compact topology" (cf. Exp. IV, 5.1). Recall that a prescheme $P$ over $S$ with operator group $G$ is called a torsor or principal homogeneous if every point of $S$ admits an open neighborhood $U$ and a faithfully flat quasi-compact morphism $S^{\prime }\to U$ such that $P^{\prime }=P{\times }_S{S}^{\prime }$ is, as a bundle with operators, isomorphic to $G^{\prime }=G{\times }_S{S}^{\prime }$ (acting on itself by right translations). Since $G$ is affine over $S$, it follows from SGA 1, VIII 5.6 that $P$ is necessarily affine over $S$. Note also that since $G$ is itself faithfully flat and quasi-compact over $S$, $P$ is principal homogeneous under $G$ if and only if it is "formally principal homogeneous", and if moreover it is faithfully flat and quasi-compact over $S$ (cf. IV, 5.1.6).

Recall on the other hand (Exp. I, 4.7.3) that giving an $S$-prescheme $P$ affine over $S$ with operator group $G=D_S(M)$ amounts to giving a quasi-coherent commutative $M$-graded algebra on $S$, i.e. a quasi-coherent algebra $\mathcal{A}$ on $S$ endowed with a direct sum decomposition (as a module):

𝓐 = ⨁_{m ∈ M} 𝓐_m,

with

𝓐_m · 𝓐_{m′} ⊆ 𝓐_{m+m′}    for m, m′ ∈ M.

This said, the answer to the problem posed above is given by:

Proposition 4.1. For the prescheme $P$ with operator group $G=D_S(M)$, defined by the $M$-graded algebra $\mathcal{A}$, to be a principal homogeneous bundle under $G$, it is necessary and sufficient that $\mathcal{A}$ satisfy the following conditions:

a) For every $m\in M$, ${\mathcal{A}}_m$ is an invertible module on $S$.

b) For $m,m^{\prime }\in M$, the homomorphism

𝓐_m ⊗_{O_S} 𝓐_{m′} → 𝓐_{m+m′}

induced by the multiplication in $\mathcal{A}$, is an isomorphism.

The necessity of the conditions is immediate by descent, since they are satisfied in the case where $P$ is the trivial principal homogeneous bundle, i.e. $\mathcal{A}=O_S(M)$. For the sufficiency, one notes that (a) already implies that $P$ is faithfully flat over $S$, it is in any case quasi-compact over $S$ (being affine over $S$), so it remains to verify that it is formally principal homogeneous under $G$, i.e. that the well-known homomorphism

P ×_S G → P ×_S P

is an isomorphism. Now on affine algebras, this homomorphism is given explicitly as the homomorphism

𝓐 ⊗ 𝓐 → 𝓐(M) = 𝓐 ⊗ O_S(M)

which in bidegree $(m,n)$ (where $m,n\in M$) is given by

x_m ⊗ y_n ↦ x_m y_n ⊗ e_n.

From the standpoint of degrees, this homomorphism is compatible with the homomorphism $M\times M\to M\times M$ given by

(m, n) ↦ (m + n, n),

which is an isomorphism. This shows that (b) expresses precisely (independently of (a)) that $P$ is formally principal homogeneous, and establishes 4.1.

Note also that one obtains, by faithfully flat descent:

Corollary 4.2. The conditions of 4.1 imply that the homomorphism

$$O_S\to {\mathcal{A}}_0$$

is an isomorphism.

If for example $M=\mathbb{Z}$, then under the conditions of 4.1 one sees that $\mathcal{A}$ is essentially known when one knows ${\mathcal{A}}_1=\mathcal{L}$, namely

𝓐 ≃ ⨁_{n ∈ ℤ} 𝓛^{⊗n}

(isomorphism of graded algebras). One thereby recovers the well-known result:

Corollary 4.3. There is an equivalence between the category of principal homogeneous bundles $P$ on $S$ with group $G_{m,S}$, and the category of invertible modules $\mathcal{L}$ on $S$ (taking as morphisms, for the definition of each category, the isomorphisms of the structures in play). One obtains two quasi-inverse functors by associating with each $P$ the degree-1 component of its $\mathbb{Z}$-graded affine algebra, and with each $\mathcal{L}$ the spectrum of the $\mathbb{Z}$-graded algebra ${⨁}_{n\in \mathbb{Z}}{\mathcal{L}}^{\otimes n}$.

In particular:

Corollary 4.4. The group of classes of principal homogeneous bundles on $S$ with group $G_{m,S}$ is isomorphic to the group $\mathrm{Pic}(S)$ of classes of invertible modules on $S$, i.e. to $H^1(S,O_S^{\times })$.

