Exposé I. Algebraic structures. Group cohomology

by M. Demazure

¹ This Exposé consists of two parts; the first gathers a certain number of general definitions and sets up notations that will often be used in the sequel, while the second treats group cohomology and culminates in Theorem 5.3.3 (vanishing of the cohomology of diagonalizable groups).

We choose once and for all a Universe.² All the definitions stated and all the constructions carried out will be relative to this Universe. We shall systematically allow ourselves the following abuse of language: in order to define a functor $f:C\to C^{\prime }$, we shall content ourselves with defining the object $f(S)$ of $C^{\prime }$ for every object $S$ of $C$, each time that there is no ambiguity about the way to define $f(h)$ for an arrow $h$ of $C$. In practice, we shall say: let $f:C\to C^{\prime }$ be the functor defined by $f(S)=\cdots$.

1. Generalities

1.1.

Let $C$ be a category. We shall denote by Ĉ the category $\mathrm{Hom}(C^{\circ },(Ens))$ of contravariant functors from $C$ to the category (Ens) of sets.³ There exists a canonical functor $h:C\to \hat{C}$ which associates to every $X\in Ob(C)$ the functor hX such that

$$hX(S)=\mathrm{Hom}(S,X).$$

For every functor $F\in Ob(\hat{C})$, one defines (cf. for example EGA 0_III, 8.1.4) a bijection

$$\mathrm{Hom}(hX,F)\stackrel{\sim }{\to }F(X).$$

⁴

In particular, for every pair X, Y of objects of $C$, the following canonical map is bijective:

Hom_C(X, Y) ⥲ Hom_Ĉ(hX, hY);

i.e. the functor $h$ is fully faithful. It thus defines an isomorphism of $C$ onto a full subcategory of Ĉ, and an equivalence of $C$ with the full subcategory of Ĉ formed by the representable functors (i.e. those isomorphic to a functor of the form hX). In the sequel, we shall often identify $X$ and hX. The following numbers aim to show that this identification can be made without danger.

Remark 1.1.1.⁵ We sometimes need the following variant. Let $D$ be a full subcategory of $C$ and let $X,Y\in Ob(D)$; denote by $h_X^{\prime }$ and $h_Y^{\prime }$ the restrictions to $D$ of hX and hY. Then one has

Hom_C(X, Y) = Hom_D(X, Y) = Hom_D̂(h′_X, h′_Y)

and therefore: to give a morphism $X\to Y$ "is the same thing" as to give, functorially in $T$, a map $\varphi (T):\mathrm{Hom}(T,X)\to \mathrm{Hom}(T,Y)$, for every $T\in Ob(D)$.

1.2.

⁶ We shall say that $F$ is a subobject (or a subfunctor) of $G$ if $F(S)$ is a subset of $G(S)$ for each $S$.

In Ĉ, "arbitrary" inverse limits exist and are computed by:

(lim←_i F_i)(S) = lim←_i F_i(S).

⁷ In particular, fiber products are defined by:

(F ×_G F′)(S) = F(S) ×_{G(S)} F′(S).

⁸

We shall choose as final object of Ĉ the functor $e$ such that $e(S)=\varnothing$⁹. Every $F\in Ob(\hat{C})$ has a unique morphism into $e$, and one sets

F × F′ = F ×_e F′.

The functor $h$ commutes with inverse limits; in particular, for $X\times X^{\prime }$ to exist ($X,X^{\prime }\in Ob(C)$), resp. for $C$ to admit a final object $e$, it is necessary and sufficient that $hX\times h{X}^{\prime }$ be representable, resp. that $e$ be representable, and one has

hX × hX′ ≃ h_{X×X′}    and    he ≃ e.

A monomorphism of Ĉ is nothing other than a morphism $F\to G$ such that, for every $S\in Ob(C)$, the corresponding map of sets $F(S)\to G(S)$ is injective.¹⁰

The functor $\Gamma$. For every $F\in Ob(\hat{C})$, one sets

$$\Gamma (F)=\mathrm{Hom}(e,F);$$

an element of $\Gamma (F)$ is therefore a family $({\gamma }_S{)}_{S\in Ob(C)}$, ${\gamma }_S\in F(S)$, such that for every arrow $f:S^{\prime }\to S^{\prime \prime }$ of $C$ one has $F(f)({\gamma }_{S^{\prime \prime }})={\gamma }_{S^{\prime }}$.

One sets $\Gamma (X)=\Gamma (hX)$ for $X\in Ob(C)$. If $C$ has a final object $e$, one therefore has an isomorphism $\Gamma (X)\simeq \mathrm{Hom}(e,X)$.

1.3.

Let $S\in Ob(C)$. We denote by $C/S$ the category of objects of $C$ over $S$, i.e. the category whose objects are the arrows $f:T\to S$ of $C$, with $\mathrm{Hom}(f,f^{\prime })$ being the subset of $\mathrm{Hom}(T,T^{\prime })$ formed by those $u$ such that $f=f^{\prime }\circ u$. If $C$ has a final object $e$, then $C/e$ is isomorphic to $C$. The category $C/S$ has a final object: the identity arrow $S\to S$.

If $f:T\to S$ is an object of $C/S$, then one can form the category $(C/S)/f$, which by abuse of language one denotes $(C/S)/T$, and one has a canonical isomorphism

$$C/T\simeq (C/S)/T.$$

This construction also applies to the category Ĉ; one defines in particular the category $\hat{C}/hS$. On the other hand, one can form the category ${\hat{C}}_{/S}$.

If $f:T\to S$ is an object of $C/S$, then $\Gamma (f)$ is identified with the set $\Gamma (T/S)$ of sections of $T$ over $S$, that is, of arrows $S\to T$ which are right inverses of $f$. Note that $hf:hT\to hS$ is then an object of $\hat{C}/hS$, and one has:

Γ(hf) ≃ Γ(hT/hS) ≃ Γ(T/S) ≃ Γ(f).

1.4.

We now propose to define an equivalence of categories ${\hat{C}}_{/S}$ and $\hat{C}/hS$, that is, to prove that "to give a functor on the category of objects of $C$ over $S$ is the same thing as to give a functor on $C$ endowed with a morphism into hS".

(i) Construction of $\alpha S:\hat{C}/hS\to {\hat{C}}_{/S}$.

Let first $H:F\to hS$ be an object of $\hat{C}/hS$. We must define a functor $\alpha S(H)$ on $C/S$. Let first $f:T\to S$ be an object of $C/S$; we define $\alpha S(H)(f)$ as the inverse image of $f\in hS(T)$ by the map $H(T):F(T)\to hS(T)$.¹¹

Next, let $u:f\to f^{\prime }$ be an arrow of $C/S$; then $F(u):F(T^{\prime })\to F(T)$ induces a map from $\alpha S(H)(f^{\prime })$ into $\alpha S(H)(f)$, which we denote $\alpha S(H)(u)$. One verifies at once that the maps

f ↦ αS(H)(f)    and    u ↦ αS(H)(u)

do define a functor on $C/S$, hence an object $\alpha S(H)$ of ${\hat{C}}_{/S}$.

Finally, let $H:F\to hS$ and $H^{\prime }:F^{\prime }\to hS$ be two objects of $\hat{C}/hS$ and $U:H\to H^{\prime }$ a morphism of $\hat{C}/hS$:

        U
   F ──────→ F′
    ╲       ╱
   H ╲     ╱ H′
      ╲   ╱
       hS

Then for every $f:T\to S$, the map $U(T):F(T)\to F^{\prime }(T)$ induces a map

$$\alpha S(U)(f):\alpha S(H)(f)\to \alpha S(H^{\prime })(f),$$

which defines a morphism of functors

$$\alpha S(U):\alpha S(H)\to \alpha S(H^{\prime }).$$

One verifies easily that the maps

H ↦ αS(H)    and    U ↦ αS(U)

do define a functor $\alpha S:\hat{C}/hS\to {\hat{C}}_{/S}$.