Taking into account that $G_{m,S}$ is the scheme of automorphisms of the module O_S, one sees that 4.4 is equivalent to the following statement, which is one of the variants of Hilbert's "Theorem 90":

Corollary 4.5. Every principal homogeneous bundle on $S$ with group $G_m$ is locally trivial (in the sense of the Zariski topology).

Remark 4.5.1. One will note that the preceding statement is no longer true in general for a group such as ${\mu }_n$, or for a "twisted form" of $G_m$; for example, the unique twisted form of $G_m$ over the field $\mathbb{R}$ of reals gives a group of 1-cohomology equal to $\mathbb{Z}/2\mathbb{Z}$.

¹⁵ Indeed, let $S^1$ be the kernel of the norm morphism $N:{\prod }_{\mathbb{C}/\mathbb{R}}{G}_{m,\mathbb{C}}\to G_{m,\mathbb{R}}$; this is a $\mathbb{C}/\mathbb{R}$-twisted form of $G_{m,\mathbb{R}}$. The equation $N(z)=-1$ in ${\prod }_{\mathbb{C}/\mathbb{R}}(G_{m,\mathbb{C}})$ defines an $S^1$-torsor $X$ over $\mathrm{Spec}(\mathbb{R})$, locally trivial for the étale topology, but not trivial since $X(\mathbb{R})=\varnothing$. Let us show that $H_{\acute{e}}^{1}t(\mathbb{R},S^1)\cong \mathbb{Z}/2\mathbb{Z}$. One has an exact sequence of smooth commutative $\mathbb{R}$-group schemes:

1 → S¹ → ∏_{ℂ/ℝ} G_{m,ℂ} → G_{m,ℝ} → 1

which gives rise to a long exact sequence of étale cohomology (or of Galois cohomology):

0 → S¹(ℝ) → ℂ^× -N→ ℝ^× → H¹_ét(ℝ, S¹) → H¹_ét(ℝ, ∏_{ℂ/ℝ} G_{m,ℂ}) → ⋯

Now (see for example XXIV, 8.4), $H_{\acute{e}}^{1}t(\mathbb{R},{\prod }_{\mathbb{C}/\mathbb{R}}{G}_{m,\mathbb{C}})\simeq H_{\acute{e}}^{1}t(\mathbb{C},G_{m,\mathbb{C}})$, and the latter is zero by 4.5 (or, here, because $\mathbb{C}$ is algebraically closed). One thus obtains an isomorphism $H_{\acute{e}}^{1}t(\mathbb{R},S^1)\simeq {\mathbb{R}}^{\times }/N({\mathbb{C}}^{\times })\simeq \pm 1$.

We shall need in the following § the following result:

Proposition 4.6. Under the conditions of 4.1, conditions (a) and (b) are equivalent to the following conditions:

a′) $O_S\to {\mathcal{A}}_0$ is an isomorphism.

b′) For every $m$ in $M$ (it suffices: in a system of generators of $M$), one has

$${\mathcal{A}}_m\cdot {\mathcal{A}}_{-m}={\mathcal{A}}_0.$$

The necessity being evident, taking 4.2 into account,¹⁶ we shall reduce to proving:

Corollary 4.7. Let $A={⨁}_{n\in \mathbb{Z}}{A}_n$ be a $\mathbb{Z}$-graded ring such that

$$A_1\cdot A_{-1}=A_0.$$

Then the $A_n$ are invertible A_0-modules, and for $n,n^{\prime }\in \mathbb{Z}$, the homomorphism

A_n ⊗_{A_0} A_{n′} → A_{n+n′}

induced by the multiplication in $A$, is an isomorphism.

By hypothesis, there exist $f_i\in A_1$, $g_i\in A_{-1}$, such that

$$(\ast )\sum _i{f}_i{g}_i=1.$$

As the conclusion to be established is local on $\mathrm{Spec}(A_0)$,¹⁷ and as, by (*), $\mathrm{Spec}(A_0)$ is covered by the affine opens $D(f_i{g}_i)$, one is reduced to the case where there exists an element $f\in A_1$ invertible in $A$. Then for every $n\in \mathbb{Z}$, $f^n$ is an element of $A_n$ invertible in $A$, hence defines an isomorphism $h\mapsto f^{n}h$ from A_0 onto $A_n$. Moreover,

this shows that one obtains an isomorphism $A_0[t,t^{-1}]\to A$ of graded A_0-algebras by sending $t$ to $f$, which completes the proof of 4.7.