(ii) Proposition 1.4.1. The functor $\alpha S:\hat{C}/hS\to {\hat{C}}_{/S}$ is an equivalence of categories.

We only indicate the principle of the construction of a quasi-inverse functor $\beta S:{\hat{C}}_{/S}\to \hat{C}/hS$. Let $G$ be a functor on $C/S$; for every object $T$ of $C$, one sets

βS(G)(T) = sum of the sets G(f) for f ∈ Hom(T, S) = hS(T),

which defines a functor $\beta S(G)$ on $C$, equipped with an obvious projection onto hS.

1.5.

The equivalence $\alpha S$ commutes with the functors $\Gamma$. In other words, if $H:F\to hS$ is an object of $\hat{C}/hS$ and $\alpha S(H)$ the corresponding object of ${\hat{C}}_{/S}$, one has

$$\Gamma (\alpha S(H))\simeq \Gamma (H)\simeq \Gamma (F/hS).$$

The equivalence $\alpha S$ commutes with the functors $h$, i.e. if $f:T\to S$ is an object of $C/S$, then $hf:hT\to hS$ is an object of $\hat{C}/hS$ whose transform by $\alpha S$ is nothing other than $h_{C/S}(f)$, where

$$h_{C/S}:C/S\to {\hat{C}}_{/S}$$

is the canonical functor.¹² As a consequence:

Proposition 1.5.1. Let $H:F\to hS$ be an object of $\hat{C}/hS$. For $\alpha S(H):(C/S{)}^{\circ }\to (Ens)$ to be representable, it is necessary and sufficient that $F:C^{\circ }\to (Ens)$ be representable; if $F\simeq hT$, then $\alpha S(H)$ is representable by the object $T\to S$ of $C/S$.

The equivalence $\alpha S$ is transitive in $S$: if $f:T\to S$ is an object of $C/S$, one has a commutative diagram of equivalences

                α_{S/hT}                    α_f
   (Ĉ/hS)/hT  ─────────→ (Ĉ_{/S})/h_T  ─────────→  Ĉ_{(C/S)/T}
        ╲                                              ╱
      ≃  ╲                                            ╱  ≃
          ╲                                          ╱
           ────────────→ Ĉ/hT  ──────────────→ Ĉ_{/T}
                                  α_T

where ${\alpha }_{S/hT}$ denotes (provisionally) the restriction (cf. 1.6) of the functor $\alpha S$ to objects above hT.

1.6. Base change in a functor

For every $S\in Ob(C)$, one has a canonical functor

$$iS:C/S\to C$$

defined by $iS(f)=T$ if $f$ is the arrow $T\to S$. If $f:T\to S$ is an object of $C/S$, one denotes by $i_{T/S}=i_f$ the functor:

$$i_{T/S}:(C/S)/T\to C/S,$$

and one has the commutative diagram:

                i_{T/S}       iS
   (C/S)/T  ─────────→ C/S  ─────→ C
        ╲                          ╱
      ≃  ╲                        ╱ iT
          ╲                      ╱
           C/T

that is, identifying $(C/S)/T$ with $C/T$ as we shall do henceforth,

$$iS\circ i_{T/S}=iT.$$

In the same way, if one identifies $C$ and $C/e$ when $C$ has a final object $e$, then $i_{S/e}:C/S\to C/e$ is identified with iS.

For $X\in Ob(C)$ (resp. $Y\in Ob(C/S)$), let $pS(X)$ (resp. $p_{T/S}(Y)$) denote the object of $C/S$ (resp. of $C/T$), when it exists, defined by $X\times S$ (resp. $Y{\times }_{S}T$) equipped with its second projection:

   X × S                  Y ×_S T
     │                        │
   pS(X) ↓     resp.     p_{T/S}(Y) ↓
     │                        │
     S                        T

The (partially defined) functor pS (resp. $p_{T/S}$) is called the base change functor. It is by definition of the product (resp. of the fiber product) the right adjoint of the functor iS (resp. $i_{T/S}$).¹³ One also denotes

pS(X) = XS    and    p_{T/S}(Y) = YT.

The functor iS defines a functor (restriction)

$$i{\ast }_S:\hat{C}\to {\hat{C}}_{/S};$$

one denotes $FS=i{\ast }_S(F)=F\circ iS$. One has obviously

$$i{\ast }_{T/S}\circ i{\ast }_S=i{\ast }_T,$$

that is, for every functor $F\in Ob(\hat{C})$,

(FS)T = FT.

The notation requires a justification, which is the following:

Proposition 1.6.1. For the functor $(hX)S:(C/S{)}^{\circ }\to (Ens)$ to be representable, it is necessary and sufficient that the product $X\times S$ exist. One then has

$$(hX)S\simeq h_{XS}.$$

This shows that FS has two interpretations: restriction of the functor $F$ to $C/S$, and functor obtained by base change $e\leftarrow S$. This leads to the following notation:

   F        FS          FT

   │         │           │
   e ←────── S ←──────── T

which renders both of the preceding interpretations. Note that one has

Γ(FS) ≃ Hom(hS, F) ≃ F(S),

in particular

$$\Gamma (XS)\simeq \mathrm{Hom}(S,X).$$

1.7.0.

¹⁴ Let $E$ be an object of Ĉ. Consider the category C_E of objects of $C$ above $E$: its objects are the pairs $(V,\rho )$ formed by an object $V$ of $C$ and a Ĉ-morphism $\rho :hV\to E$, i.e. $\rho \in E(V)$; a morphism from $(V,\rho )$ to $(V^{\prime },{\rho }^{\prime })$ is

the datum of a $C$-morphism $f:V\to V^{\prime }$ such that ${\rho }^{\prime }\circ f=\rho$ (i.e. $E(f)({\rho }^{\prime })=\rho$). Denote by $L$ the functor

$$\lim {\to }_{(V,\rho )\in C_E}hV,$$

i.e. for every $S\in Ob(C)$, $L(S)=\lim {\to }_{(V,\rho )}hV(S)$ is the set of equivalence classes of triples $(V,\rho ,v)$, where $v:S\to V$ is a $C$-morphism, and where one identifies $(V,\rho ,v)$ with $(V^{\prime },{\rho }^{\prime },f\circ v)$ for every $C$-morphism $f:V\to V^{\prime }$ such that ${\rho }^{\prime }\circ f=\rho$.

Then the map ${\varphi }_E(S)$ which to the class of $(V,\rho ,v)$ associates the element $\rho \circ v$ of $E(S)$ is well defined, and defines a morphism of functors

φ_E :  lim→_{(V,ρ)∈C_E} hV → E.

Lemma. ${\varphi }_E$ is an isomorphism.

Indeed, let $S\in Ob(C)$. Every $x\in E(S)$ is the image under ${\varphi }_E(S)$ of the triple $(S,x,i{d}_S)$; this shows that ${\varphi }_E(S)$ is surjective. On the other hand, let ${\ell }_1=(V_1,{\rho }_1,v_1)$ and ${\ell }_2=(V_2,{\rho }_2,v_2)$ be two elements of $L(S)$ having the same image in $E(S)$; set $\gamma ={\rho }_1\circ v_1={\rho }_2\circ v_2$. Then ${\ell }_1$ and ${\ell }_2$ are both equal, in $L(S)$, to the class of the triple $(S,\gamma ,i{d}_S)$. This shows that ${\varphi }_E(S)$ is injective.

Corollary. For every object $F$ of Ĉ, one has

Hom(E, F) = lim←_{(V,ρ)∈C_E} F(V).

1.7. Objects Hom, Isom, etc.

Let $F$ and $G$ be two objects of Ĉ. We shall define another object of Ĉ in the following way:

Hom(F, G)(S) = Hom_{Ĉ_{/S}}(FS, GS) ≃ Hom_{Ĉ/hS}(F × hS, G × hS) ≃ Hom_Ĉ(F × hS, G).

The object $\mathrm{Hom}(F,G)$ defined above has the following properties:

(i) $\mathrm{Hom}(e,G)\simeq G$.¹⁵

(ii) The formation of Hom commutes with base extension:

Hom(FS, GS) ≃ Hom(F, G)_S.

(iii) $(F,G)\mapsto \mathrm{Hom}(F,G)$ is a bifunctor, contravariant in $F$ and covariant in $G$.

These three properties are obvious from the definitions.

We shall show that, for every object $E$ of Ĉ, one has

Hom(E, Hom(F, G)) ≅ Hom(E × F, G).

Let $\varphi :E\times F\to G$; we must associate to it a morphism of $E$ into $\mathrm{Hom}(F,G)$. So let $S^{\prime }\to S$ be an arrow of $C$. One has maps

E(S) × F(S′) → E(S′) × F(S′) ──φ(S′)──→ G(S′).

Every element $e$ of $E(S)$ therefore defines, for every $S^{\prime }\to S$, a map $F(S^{\prime })\to G(S^{\prime })$ functorial in $S^{\prime }$, i.e. an element ${\theta }_{\varphi }(e)$ of $\mathrm{Hom}(F,G)(S)$. One has thus obtained a map

φ ↦ θ_φ,    Hom(E × F, G) → Hom(E, Hom(F, G)),

which is "functorial in $E$".

Proposition 1.7.1.¹⁶ Let $E,F,G\in Ob(\hat{C})$.

(a) The map $\varphi \mapsto {\theta }_{\varphi }$ is a bijection:

Hom(E × F, G) ⥲ Hom(E, Hom(F, G)).

(b) Moreover, one has an isomorphism of functors:

Hom(E, Hom(F, G)) ≃ Hom(E × F, G).

(a) Consider both members as functors in $E$. The asserted result is true if $E=hX$; indeed, in that case it is nothing other than the definition of the functor $\mathrm{Hom}(F,G)$. On the other hand, both members as functors in $E$ transform direct limits into inverse limits. Finally, by Lemma 1.7.0, every object $E$ of Ĉ is isomorphic to the direct limit of the hX, where $X$ runs over the category C_E. This proves (a).

¹⁷ Let us sketch a direct proof of (a). To every $\theta \in \mathrm{Hom}(E,\mathrm{Hom}(F,G))$, one associates the element ${\varphi }_{\theta }$ of $\mathrm{Hom}(E\times F,G)$ defined as follows. For every $S\in Ob(C)$, one has a map

θ(S) : E(S) → Hom(F × S, G),

functorial in $S$. If $(e,f)\in E(S)\times F(S)$, then $f$ is a morphism $S\to F$, hence $f\times i{d}_S$ is a morphism $S\to F\times S$; on the other hand, $\theta (S)(e)$ is a morphism $F\times S\to G$, so by composition one obtains a morphism:

θ(S)(e) ∘ (f × id_S) : S → G,

i.e. an element ${\varphi }_{\theta }(S)(e,f)$ of $G(S)$. One verifies easily that the correspondence $S\mapsto {\varphi }_{\theta }(S)$ is functorial in $S$, hence defines a morphism ${\varphi }_{\theta }$ from $E\times F$ to $G$. We leave to the reader the verification that the maps $\theta \mapsto {\varphi }_{\theta }$ and $\varphi \mapsto {\theta }_{\varphi }$ are mutually inverse bijections.

Let us prove (b). If $S\in Ob(C)$, one has, by 1.7 (ii) and (a) applied to $C/S$:

Hom(E, Hom(F, G))(S) ≃ Hom_S(ES, Hom_S(FS, GS))
                     ≅ Hom_S(ES ×_S FS, GS)
                     ≅ Hom(E × F × S, G)
                     ≅ Hom(E × F, G)(S)

and these isomorphisms are functorial in $S$.

Corollary 1.7.2. One has:

Hom(E, Hom(F, G)) ≃ Hom(F, Hom(E, G)),
Hom(E, Hom(F, G)) ≃ Hom(F, Hom(E, G)).

In particular, taking $E=e$, and taking into account $\mathrm{Hom}(e,G)\simeq G$, one has

Γ(Hom(F, G)) ≃ Hom(F, G).

Note that the composition of Hom's provides functorial morphisms

Hom(F, G) × Hom(G, H) → Hom(F, H).

If $F$ and $G$ are two objects of Ĉ, one denotes by $Isom(F,G)$ the subset of $\mathrm{Hom}(F,G)$ formed by the isomorphisms of $F$ onto $G$. One then defines a subobject $Isom(F,G)$ of $\mathrm{Hom}(F,G)$ by:

Isom(F, G)(S) = Isom(FS, GS).

One then has isomorphisms

Γ(Isom(F, G)) ≃ Isom(F, G),
Isom(F, G) ≃ Isom(G, F).

In the particular case where $F=G$, one sets

End(F) = Hom(F, F),    End(F) = Hom(F, F) ≃ Γ(End(F)),
Aut(F) = Isom(F, F),    Aut(F) = Isom(F, F) ≃ Γ(Aut(F)).

The formation of the objects Hom, Isom, Aut, End commutes with base change.

Remark 1.7.3.¹⁸ One can construct an object isomorphic to $Isom(F,G)$ in the following manner: one has a morphism

Hom(F, G) × Hom(G, F) → End(F);

permuting $F$ and $G$, one deduces a morphism

Hom(F, G) × Hom(G, F) → End(F) × End(G).

On the other hand, the identity morphism of $F$ is an element of $\mathrm{End}(F)$ and therefore defines a morphism $e\to \mathrm{End}(F)$. Doing the same with $G$ and forming the product, one finds a morphism

$$e\to \mathrm{End}(F)\times \mathrm{End}(G).$$

It is then immediate that the fiber product of $e$ and of $\mathrm{Hom}(F,G)\times \mathrm{Hom}(G,F)$ over $\mathrm{End}(F)\times \mathrm{End}(G)$ is isomorphic to $Isom(F,G)$.

All these definitions apply in particular to the case where $F=hX,G=hY$. In the case where $\mathrm{Hom}(hX,hY)$ is representable by an object of $C$, one denotes this object by $\mathrm{Hom}(X,Y)$. It has the following property: if $Z\times X$ exists, then

Hom(Z, Hom(X, Y)) ≃ Hom(Z × X, Y).

This property characterizes it when the products exist in $C$.

One defines similarly (when they happen to exist) objects

Isom(X, Y),    End(X),    Aut(X);

we simply note that, by the construction given above, $Isom(X,Y)$ exists whenever fiber products exist in $C$ and Hom(X, Y), Hom(Y, X), End(X) and $\mathrm{End}(Y)$ exist.

All that precedes applies equally to categories of the form $C/S$. The corresponding objects will be denoted as explicitly as possible by appropriate symbols: for example, if $T$ and $T^{\prime }$ are two objects of $C$ above $S$, one will denote by ${\mathrm{Hom}}_S(T,T^{\prime })$ the object ${\mathrm{Hom}}_{C/S}(T/S,T^{\prime }/S)$.

1.8. Constant objects

Let $C$ be a category in which direct sums and fiber products exist, and in which direct sums commute with base change (for example, the category of schemes¹⁹). For every set $E$ and every object $S$ of $C$, we set

(1.8.1)    ES =  the direct sum of a family (S_i)_{i∈E}
                 of objects of C all isomorphic to S.

This object is characterized by the formula:

(1.8.2)    Hom_C(ES, T) = Hom(E, Hom_C(S, T)),

for every $T\in Ob(C)$, where the second Hom is taken in the category of sets.

The object ES is equipped with a canonical projection onto $S$, in such a way that $E\mapsto ES$ is in fact a functor from (Ens) to $C/S$.

²⁰ If $S^{\prime }\to S$ is an arrow of $C$, one has, since direct sums commute with base change,

$$E_{S^{\prime }}=(ES{)}_{S^{\prime }}.$$

In particular, if $C$ has a final object $e$, one has

$$ES=(Ee{)}_S.$$

The functor $E\mapsto ES/S$, from (Ens) to $C/S$, commutes with finite products. For this, it suffices to see that

(×)    ES ×_S FS = (E × F)_S.