Then, under the conditions of 4.6, 4.7 already implies that the ${\mathcal{A}}_m$ ($m\in M$) are invertible. To prove condition 4.1 (b), one may therefore suppose that ${\mathcal{A}}_m$ and ${\mathcal{A}}_{m^{\prime }}$ admit bases $f_m$ and $f_{m^{\prime }}$, having inverses $f_m^{-1}\in \Gamma ({\mathcal{A}}_{-m})$ and $f_{m^{\prime }}^{-1}\in \Gamma ({\mathcal{A}}_{-m^{\prime }})$. Then the product by $f_m^{-1}{f}_{m^{\prime }}^{-1}\in \Gamma ({\mathcal{A}}_{-m-m^{\prime }})$ defines a homomorphism ${\mathcal{A}}_{m+m^{\prime }}\to {\mathcal{A}}_0\simeq O_S$, sending the image of $f_m\otimes f_{m^{\prime }}$ to the section 1 of O_S. In the diagram

                     w
                   ──────────────────────
                 ↗                       ↘
𝓐_m ⊗ 𝓐_{m′}  ─u→  𝓐_{m+m′}  ─v→  𝓐_0 ≃ O_S

$w$ and $v$ are therefore epimorphisms of invertible sheaves, hence isomorphisms, hence $u$ is an isomorphism. QED.

5. Quotient of an affine scheme by a diagonalizable group acting freely

¹⁸ We denote by $M$ the set of faithfully flat quasi-compact morphisms, and we recall that torsors are considered in the sense of the (fpqc) topology.

Theorem 5.1. Let $S$ be a prescheme, $M$ an ordinary commutative group, $G=D_S(M)$ the diagonalizable group over $S$ defined by it, $P$ an $S$-prescheme affine over $S$ on which $G$ acts freely on the right.

Then the equivalence relation defined by $G$ in $P$ is $M$-effective (cf. IV, 3.4), i.e. the quotient $X=P/G$ exists and $P$ is a torsor over $X$ with group $G_X=D_X(M)$. Moreover, $P/G$ is affine over $S$; more precisely, if $P$ is defined by the $M$-graded algebra $\mathcal{A}$, then $P/G$ is isomorphic to $\mathrm{Spec}({\mathcal{A}}_0)$, where ${\mathcal{A}}_0={\mathcal{A}}^G$ is the degree-0 component of $\mathcal{A}$.

Proof. Set $X=\mathrm{Spec}({\mathcal{A}}_0)$. Then one has a natural morphism $P\to X$, deduced from ${\mathcal{A}}_0\to \mathcal{A}$, which is invariant under the action of $G$. In this way, $P$ becomes an $X$-prescheme, affine over $X$, with operator group $G_X=D_X(M)$, and the hypothesis that $G$ acts freely on $P/S$ implies that G_X acts freely on $P/X$. Everything reduces to showing that $P$ is a principal homogeneous bundle under G_X, using the fact that ${\mathcal{B}}_0=O_X$, where $\mathcal{B}$ is the $M$-graded algebra on $X$ defining $P/X$. We may then suppose $X=S$ and $S$ affine, so $P$ is affine, given by an $M$-graded ring $A$ whose homogeneous part of degree $m$ will be denoted $A_m$, so that $S=\mathrm{Spec}(A_0)$. Taking 4.6 into account, it remains to verify that one has:

(*)    A_m · A_{−m} = A_0    for every m ∈ M.

One observes moreover by a direct computation that (*) is equivalent to saying that $P{\times }_{S}G\to P{\times }_{S}P$ is a closed immersion, and not only a monomorphism (under the hypothesis that $G$ acts freely), i.e. that the homomorphism on affine rings

θ : A ⊗_{A_0} A → A(M)

is surjective¹⁹. This gives 5.1 when one supposes that the equivalence relation defined by $G$ in $P$ is closed. We shall, however, show that this hypothesis is already a consequence of the fact that $G$ acts freely (which is moreover implicitly contained in Theorem 5.1, since $G{\times }_{S}P$ must be isomorphic to $P{\times }_{X}P$, which is closed in $P{\times }_{S}P$ since $X$ is affine over $S$ hence separated over $S$).

Let $R=P{\times }_{S}G$. The hypothesis that $G$ acts freely, i.e. that $R\to P{\times }_{S}P$ is a monomorphism, can be written as saying that the diagonal morphism

R → R ×_{(P ×_S P)} R = R₀

is an isomorphism. One has $R=\mathrm{Spec}(A(M))$ and

$$R_0=\mathrm{Spec}(A(M\times M)/K)$$

²⁰ where $K$ is the ideal generated by the elements of the form

x_m(e_{m,0} − e_{0,m}),    with m ∈ M, x_m ∈ A_m;

let $\varphi :A(M\times M)\to A(M)$ be the surjective ring homomorphism defined by

x e_{m,n} ↦ x e_{m+n}    (m, n ∈ M, x ∈ A)

(where the $e_m$, resp. $e_{m,n}=e_m\otimes e_n$, are the elements of the canonical basis of $A(M)$, resp. $A(M\times M)$). Then the diagonal morphism $R\to R_0$ corresponds to the homomorphism

φ̄ : A(M × M)/K → A(M)

obtained by passing to the quotient by $K$. Now the kernel of $\varphi$ is the ideal $K^{\prime }$ generated by the

$$d_m=e_{m,0}-e_{0,m}.$$

One has $K\subseteq K^{\prime }$, and the hypothesis that $G$ acts freely on $P$, i.e. that $\bar{\varphi }$ is an isomorphism, is equivalent to the equality $K^{\prime }=K$, which is expressed by the relations

(**)   d_m ∈ K = ∑_p A(M × M) A_p d_p,    for every m ∈ M.

Using the natural tri-grading of $A(M\times M)$, and the fact that the first degree of $d_m$ is zero, this means that one can write $d_m$ as a sum of elements of the form

f e_{r,s} (e_{p,0} − e_{0,p})    with f ∈ A_{−p} · A_p,

and using the fact that the total degree of $d_m$ is $m$, one can restrict to terms such that

r + s + p = m.

Hence one must have, for every $m\in M$, an expression:

(***)   { d_m = e_{m,0} − e_{0,m} = ∑_{r,s} λ_{r,s}(e_{m−s,s} − e_{r,m−r})
         with λ_{r,s} ∈ J_p = A_p · A_{−p} ⊆ A_0, p = m − (r + s).

One must conclude the relation (*), i.e. the relations

(****)   J_n = A_0    for every n ∈ M.

Now for this, it suffices to establish the same relation modulo every maximal ideal of A_0. As the hypotheses made are invariant under such a reduction, one may already suppose that A_0 is a field.

Lemma 5.2. Under the preceding conditions (with A_0 a field), if $M\ne 0$, there exists a $p\in M-0$ such that $J_p=A_0$.

Indeed, if this were not so, the sum in the right-hand side of (***) would be zero for every $m\in M$, which is absurd.

Corollary 5.3. Under the preceding conditions, but without supposing any longer that A_0 is a field, there exist finitely many elements $p_i\in M-0$ such that ${\sum }_i{J}_{p_i}=A_0$.

Indeed, one applies the result 5.2 to the situations deduced from $A/A_0$ by reduction modulo the maximal ideals of A_0.²¹

Corollary 5.4. Suppose again that A_0 is a field. Then for every subgroup $N$ of $M$ such that $N\ne M$, there exists a $p\in M-N$ such that $J_p=A_0$.

Indeed, let $M^{\prime }=M/N$, and consider the $M^{\prime }$-graded ring $A^{\prime }$, whose underlying ring is $A$, and whose grading is given by

A′_{m′} = ⨁_{m ∈ h^{−1}(m′)} A_m,

where $h:M\to M^{\prime }=M/N$ is the canonical homomorphism. Geometrically, this construction amounts to considering the subgroup $G^{\prime }=D_S(M^{\prime })$ of $G$, and the structure on $P$ of scheme with operator group $G^{\prime }$ induced by the action of $G$. It is then evident that $G^{\prime }$ acts freely on $P^{\prime }$, i.e. the pair $(M^{\prime },A^{\prime })$ satisfies the hypotheses of 5.3. One thus obtains

1 = ∑_i f_i g_i    with f_i ∈ A′_{m′_i}, g_i ∈ A′_{−m′_i} and m′_i ∈ M′ − {0},

whence at once the conclusion 5.4 by taking the components of the right-hand side along A_0, and using that A_0 is a field.

Let us now note that

J_p · J_q ⊆ J_{p+q}    and J_p = J_{−p},

so if $N$ denotes the set of $m\in M$ such that $J_p=A_0$, one sees that $N$ is a subgroup of $M$. Using 5.4 one sees that it is equal to $M$. This completes the proof of Theorem 5.1.

As we have pointed out in the course of the proof, Theorem 5.1 implies:

Corollary 5.5. Under the conditions of 5.1, the graph morphism

P ×_S G → P ×_S P

is a closed immersion.

One concludes at once:

Corollary 5.6. Let $\sigma$ be a section of $P$ over $S$. Then the morphism $g\mapsto \sigma \cdot g$ from $G$ to $P$ defined by $\sigma$ is a closed immersion.

Corollary 5.7. Let $G$, $H$ be two $S$-groups, with $G$ diagonalizable, $H$ affine over $S$, and let $u:G\to H$ be a homomorphism of $S$-groups that is a monomorphism. Then $u$ is a closed immersion, $H/G=X$ exists and $H$ is a principal homogeneous bundle over $X$ with group G_X; finally, $X$ is affine over $S$.