Now, by the results of 1.7 applied to $C/S$, one has, for every $T\in Ob(C/S)$, natural isomorphisms (all unspecified Hom's being taken in $C/S$):

Hom((E × F)_S, T) ≅ Hom_{(Ens)}(E × F, Hom(S, T)) ≅
        Hom_{(Ens)}(E, Hom_{(Ens)}(F, Hom(S, T))) ≅ Hom_{(Ens)}(E, Hom(FS, T))

and

Hom(ES ×_S FS, T) ≅ Hom(ES, Hom(FS, T)) ≅ Hom_{(Ens)}(E, Hom(S, Hom(FS, T))).

Now Hom(S, Hom(FS, T)) ≅ Hom(FS, T), hence $(\times )$.

Suppose that, with $\varnothing$ denoting an initial object of $C$, the diagram

   ∅ ────→ S
   │         │
   ↓         ↓
   S ────→ S ⊔ S

$(\ast )$ is cartesian.

(This is the case for the category of schemes.)²¹ Then the functor $E\mapsto ES$ commutes with finite inverse limits.

Indeed, taking $(\times )$ into account, it suffices to see that $E\mapsto ES$ commutes with fiber products. Let $u:E\to G$ and $v:F\to G$ be two maps of sets. Since in $C$ direct sums commute with base change, one has

(1)    FS ×_{GS} ES ≅ ⨆_{f∈F} S_f ×_{GS} ES ≅ ⨆_{f∈F, x∈E} S_f ×_{GS} S_x.

If $v(f)\ne u(x)$, there exists in $C/S$ a morphism

S_f ×_{GS} S_x → S_f ×_{S_{v(f)} ⊔ S_{u(x)}} S_x;

now by hypothesis $(\ast )$ the right-hand term is $\varnothing$. Consequently,

(2)    S_f ×_{GS} S_x ≅ ∅    if v(f) ≠ u(x).

On the other hand, if $v(f)=u(x)$, there exists in $C/S$ a morphism

S → S_f ×_{GS} S_x;

since $S\stackrel{id}{\to }S$ is the final object of $C/S$, it follows that

(3)    S_f ×_{GS} S_x ≅ S    if v(f) = u(x).

Combining (1), (2), and (3) one obtains an isomorphism functorial in $S$

FS ×_{GS} ES ≅ ⨆_{F ×_G E} S = (F ×_G E)_S.

An object of the form ES will be called a constant object. Note that one has a morphism functorial in $E$:

$$E\to \Gamma (ES/S)$$

which associates to each $i\in E$ the section of ES over $S$ defined by the isomorphism of $S$ onto $S_i$. Suppose condition $(\ast )$ is satisfied for every object $S$ of $C$; then the morphism $E\to \Gamma (ES/S)$ is a monomorphism for every S ≄ ∅.

If $C$ is the category of schemes, then $\Gamma (ES/S)$ is identified with the locally constant maps from the topological space $S$ to the set $E$, the preceding map associating to each element of $E$ the corresponding constant map. Note that, by what was just said, ES may also be defined as representing the functor which to every $S^{\prime }$ over $S$ associates the set of locally constant functions from the topological space $S^{\prime }$ to the set $E$.²²

2. Algebraic structures

Given a species of algebraic structure in the category of sets, we propose to extend it to the category $C$. Let us first treat one example: the case of groups.

2.1. Group structures

We retain the notations of the preceding paragraph.

Definition 2.1.1. Let $G\in Ob(\hat{C})$. By a Ĉ-group structure on $G$, we mean the datum, for every $S\in Ob(C)$, of a group structure on the set $G(S)$, in such a way that for every arrow $f:S^{\prime }\to S^{\prime \prime }$ of $C$, the map $G(f):G(S^{\prime \prime })\to G(S^{\prime })$ is a group homomorphism. If $G$ and $H$ are two Ĉ-groups, by a morphism of Ĉ-groups from $G$ into $H$ we mean any morphism $u\in \mathrm{Hom}(G,H)$ such that for every $S\in Ob(C)$, the map of sets $u(S):G(S)\to H(S)$ is a group homomorphism.

One denotes by ${\mathrm{Hom}}_{\hat{C}-Gr.}(G,H)$ the set of morphisms of Ĉ-groups from $G$ into $H$, and by $(\hat{C}-Gr.)$ the category of Ĉ-groups.

Examples. Let $E\in Ob(\hat{C})$; the object $\mathrm{Aut}(E)$ is endowed in an obvious manner with a Ĉ-group structure. The final object $e$ has a unique Ĉ-group structure which makes it a final object of $(\hat{C}-Gr.)$.

For every $S\in Ob(C)$, let $e_G(S)$ be the identity element of $G(S)$. The family of the $e_G(S)$

defines an element $e_G\in \Gamma (G)=\mathrm{Hom}(e,G)$ which is a morphism of Ĉ-groups $e\to G$, and which is called the unit section of $G$.

Note that to give a Ĉ-group structure on $G$ amounts to giving a composition law on $G$, i.e. a Ĉ-morphism

π_G : G × G → G

such that, for every $S\in Ob(C)$, ${\pi }_G(S)$ endows $G(S)$ with a group structure.

In the same way, $f:G\to H$ is a morphism of Ĉ-groups if and only if the following diagram is commutative:

              π_G
   G × G ─────────→ G
     │              │
   (f,f)           f
     ↓     π_H      ↓
   H × H ─────────→ H

A subobject $H$ of $G$ such that, for every $S\in Ob(C)$, $H(S)$ is a subgroup of $G(S)$, obviously has a Ĉ-group structure induced by that of $G$: it is the only one for which the monomorphism $H\to G$ is a morphism of Ĉ-groups. The Ĉ-group $H$ endowed with this structure is called a sub-Ĉ-group of $G$.

If $G$ and $H$ are two Ĉ-groups, the product $G\times H$ is endowed with an obvious Ĉ-group structure: for every $S\in Ob(C)$, one endows $G(S)\times H(S)$ with the product group structure of the group structures given on $G(S)$ and $H(S)$. The Ĉ-group $G\times H$ endowed with this structure will be called the product Ĉ-group of $G$ and $H$ (it is indeed the product in the category of Ĉ-groups).

If $G$ is a Ĉ-group, then for every $S\in Ob(C)$, GS is a ${\hat{C}}_{/S}$-group. If $G$ and $H$ are two Ĉ-groups, one defines the object ${\mathrm{Hom}}_{\hat{C}-Gr.}(G,H)$ of Ĉ by:

Hom_{Ĉ-Gr.}(G, H)(S) = Hom_{Ĉ_{/S}-Gr.}(GS, HS)

(Note: ${\mathrm{Hom}}_{\hat{C}-Gr.}$ is not in general a Ĉ-group, nor a fortiori the Hom object in

the category $(\hat{C}-Gr.)$).

One defines similarly the objects

Isom_{Ĉ-Gr.}(G, H),    End_{Ĉ-Gr.}(G),    Aut_{Ĉ-Gr.}(G).

Definition 2.1.2. Let $G\in Ob(C)$. By a $C$-group structure on $G$, we mean a Ĉ-group structure on $hG\in Ob(\hat{C})$. By a morphism of the $C$-group $G$ into the $C$-group $H$, we mean an element $u\in \mathrm{Hom}(G,H)\simeq \mathrm{Hom}(hG,hH)$ which defines a morphism of Ĉ-groups from hG into hH.

One denotes by (C-Gr.) the category of $C$-groups. Note that there exists in (Cat) a cartesian square

   (C-Gr.) ─────────→ (Ĉ-Gr.)
      │                  │
      ↓        h         ↓
      C ─────────────→  Ĉ

All the preceding definitions and constructions therefore transport at once to (C-Gr.) whenever the functors they involve (products, Hom objects, etc.) are representable. They also apply to the categories $C/S$. In that case, we shall write ${\mathrm{Hom}}_{S-Gr.}$ for ${\mathrm{Hom}}_{{\hat{C}}_{/S}-Gr.}$, etc.