Corollary 5.8. Under the conditions of 5.1, if $P$ is of finite type (resp. of finite presentation) over $S$, the same holds for $X=P/G$.

Indeed, it follows from the hypothesis that the fibers of G_X are of finite type, hence G_X is of finite presentation over $X$ by 2.1 (b), hence $P$ being a torsor under G_X is of finite presentation over $X$²². Since it is also faithfully flat over $X$, our conclusion then follows from Exp. V, Prop. 9.1.

6. Essentially free morphisms, and representability of certain functors of the form ${\prod }_{Y/S}Z/Y$23

Definition 6.1. Let $f:X\to S$ be a morphism of preschemes. We say that $f$ is essentially free*, or also that $X$ is* essentially free over $S$, if one can find a covering of $S$ by affine opens $S_i$, for each $i$ an $S_i$-prescheme $S_i^{\prime }$ affine and faithfully flat over $S_i$, and a covering $(X_{ij}^{\prime }{)}_j$ of $X_i^{\prime }=X{\times }_S{S}_i^{\prime }$ by affine opens $X_{ij}^{\prime }$, such that for every $(i,j)$, the ring of $X_{ij}^{\prime }$ is a free module over the ring of $S_i^{\prime }$.²⁴

Proposition 6.2. a) If $X$ is essentially free over $S$, it is flat over $S$, the converse being true if $S$ is Artinian.

b) If $S$ is the spectrum of a field, every $S$-prescheme is essentially free over $S$.

c) If $X$ is essentially free over $S$, and if $S^{\prime }\to S$ is a base change morphism, then $X^{\prime }=X{\times }_S{S}^{\prime }$ is essentially free over $S^{\prime }$. The converse is true if $S^{\prime }\to S$ is faithfully flat and quasi-compact.

The proof is immediate, using for the converse in (a) the fact that a flat module over an Artinian local ring is free.²⁵

Proposition 6.3. Let $H$ be an $S$-prescheme in groups that is diagonalizable (more generally, that becomes diagonalizable by suitable faithfully flat quasi-compact extension of every affine open of $S$, i.e. $H$ is "of multiplicative type", cf. IX 1.1). Then $H$ is essentially free over $S$.

Indeed, if $H$ is diagonalizable, it is affine over $S$ and defined by an algebra which is a free O_S-module.

The introduction of Definition 6.1 is justified by the following:

Theorem 6.4. Let $S$ be a prescheme, $Z$ an essentially free $S$-prescheme, $Y$ a closed subprescheme of $Z$. Consider the following functor²⁶

F = ∏_{Z/S} Y/Z : (Sch)°/S → (Ens),
F(S′) = Γ(Y_{S′}/Z_{S′}) = { ∅           if Z_{S′} ≠ Y_{S′};
                             {id_{Z_{S′}}}  if Z_{S′} = Y_{S′}.

This functor is representable by a closed subprescheme of $S$.²⁷

²⁸ Let us first note that $F$ is a sheaf for the (fpqc) topology: since $F(S^{\prime })=\varnothing$ or {pt} for every $S^{\prime }$, this reduces to verifying that if $(S_i)$ is an open covering of $S$ (resp. $S^{\prime }\to S$ a faithfully flat quasi-compact morphism), and if each $Y_{S_i}\to Z_{S_i}$ (resp. if $Y_{S^{\prime }}\to Z_{S^{\prime }}$) is an isomorphism, the same holds for $Y\to Z$; now this is clear (resp. follows from SGA 1, VIII 5.4 or EGA IV₂, 2.7.1).

Moreover, by SGA 1, VIII 2.1 and 5.5, faithfully flat quasi-compact morphisms are of effective descent for the fibered category of closed immersion arrows. This allows us to restrict ourselves, with the notation of 6.1, to the case where $S=S_i^{\prime }$.

Let $(Z_j)$ be a covering of $Z$ by affine opens such that $O(Z_j)$ is a free module over $A=O(S)$, and let $Y_j=Y\cap Z_j$ and $F_j:(Sch{)}^{\circ }/S\to (Ens)$ be the functor defined in terms of $(Z_j,Y_j)$ as $F$ in terms of $(Z,Y)$. It is a subfunctor of the final functor, and one evidently has $F={\bigcap }_j{F}_j$, which reduces us to

proving that each $F_j$ is representable by a closed subscheme $T_j$ of $S$ (for then $F$ will be representable by the closed subscheme $T$ intersection of the $T_j$). One may therefore suppose $Z$ also affine, $Z=\mathrm{Spec}(B)$, where $B$ is a free $A$-module. Let $J$ be a subset of $B$ defining the subscheme $Y$ of $Z$, and let $K$ be the ideal in $A$ generated by the $u_i(J)\subseteq A$, where the $u_i:B\to A$ are the coordinate forms with respect to the chosen basis. One observes at once that $T=V(K)=\mathrm{Spec}(A/K)$ satisfies the desired condition, which completes the proof.