2.2.

More generally, if (T) is a species of structure on $n$ base sets defined by finite inverse limits (for example, by commutativities of diagrams constructed with cartesian products: structures of monoid, group, set with operators, module over a ring, Lie algebra over a ring, etc.), the preceding construction permits one to define the notion of "structure of species (T) on $n$ objects $F_1,\cdots ,F_n$ of Ĉ":

such a structure will be the datum, for each $S$ of $C$, of a structure of species (T) on the sets $F_1(S),\cdots ,F_n(S)$ in such a way that for every arrow $S^{\prime }\to S^{\prime \prime }$ of $C$, the family of maps $(F_i(S^{\prime \prime }))\to (F_i(S^{\prime }))$ is a polyhomomorphism for the species of structure (T). One defines similarly the morphisms of the species of structure (T), whence a category $(\hat{C}\times \hat{C}\cdots \times \hat{C}{)}^{(T)}$. The fully faithful functor $(h\times h\times \cdots \times h)$ then allows one to define by inverse image the category $(C\times C\times \cdots \times C{)}^{(T)}$, and then, as it commutes with inverse limits, to transport to it all the properties, notions, and notations of a functorial kind introduced in Ĉ. Suppose now that in $C$ fiber products exist, and let (T) be a species of algebraic structure defined by the datum of certain morphisms between cartesian products satisfying axioms consisting of certain commutativities of diagrams constructed using the preceding arrows. A structure of species (T) on a family of objects of $C$ will therefore be defined by certain morphisms between cartesian products satisfying certain commutation conditions. It follows that if $C$ and $C^{\prime }$ are two categories possessing products and $f:C\to C^{\prime }$ is a functor commuting with products, then for every family of objects $(F_i)$ of $C$ endowed with a structure of species (T), the family $(f(F_i))$ of objects of $C^{\prime }$ will thereby also be endowed with a structure of species (T). Every $C$-group will be transformed into a $C^{\prime }$-group, every pair ($C$-ring, $C$-module over this $C$-ring) into an analogous pair in $C^{\prime }$, etc.

In particular, let $C$ be a category satisfying the conditions of 1.8;²³ the functor $E\mapsto ES$ defined in loc. cit. commutes with finite inverse limits; it therefore transforms group into $S$-group (i.e. $C/S$-group), ring into $S$-ring, etc.

Remark. It is good to note that the preceding construction procedure applied to the category Ĉ does give back the notions already defined there; in other words, it amounts to the same thing to give on an object of Ĉ a structure of species (T) when one considers this object as a functor on $C$, or to give a structure of species (T) on the representable functor on Ĉ defined by this object.²⁴

We shall still treat two particular cases of the preceding construction, the case of structures with operator groups and the case of modules.

2.3. Structures with operator groups

Definition 2.3.1. Let $E\in Ob(\hat{C})$ and $G\in Ob(\hat{C}-Gr.)$. A structure of object with Ĉ-operator group $G$ (or of $G$-object) on $E$ is the datum, on $E(S)$, for every $S\in Ob(C)$, of a structure of set with operator group $G(S)$ in such a way that, for every arrow $S^{\prime }\to S^{\prime \prime }$ of $C$, the map of sets $E(S^{\prime \prime })\to E(S^{\prime })$ is compatible with the operator homomorphism $G(S^{\prime \prime })\to G(S^{\prime })$.

As usual, it amounts to the same thing to give a morphism

μ : G × E → E

which for each $S$ endows $E(S)$ with a structure of set with operators $G(S)$. But Hom(G × E, E) ≃ Hom(G, End(E)), so $\mu$ defines a morphism $G\to \mathrm{End}(E)$, and it is immediate that this maps $G$ into $\mathrm{Aut}(E)$ and is a morphism of Ĉ-groups. Consequently: to give on $E$ a structure of object with Ĉ-operator group $G$ is equivalent to giving a morphism of Ĉ-groups

$$\rho :G\to \mathrm{Aut}(E).$$

In particular, every element $g\in G(S)$ defines an automorphism $\rho (g)$ of the functor ES, that is, an automorphism of $E\times hS$ commuting with the projection $E\times hS\to hS$, and in particular an automorphism of the set $E(S^{\prime })$ for every $S^{\prime }\to S$.

Definition 2.3.2. One denotes by $E^G$ the subobject of $E$ defined as follows:

E^G(S) = {x ∈ E(S) | x_{S′} invariant under G(S′) for every S′ → S},

where $x_{S^{\prime }}$ denotes the image of $x$ by $E(S)\to E(S^{\prime })$.

Then $E^G$ ("subobject of invariants of $G$")

is the largest subobject of $E$ on which $G$ operates trivially.

Definition 2.3.3. Let $F$ be a subobject of $E$. One denotes by $Nor{m}_{G}F$ and $Cent{r}_{G}F$ the sub-Ĉ-groups of $G$ defined by

²⁵

(Norm_G F)(S) = {g ∈ G(S) | ρ(g) FS = FS}
              = {g ∈ G(S) | ρ(g) F(S′) = F(S′), for every S′ → S},

(Centr_G F)(S) = {g ∈ G(S) | ρ(g)|FS = identity}
               = {g ∈ G(S) | ρ(g)|F(S′) = identity, for every S′ → S},

where the vertical bar after $\rho (g)$ denotes restriction.

Scholium 2.3.3.1.²⁶ In particular, let $x\in \Gamma (E)$, i.e. (cf. 1.2) a collection of elements $x_S\in E(S)$, $S\in Ob(C)$, such that for every arrow $f:S^{\prime }\to S$ one has $E(f)(x_S)=x_{S^{\prime }}$ (if $C$ has a final object S_0 one has $\Gamma (E)=E(S_0)$). Then $x$ defines a subfunctor of $E$, which we shall denote $\bar{x}$, and one has $Nor{m}_G\bar{x}=Cent{r}_G\bar{x}$. We shall denote by $Sta{b}_G(x)$ and call stabilizer of $x$ this functor; for every $S\in Ob(C)$ one therefore has:

Stab_G(x)(S) = {g ∈ G(S) | ρ(g) x_S = x_S}.

Suppose that fiber products exist in $C$; if $G=hG$ (resp. $E=hE$), where $G$ is a $C$-group (resp. $E\in Ob(C)$), and if $C$ has a final object S_0, so that $x$ is a morphism $S_0\to E$, then $Sta{b}_G(x)$ is representable by the fiber product $G{\times }_E{S}_0$, where $G\to E$ is the composite of $i{d}_G\times x:G=G\times S_0\to G\times E$ and of $\mu :G\times E\to E$.

Remark 2.3.3.2.²⁶ The formation of $E^G$, $Nor{m}_{G}F$ and $Cent{r}_{G}F$ commutes with base change, i.e. for every $S\in Ob(C)$ one has

(E^G)_S = (ES)^{GS},    (Norm_G F)_S ≃ Norm_{GS} F_S,    (Centr_G F)_S ≃ Centr_{GS} F_S.

Definition 2.3.4. If $G$ is a $C$-group and $E$ an object of Ĉ (resp. $E$ an object of $C$) a structure of $G$-object on $E$ (resp. on $E$) is a structure of hG-object on $E$ (resp. hE).

In view of this definition, all the notions and notations defined above transport to $C$ when they involve only representable functors: for example, if $Nor{m}_{hG}(hF)$ is representable, then there exists one and only one subobject of $G$ that represents it, and which is then a sub-$C$-group of $G$; one denotes it $Nor{m}_G(F)$, etc.

Definition 2.3.5. a) One says that the Ĉ-group $G$ operates on the Ĉ-group $H$ if $H$ is endowed with a structure of $G$-object such that, for every $g\in G(S)$, the automorphism of $H(S)$ defined by $g$ is a group automorphism.