Examples 6.5. Let us give some important examples of functors that reduce to functors ${\prod }_{Z/S}Y/Z$ of the type envisaged in 6.4 and for which it is useful in the sequel to have criteria of representability. We denote by $S$ a prescheme, by $X$, $Y$, $Z$, etc., preschemes over $S$.

a) Suppose given an $S$-morphism

(*)    q : X → Hom_S(Y, Z),

("$X$ acts on $Y$ with values in $Z$"), i.e. a morphism

(**)   r : X ×_S Y → Z.

Consider a subprescheme $Z^{\prime }$ of $Z$, whence a monomorphism

Hom_S(Y, Z′) → Hom_S(Y, Z)

which makes the first functor a subfunctor of the second; let $X^{\prime }$ be the inverse image of this subfunctor by (*), that is the subfunctor of $X$ such that $X^{\prime }(T)$ is the set of $x\in X(T)$ such that $q(x):Y_T\to Z_T$ factors through $Z_T^{\prime }$. This functor $X^{\prime }$ can be described as follows: set $P=X{\times }_{S}Y$, let $P^{\prime }$ be the inverse image of $Z^{\prime }$ by $r:P\to Z$; then one has an evident isomorphism

$$(\ast \ast \ast )X^{\prime }\simeq \prod _{P/X}{P}^{\prime }/P.$$

One thus obtains: if $Y$ is essentially free over $S$ and $Z^{\prime }$ closed in $Z$, the subfunctor $X^{\prime }$ of $X$ is representable by a closed subprescheme of $X$.

b) Suppose given two ways of making $X$ act on $Y$ with values in $Z$, i.e. two morphisms

q_1, q_2 : X ⇒ Hom_S(Y, Z),

and set $X^{\prime }=Ker(q_1,q_2)$: this is the subfunctor of $X$ such that $X^{\prime }(T)$ is the set of $x\in X(T)$ such that the two morphisms $q_1(x),q_2(x):Y_T\Rightarrow Z_T$ are equal. Now giving $q_1,q_2$ is equivalent to giving a morphism

q : X → Hom_S(Y, Z ×_S Z),

or, again, a morphism $r:X{\times }_{S}Y\to Z{\times }_{S}Z$; setting $U=Z{\times }_{S}Z$, let $U^{\prime }$ be the diagonal subprescheme of $Z{\times }_{S}Z$; then $X^{\prime }$ is none other than the inverse image of the subfunctor ${\mathrm{Hom}}_S(Y,U^{\prime })\to {\mathrm{Hom}}_S(Y,U)$ by $q$, hence can be put in the form (***), with $P=X{\times }_{S}Y$ and $P^{\prime }$ = inverse image of the diagonal by $r$, i.e. kernel of

X ×_S Y ⇒ Z.

One is thus in the conditions of (a). One sees consequently that: if $Y$ is essentially free over $S$ and $Z$ separated over $S$, the subfunctor $X^{\prime }$ of $X$ is representable by a closed subprescheme of $X$.

c) Suppose given a morphism

q : X → Hom_S(Y, Y),

i.e. "$X$ acts on $Y$". Let $X^{\prime }$ be the "kernel" of this morphism, i.e. the subfunctor $X^{\prime }$ of $X$ such that $X^{\prime }(T)$ is the set of $x\in X(T)$ such that $q(x):Y_T\to Y_T$ is the identity. This functor is amenable to (b), as one sees by introducing a second homomorphism

q′ : X → Hom_S(Y, Y)

"by making $X$ act trivially on $Y$". Hence: if $Y$ is essentially free over $S$ and separated over $S$, the kernel subfunctor of $q$ is representable by a closed subprescheme of $X$.

d) Under the conditions of (c), consider the subfunctor $Y^{\prime }$ of $Y$ "of invariants under $X$", so $Y^{\prime }(T)$ is the set of $y\in Y(T)$ such that the corresponding morphism $q(y):X_T\to Y_T$ is "the $T$-constant morphism with value $y$". Introducing $q^{\prime }$ as in (c), and the homomorphisms corresponding to $q$ and $q^{\prime }$:

q, q′ : Y ⇒ Hom_S(X, Y),

one sees that $Y^{\prime }$ is precisely $Ker(q,q^{\prime })$, and is therefore amenable again to (b) (with the roles of $X$, $Y$ reversed and $Z=Y$).