It amounts to the same thing to say that for every $g\in G(S)$, the automorphism $\rho (g)$ of HS

is an automorphism of ${\hat{C}}_{/S}$-groups, or that the morphism of Ĉ-groups $G\to \mathrm{Aut}(H)$ maps $G$ into ${\mathrm{Aut}}_{\hat{C}-Gr.}(H)$.

b) In the situation above, there exists on the product $H\times G$ a unique Ĉ-group structure such that, for every $S$, $(H\times G)(S)$ is the semidirect product of the groups $H(S)$ and $G(S)$ relative to the given operation of $G(S)$ on $H(S)$. We shall denote this Ĉ-group $H\cdot G$ and call it the semidirect product of $H$ by $G$. One therefore has by definition

(H ⋅ G)(S) = H(S) ⋅ G(S).

Let $G$ be a Ĉ-group. For every arrow $S^{\prime }\to S$ of $C$ and every $g\in G(S)$, let $Int(g)$ be the automorphism of $G(S^{\prime })$ defined by $Int(g)h=gh{g}^{-1}$. This definition extends into that of a morphism of Ĉ-groups

Int : G → Aut_{Ĉ-Gr.}(G) ⊂ Aut(G).

Definition 2.3.3 therefore applies, and one has sub-Ĉ-groups of $G$

Norm_G(E) and Centr_G(E)

for every subobject $E$ of $G$.

Definition 2.3.6. One calls center of $G$ and one denotes by $Centr(G)$ the sub-Ĉ-group $Cent{r}_G(G)$ of $G$. One says that $G$ is commutative if $Centr(G)=G$, or, what amounts to the same thing, if $G(S)$ is commutative for every $S$.

One says that the sub-Ĉ-group $H$ of $G$ is invariant in $G$ (or normal*) if $Nor{m}_G(H)=G$, or, what amounts to the same thing, if $H(S)$ is invariant in $G(S)$ for every $S$.*²⁷

Definition 2.3.6.1.²⁸ Let $f:G\to G^{\prime }$ be a morphism of Ĉ-groups. One calls kernel of $f$, and one denotes by Ker f, the sub-Ĉ-group of $G$ defined by

(Ker f)(S) = {x ∈ G(S) | f(S)(x) = 1} = Ker f(S)

for every $S\in Ob(C)$; it is an invariant sub-Ĉ-group.

Note that if $G=hG$ and $G^{\prime }=h{G}^{\prime }$, if $C$ has a final object S_0 and if fiber products exist in $C$, then $Ker(f)$ is representable by $S_0{\times }_{G^{\prime }}G$.

Definition 2.3.6.2.²⁸ Let $E\in \hat{C}$ and $G$ a Ĉ-group operating on $E$. One says that the operation of $G$ on $E$ is faithful if the kernel of the morphism $G\to \mathrm{Aut}(E)$ is trivial, i.e. if for every $S\in Ob(C)$ and $g\in G(S)$, the condition $g_{S^{\prime }}\cdot x=x$ for every $S^{\prime }\to S$ and $x\in E(S^{\prime })$ entails $g=1$.

Many definitions and propositions of the elementary theory of groups transpose easily. Let us simply point out the following, which will be useful to us:

Proposition 2.3.7. Let $f:W\to G$ be a morphism of Ĉ-groups. Set $H(S)=Kerf(S)$. Let $u:G\to W$ be a morphism of Ĉ-groups which is a section of $f$

(and which is then necessarily a monomorphism). Then $W$ is identified with the semidirect product of $H$ by $G$ for the operation of $G$ on $H$ defined by $(g,h)\mapsto Int(u(g))h$ for $g\in G(S)$, $h\in H(S)$, $S\in Ob(C)$.

The set of these definitions and propositions transports as usual to $C$. One defines in particular the semidirect product of two $C$-groups $H$ and $G$ (with $G$ operating on $H$) when the cartesian product $H\times G$ exists, and one has the analogue of Proposition 2.3.7 in the following form:

Proposition 2.3.8. Let $H\stackrel{i}{\to }W\stackrel{f}{\to }G$ be a sequence of morphisms of $C$-groups such that for every $S\in Ob(C)$, (H(S), i(S)) is a kernel of $f(S):W(S)\to G(S)$. Let $u:G\to W$ be a morphism of $C$-groups which is a section of $f$. Then $W$ is identified with the semidirect product of $H$ by $G$ for the operation of $G$ on $H$ such that if $S\in Ob(C)$, $g\in G(S)$ and $h\in H(S)$, one has $Int(u(g))i(h)=i({}^{g}h)$.

3. The category of O-modules, the category of G-O-modules

Definition 3.1. Let $O$ and $F$ be two objects of Ĉ. One says that $F$ is a Ĉ-module over the Ĉ-ring $O$, or in abbreviated form an $O$-module, if, for every $S\in Ob(C)$, one has endowed $O(S)$ with a ring structure and $F(S)$ with a structure of module over this ring in such a way that, for every arrow $S^{\prime }\to S^{\prime \prime }$ of $C$, $O(S^{\prime \prime })\to O(S^{\prime })$ is a ring homomorphism and $F(S^{\prime \prime })\to F(S^{\prime })$ is a homomorphism of abelian groups compatible with this ring homomorphism.

If $O$ is fixed, one defines in the usual way a morphism of the $O$-modules $F$ and $F^{\prime }$, whence the commutative group ${\mathrm{Hom}}_O(F,F^{\prime })$, and the category of $O$-modules, denoted (O-Mod.).

Lemma 3.1.1.²⁹ (O-Mod.) is endowed with a structure of abelian category, defined "argument by argument". Moreover, (O-Mod.) verifies the axiom (AB 5) (cf. [Gr57, 1.5]), i.e. arbitrary direct sums exist, and if $M$ is an $O$-module, $N$ a submodule, and $(F_i{)}_{i\in I}$ a directed family of submodules of $M$, then

⋃_{i∈I} (F_i ∩ N) = (⋃_{i∈I} F_i) ∩ N.

Indeed, let $f:F\to F^{\prime }$ be a morphism of $O$-modules. One defines the $O$-modules Ker f (resp. Im f and Coker f) by setting, for every $S\in Ob(C)$, $(Kerf)(S)=Kerf(S)$ (resp. ⋯). Then Ker f (resp. Coker f) is a kernel (resp. cokernel) of $f$, and one has an isomorphism of $O$-modules $F/Kerf\stackrel{\sim }{\to }Imf$. This proves that (O-Mod.) is an abelian category.

Arbitrary direct sums exist and are defined "argument by argument". Finally, if $M$ is an $O$-module, $N$ a submodule, and $(F_i{)}_{i\in I}$ a directed family of submodules of $M$, then the inclusion

⋃_{i∈I} (F_i ∩ N) ⊂ (⋃_{i∈I} F_i) ∩ N

is an equality: indeed, if $S\in Ob(C)$ and $x\in N(S)\cap {\bigcup }_i{F}_i(S)$, there exists $i\in I$ such that $x\in N(S)\cap F_i(S)$.

Proposition 3.1.2.²⁹ Suppose the category $C$ is small, i.e. that $Ob(C)$ is a set. Then (O-Mod.) has a generator, and therefore enough injective objects.

Proof. For every object $S$ of $C$, one defines the $O$-module O_S as follows. For every object $S^{\prime }$, $O_S(S^{\prime })$ is a direct sum of copies of $O(S^{\prime })$ indexed by $\mathrm{Hom}(S^{\prime },S)$, i.e.

O_S(S′) = ⊕_{g : S′ → S} O(S′) g.

For every morphism $f:S^{\prime \prime }\to S^{\prime }$, denoting $f\ast$ the ring homomorphism $O(S^{\prime })\to O(S^{\prime \prime })$, the morphism $O_S(f):O_S(S^{\prime })\to O_S(S^{\prime \prime })$ sends each element a g of $O_S(S^{\prime })$ (where

$a\in O(S^{\prime })$ and $g:S^{\prime }\to S$) to the element $f\ast (a)g\circ f$ of $O_S(S^{\prime \prime })$. One verifies easily that this does define a functor in $O$-modules.

Since, by hypothesis, $Ob(C)$ is a set, one can consider the direct sum $G={\oplus }_{S\in Ob(C)}{O}_S$.

Lemma 3.1.2.1. Let $F$ be an $O$-module. For every $S\in Ob(C)$, every element $x$ of $F(S)$ defines a unique morphism of $O$-modules $x:O_S\to F$ such that $x(1i{d}_S)=x$.

Proof. Let $\varphi$ be a morphism of $O$-modules $O_S\to F$ such that $\varphi (1i{d}_S)=x$. Let $g:S^{\prime }\to S$ and $a\in O(S^{\prime })$. The $O(S^{\prime })$-linearity and the commutativity of the diagram

              φ
   O_S(S) ────────→ F(S)
     │              │
   O_S(g)          F(g)
     ↓     φ        ↓
   O_S(S′) ────────→ F(S′)

entail that $\varphi (ag)=a\varphi (1g)=aF(g)(\varphi (1i{d}_S))=aF(g)(x)$. So $\varphi$, if it exists, is determined by the previous equality. Conversely, if one defines $\varphi$ thus, then for every $f:S^{\prime \prime }\to S^{\prime }$ one has

φ ∘ O_S(f)(ag) = φ(f*(a) g ∘ f) = f*(a) F(g ∘ f)(x) = F(f)(a F(g)(x)) = F(f) ∘ φ(ag)

i.e. the diagram

              φ
   O_S(S′) ────────→ F(S′)
     │              │
   O_S(f)          F(f)
     ↓     φ        ↓
   O_S(S″) ────────→ F(S″)

is commutative. This proves the lemma.

Now, let $F$ be an $O$-module. For every $S\in Ob(C)$, let $F_0(S)$ be a system of generators of the $O(S)$-module $F(S)$, and let L_S be the direct sum indexed by $F_0(S)$ of copies of O_S. By the lemma, one obtains a morphism of $O$-modules $L_S\to F$, such that the morphism $L_S(S)\to F(S)$ is surjective, hence an epimorphism in the category of $O(S)$-modules.

Set $I={⨆}_{S\in Ob(C)}{F}_0(S)$. Then one obtains a morphism of $O$-modules

G^{⊕I} ──→ ⊕_{S∈Ob(C)} L_S ──→ F

which is "argument by argument" an epimorphism, hence an epimorphism of (O-Mod.). This shows that $G$ is a generator of (O-Mod.) (cf. [Gr57, 1.9.1]). Since (O-Mod.) verifies (AB 5), it then follows from [Gr57, 1.10.1] that (O-Mod.) has enough injective objects.

Remark 3.1.3. If O_0 is the Ĉ-ring defined by $O_0(S)=\mathbb{Z}$ (which should not be confused with the functor associated to the constant object $\mathbb{Z}$), then the category of O_0-modules is isomorphic to the category of commutative Ĉ-groups.

Definition 3.1.4. Note that, if $F$ is an $O$-module, then for every $S\in Ob(C)$, FS is an O_S-module. One can therefore define a Ĉ-abelian group ${\mathrm{Hom}}_O(F,F^{\prime })$ by

Hom_O(F, F′)(S) = Hom_{O_S}(FS, F′_S).

One defines similarly the objects

Isom_O(F, F′),    End_O(F)    and    Aut_O(F),

the last being a Ĉ-group for the structure induced by the composition of automorphisms.

Definition 3.2. Let $O$ be a Ĉ-ring, $F$ an $O$-module and $G$ a Ĉ-group. By a structure of $G$-$O$-module on $F$ we mean a structure of $G$-object such that for every $S\in Ob(C)$ and every $g\in G(S)$, the automorphism of $F(S)$ defined by $g$ is an automorphism of its $O(S)$-module structure.

It amounts to the same thing to say that the morphism of Ĉ-groups

$$\rho :G\to \mathrm{Aut}(F)$$

defined in 2.3 maps $G$ into the sub-Ĉ-group ${\mathrm{Aut}}_O(F)$ of $\mathrm{Aut}(F)$. To give a structure of $G$-$O$-module on the $O$-module $F$ is therefore to give a morphism of Ĉ-groups

$$\rho :G\to {\mathrm{Aut}}_O(F).$$

One defines in a natural way the abelian group ${\mathrm{Hom}}_{G-O}(F,F^{\prime })$, hence the additive category of $G$-$O$-modules denoted (G-O-Mod.).

Remark 3.2.1. The reader will note that (G-O-Mod.) may also be defined as the category of O[G]-modules, where O[G] is the algebra of the Ĉ-group $G$ over the Ĉ-ring $O$, whose definition is clear.³⁰ Consequently, by 3.1.1 and 3.1.2, (G-O-Mod.) is an abelian category verifying the axiom (AB 5); moreover, if $C$ is small, (G-O-Mod.) has enough injective objects.

All the constructions of this paragraph specialize at once to the case where $G$ (or $O$, or both) are representable by objects of $C$, which are thereby endowed with the corresponding algebraic structures.

We have treated succinctly the case of the principal algebraic structures encountered in the rest of this seminar. For the others (structure of $O$-Lie algebra for example), we believe that the reader will have had enough examples in this paragraph to make the general mechanism sketched in 2.2 work himself in each particular case.

We shall now apply what we have just done to the category of schemes, denoted (Sch), and more generally to the categories $(Sch)/S$ (also denoted $(Sch/S)$).

4. Algebraic structures in the category of schemes

We shall allow ourselves, whenever there is no ambiguity, the following abuses of language: one will designate by $T$ the object $T\stackrel{f}{\to }S$ of $C/S$, the datum of $f$ ("structural morphism of $T$") being understood, and one will identify $C$ with a subcategory of Ĉ. Given the compatibilities stated in the preceding paragraphs, these identifications can be made without danger.

We shall further simplify the terminology along the following model: a (Sch)-group will also be called a group scheme, a $(Sch)/S$-group a group scheme over $S$, or $S$-group scheme, or $S$-group, or $A$-group when $S$ is the spectrum of the ring $A$.

4.1. Constant schemes

The category of schemes satisfies the conditions required in 1.8. One can therefore define constant objects in it; for every set $E$, one has a scheme $E_{\mathbb{Z}}$, and for every scheme $S$, an $S$-scheme $ES=(E_{\mathbb{Z}}{)}_S$. Recall (cf. loc. cit.) that for every $S$-scheme $T$, ${\mathrm{Hom}}_S(T,ES)$ is the set of locally constant maps

from the topological space $T$ to $E$.

The functor $E\mapsto ES$ commutes with finite inverse limits. In particular if $G$ is a group, GS is an $S$-group scheme; if $O$ is a ring, OS is an $S$-ring scheme, etc.

4.2. Affine S-groups

Let us recall a certain number of things about affine $S$-schemes (EGA II, § 1). One says that the $S$-scheme $T$ is affine over $S$ if the inverse image of every affine open subset of $S$ is affine. The OS-algebra $f_{\ast }(O_T)$, which one denotes $A(T)$, is then quasi-coherent ($f$ denotes the structural morphism of $T$). Conversely, to every quasi-coherent OS-algebra $A$, one can associate an $S$-scheme affine over $S$, denoted $\mathrm{Spec}(A)$. These functors $T\mapsto A(T)$ and $A\mapsto \mathrm{Spec}(A)$ are quasi-inverse equivalences between the category of $S$-schemes affine over $S$ and the category opposite to that of quasi-coherent OS-algebras.

It follows that to give an algebraic structure on an $S$-scheme $T$ affine over $S$ is equivalent to giving the corresponding structure on $A(T)$ in the category opposite to that of quasi-coherent OS-algebras. In particular, if $G$ is an $S$-group affine over $S$, $A(G)$ is endowed with a structure of augmented OS-bialgebra, that is, one has two morphisms of OS-algebras

Δ : A(G) → A(G) ⊗_{OS} A(G)    and    ε : A(G) → OS

corresponding to the morphisms of $S$-schemes

π : G × G → G    and    e_G : S → G.