Consequently, if $X$ is essentially free over $S$, $Y$ separated over $S$, then the subfunctor $Y^{\prime }$ of $Y$ of invariants under $X$ is representable by a closed subprescheme of $Y$.

e) Constructions of the type explained in the preceding examples are especially frequent in group theory. Thus, when $G$ is an $S$-prescheme in groups acting on the $S$-prescheme $X$:

$$q:G\to {\mathrm{Aut}}_S(X),$$

the kernel of $q$ ("the subgroup of $G$ acting trivially") is a closed subscheme of $G$ provided that $X$ is essentially free and separated over $S$ (example (c)), and the subobject $X^G$ of invariants is a closed subprescheme of $X$, provided that $G$ is essentially free over $S$ and $X$ separated over $S$²⁹ (example (d)).

Let $Y$, $Z$ be subpreschemes of $X$; consider the subfunctor $Trans{p}_G(Y,Z)$ of $G$ ("transporter of $Y$ into $Z$") whose points with values in a $T$ over $S$ are the $g\in G(T)$ such that the corresponding automorphism of X_T satisfies $g(Y_T)\subseteq Z_T$ i.e. induces a morphism $Y_T\to X_T$ factoring through $Y_T\to Z_T$. Hence: if $Y$ is essentially free over $S$, and $Z$ closed in $X$, then $Trans{p}_G(Y,Z)$ is a closed subprescheme of $G$ (example (a)).

One may also consider the strict transporter of $Y$ into $Z$,³⁰ whose points with values in a $T$ over $S$ are the $g\in G(T)$ such that $g(Y_T)=Z_T$, which is nothing but $Trans{p}_G(Y,Z)\cap \sigma (Trans{p}_G(Z,Y))$, where $\sigma$ is the symmetry of $G$. Consequently, if $Y$ and $Z$ are essentially free over $S$ and closed in $X$, the strict transporter of $Y$ into $Z$ is a closed subprescheme of $G$.

An important case is that where $X=G$, with $G$ acting on itself by inner automorphisms. If $H$ is a subprescheme of $G$, the strict transporter of $H$ into $H$ is also called the normalizer of $H$ in $G$, and denoted $Nor{m}_{G}H$. Hence: if $H$ is a closed subprescheme of $G$ in groups, essentially free over $S$, then $Nor{m}_{G}H$ is representable by a closed subprescheme of $G$ in groups.

Let finally $Z$ be a subprescheme of $G$; then its centralizer $Cent{r}_G(Z)$ in $G$ is the subfunctor of $G$ in groups defined by the procedure of (d), considering "$Z$ acts on $G$" via the action induced by that of $G$; hence if $Z$ is essentially free over $S$ and $G$ separated over $S$, $Cent{r}_G(Z)$ is a closed subprescheme of $G$ in groups.

In particular, if $G$ is essentially free and separated over $S$, then the center $C$ of $G$, which is none other than $Cent{r}_G(G)$, is a closed subprescheme of $G$ in groups.

When $S$ is the spectrum of a field, 6.3 (b) shows that in examples (a) to (e) above, the conditions "essentially free" are automatically satisfied; only separation conditions remain. Recalling that a prescheme in groups over a field is necessarily separated, one finds for example:

Corollary 6.7.³¹ Let $G$ be a prescheme in groups over a field $k$. Then:

– For every subprescheme $Z$ of $G$, the centralizer of $Z$ in $G$ is a closed subprescheme of $G$ in groups; this is in particular the case for the center $Cent{r}_G(G)$ of $G$.

– More generally, if $u,v:X\to G$ are morphisms of preschemes, $Trans{p}_G(u,v)$ is representable by a closed subprescheme of $G$.

– For two subpreschemes Y, Z of $G$, with $Z$ closed, $Trans{p}_G(Y,Z)$ is a closed subprescheme of $G$. If $Y$ is also closed, the same conclusion holds for $Trans{p}_G^{str}(Y,Z)$.

– For every subprescheme in groups³² $H$ of $G$, the normalizer $Nor{m}_G(H)$ is a closed subscheme of $G$ in groups.

Remark 6.8.³³ Let $A$ be a commutative ring, $M$ an $A$-module, $M^{\vee }={\mathrm{Hom}}_A(M,A)$; endow the $A$-module ${\mathrm{End}}_A(M)$ with the topology of pointwise convergence (discrete), i.e. a basis of neighborhoods of 0 is formed by the following $A$-submodules, where $n\in \mathbb{N}$ and $m_1,\cdots ,m_n\in M$:

K(m_1, …, m_n) = { u ∈ End_A(M) | u(m_i) = 0  for i = 1, …, n }.

We say that $M$ is a quasi-free $A$-module if the image of the canonical morphism $\Theta :M{\otimes }_A{M}^{\vee }\to {\mathrm{End}}_A(M)$ contains $i{d}_M$ in its closure, i.e. if the following condition holds:

(*)  { for all m_1, …, m_n ∈ M, there exist x_1, …, x_r ∈ M and f_1, …, f_r ∈ M^∨
     { such that m_i = Θ(∑_{s=1}^r x_s ⊗ f_s)(m_i) = ∑_{s=1}^r f_s(m_i) x_s for i = 1, …, n.

(In this case, $Im\Theta$ is dense in ${\mathrm{End}}_A(M)$, since for every $u\in {\mathrm{End}}_A(M)$ one has

u(m_i) = ∑_{s=1}^r f_s(m_i) u(x_s) = Θ(∑_{s=1}^r u(x_s) ⊗ f_s)(m_i).)

Let us note first that this property is stable under base change. Indeed, let $\varphi :A\to A^{\prime }$ be a morphism of rings, $M^{\prime }=M{\otimes }_A{A}^{\prime }$, and $m_1^{\prime },\cdots ,m_n^{\prime }\in M^{\prime }$. Then $m_i^{\prime }={\sum }_j{m}_{ij}\otimes b_{ij}$ ($m_{ij}\in M$, $b_{ij}\in A^{\prime }$); by hypothesis, there exist $x_1,\cdots ,x_r\in M$ and $f_1,\cdots ,f_r\in {\mathrm{Hom}}_A(M,A)$ such that $m_{ij}={\sum }_s{x}_s{f}_s(m_{ij})$ for all i, j. Denote by $\varphi \circ f_s$ the image of $f_s$ in ${\mathrm{Hom}}_A(M,A^{\prime })={\mathrm{Hom}}_{A^{\prime }}(M^{\prime },A^{\prime })$; then for every $i=1,\cdots ,n$ one has:

(∑_s x_s ⊗ φ ∘ f_s)(m′_i) = ∑_{s,j} x_s ⊗ φ(f_s(m_{ij})) b_{ij} = ∑_j (∑_s x_s f_s(m_{ij})) ⊗ b_{ij} = m′_i,

which proves that $M^{\prime }$ is quasi-free over $A^{\prime }$.

Let us also note that every projective $A$-module $P$ is quasi-free (there exist $A$-morphisms $A^{(I)}-\pi \to P-\tau \to A^{(I)}$ such that $\pi \circ \tau =i{d}_P$; denote by $(e_i)$ the canonical basis of $A^{(I)}$ and $f_i$ the linear form $e_i^{\ast }\circ \tau$ on $P$; if $m_1,\cdots ,m_n\in P$, there exists a finite subset $J$ of $I$ such that $m_k={\sum }_{i\in J}{f}_i(m_k)\pi (e_i)$ for $k=1,\cdots ,n$).

Then, Theorem 6.4 remains valid if in the statement of Definition 6.1 the word "free" is replaced by "projective" or, more generally, by "quasi-free". Indeed, proceeding as in the proof of 6.4, one is reduced to proving:

Lemma 6.8.1. Let $M$ be a quasi-free $A$-module, $N$ a submodule, $F$ the covariant functor $(A-algebras)\to (Ens)$ such that $F(B)=pt$ if $M_B=(M/N{)}_B$, and $F(B)=\varnothing$ otherwise. Then there exists an ideal $K$ of $A$ such that $F(B)=pt$ if and only if the morphism $A\to B$ factors through $A/K$, i.e. one has a functorial isomorphism in $B$:

$$F(B)\simeq {\mathrm{Hom}}_{A-alg.}(A/K,B).$$

Proof. Let $(n_j)$ be a system of generators of $N$, and let $F_j$ be the subfunctor of the final functor $e$ ($e(B)=pt$ for every $B$) corresponding to the submodule $A{n}_j$. One has $F(B)=pt$ if and only if the image of each $n_j$ in M_B is zero, hence $F$ is the intersection of the functors $F_j$. This reduces us to the case where $N$ is generated by one element $n$.

Let $\varphi :A\to B$ be an $A$-algebra; if the image $n\otimes 1$ of $n$ in M_B is zero, then for every $f\in M^{\vee }={\mathrm{Hom}}_A(M,A)$ one has $0=f(n)\otimes 1=\varphi (f(n))$. On the other hand, since $M$ is quasi-free, there exist $x_1,\cdots ,x_r\in M$ and $f_1,\cdots ,f_r\in M^{\vee }$ such that $n={\sum }_s{f}_s(n)x_s$, whence $n\otimes 1={\sum }_s{x}_s\otimes \varphi (f_s(n))$. It follows that $n\otimes 1=0$ if and only if $\varphi$ factors through $A/K$, where $K$ is the ideal generated by the $f_s(n)$ for $s=1,\cdots ,r$. This proves the lemma.