The maps $\Delta$ and $ϵ$ satisfy the following conditions (which express that $G$ is an $S$-monoid):³¹

(HA 1) $\Delta$ is co-associative: the following diagram is commutative

                       A(G) ⊗_{OS} A(G)
                        ╱            ╲
                Δ      ╱              ╲  id ⊗ Δ
                      ╱                ╲
                 A(G)                  A(G) ⊗_{OS} A(G) ⊗_{OS} A(G)
                      ╲                ╱
                       ╲              ╱  Δ ⊗ id
                Δ       ╲            ╱
                       A(G) ⊗_{OS} A(G)

(HA 2) $\Delta$ is compatible with $ϵ$: the two following composites are the identity

        Δ                       id ⊗ ε                ∼
A(G) ────→ A(G) ⊗_{OS} A(G) ─────────→ A(G) ⊗_{OS} OS ──→ A(G),

        Δ                       ε ⊗ id                ∼
A(G) ────→ A(G) ⊗_{OS} A(G) ─────────→ OS ⊗_{OS} A(G) ──→ A(G).

Let us take this opportunity to note once more that it follows from the definition of an $S$-group structure that to give such a structure on an $S$-scheme $G$ affine over $S$, it is not necessary to verify anything on $A(G)$, but simply to endow each $G(S^{\prime })$ for $S^{\prime }$ above $S$ with a group structure functorial in $S^{\prime }$.

This remark applies mutatis mutandis to morphisms.

4.3. The groups G_a and G_m. The ring O

4.3.1.

Let $G_a$ be the functor from $(Sch{)}^{\circ }$ to (Ens) defined by

$$G_a(S)=\Gamma (S,OS)$$

endowed with the $\widehat{Sch}$-group structure defined by the additive group structure of the ring $\Gamma (S,OS)$. It is representable by an affine scheme, which we shall denote $G_a$, which is therefore a group scheme

$$G_a=\mathrm{Spec}\mathbb{Z}[T].$$

Indeed, G_a(S) = Hom(S, G_a) = Hom_{Alg.}(ℤ[T], Γ(S, OS)) ≃ Γ(S, OS).

For every scheme $S$, one therefore has an $S$-group affine over $S$, denoted $G_{a,S}$, which corresponds to the OS-bialgebra OS[T], with the diagonal map and the augmentation defined by:

Δ(T) = T ⊗ 1 + 1 ⊗ T,    ε(T) = 0.

4.3.2.

Let $G_m$ be the functor from $(Sch{)}^{\circ }$ to (Ens) defined by

$$G_m(S)=\Gamma (S,OS{)}^{\times },$$

where $\Gamma (S,OS{)}^{\times }$ denotes the multiplicative group of invertible elements of the ring $\Gamma (S,OS)$, endowed with its natural $\widehat{Sch}$-group structure. It is representable by an affine scheme denoted $G_m$

G_m = Spec ℤ[T, T^{-1}] = Spec ℤ[ℤ],

where $\mathbb{Z}[\mathbb{Z}]$ denotes the algebra of the group $\mathbb{Z}$ over the ring $\mathbb{Z}$. Indeed,

G_m(S) = Hom_{Alg.}(ℤ[T, T^{-1}], Γ(S, OS)) ≃ Γ(S, OS)^×.

For every scheme $S$, one therefore has an $S$-group affine over $S$ denoted $G_{m,S}$, which corresponds to the OS-algebra $OS[\mathbb{Z}]$, with the diagonal map and augmentation defined by:

Δ(x) = x ⊗ x    and    ε(x) = 1,    for x ∈ ℤ.

4.3.3.

Let $O$ be the functor $G_a$ endowed with its $\widehat{Sch}$-ring structure. It is represented by the scheme $\mathrm{Spec}\mathbb{Z}[T]$, which we shall denote $O$ when considering it as endowed with its ring scheme structure.

For every scheme $S$, OS = S ×_{Spec ℤ} Spec ℤ[T] = Spec(OS[T]) is therefore an $S$-ring scheme, affine over $S$. (Note: in EGA II 1.7.13, OS is denoted S[T]).

4.3.3.1.

³² For every object $X$ of $\widehat{Sch}$, $O(X)=\mathrm{Hom}(X,O)$ is endowed with a ring structure, functorial in $X$. In particular, if $X^{\prime }$ is a scheme and one is given morphisms $x:X^{\prime }\to X$ and $f:X\to O$ (i.e. $f\in O(X)$), then $f(x)=f\circ x$ is an element of $O(X^{\prime })=\Gamma (X^{\prime },O_{X^{\prime }})$.

Definition. Let $\pi :M\to X$ be a morphism of $\widehat{Sch}$, and let $OX=O\times X$. One says that $M$ is an OX-module if one has given, for every $X$-scheme $X^{\prime }$, a structure of $O(X^{\prime })$-module on ${\mathrm{Hom}}_X(X^{\prime },M)$, functorial in $X^{\prime }$.

This amounts to giving a law of $X$-abelian group $\mu :M{\times }_{X}M\to M$ and an "external law"

O × M = OX ×_X M → M,    (f, m) ↦ f ⋅ m

which is an $X$-morphism (i.e. $\pi (f\cdot m)=\pi (m)$) and which, for every $x\in X(S)$, endows $M(x)=m\in M(S)|\pi (m)=x$ with a structure of $O(S)$-module.

In this case, for every $Y\in Ob{\widehat{Sch}}_{/X}$ (not necessarily representable), ${\mathrm{Hom}}_X(Y,M)=\Gamma (M_Y/Y)$ is an $O(Y)$-module, functorially in $Y$.

4.4. Diagonalizable groups

4.4.1.

The construction of $G_m$ generalizes as follows: let $M$ be a commutative group and $M_{\mathbb{Z}}$ the group scheme associated to it by the procedure of 4.1. Consider the functor defined by

D(M)(S) = Hom_{groups}(M, G_m(S)) ≃ Hom_{S-Gr.}(MS, G_{m,S}).

It is a commutative $\widehat{Sch}$-group representable by a group scheme that one will denote $D(M)$; one will therefore have by definition:

$$D(M)\simeq {\mathrm{Hom}}_{(Sch)-Gr.}(M_{\mathbb{Z}},G_m).$$

Set in fact

$$D(M)=\mathrm{Spec}\mathbb{Z}[M],$$

where $\mathbb{Z}[M]$ is the algebra of the group $M$ over the ring $\mathbb{Z}$; one has

D(M)(S) = Hom_{Alg.}(ℤ[M], Γ(S, OS)) ≃ Hom_{groups}(M, Γ(S, OS)^×)

by the very definition of the algebra $\mathbb{Z}[M]$.

4.4.2.

For every scheme $S$ one therefore has an $S$-group scheme affine over $S$

D_S(M) = D(M)_S = Hom_{(Sch)-Gr.}(M_ℤ, G_m)_S = Hom_{S-Gr.}(MS, G_{m,S}).

It is associated to the OS-bialgebra OS[M], endowed with the diagonal map and augmentation defined by

Δ(x) = x ⊗ x    and    ε(x) = 1,    for x ∈ M.

4.4.3.

If $f:M\to N$ is a homomorphism of commutative groups, one defines in an obvious manner a morphism of $S$-groups

$$D_S(f):D_S(N)\to D_S(M),$$

whence a functor

$$D_S:M\mapsto D_S(M)$$

from the category of abelian groups to the category of $S$-groups affine over $S$, which one may also define as the composite of the functor $M\mapsto MS$ and of the functor $MS\mapsto {\mathrm{Hom}}_{S-Gr.}(MS,G_{m,S})$. This functor commutes with extensions of the base.

An $S$-group isomorphic to a group of the form $D_S(M)$ is said to be diagonalizable. Note that the elements of $M$ are interpreted as certain characters of $D_S(M)$, that is, certain elements of ${\mathrm{Hom}}_{S-Gr.}(D_S(M),G_{m,S})$. (Confer VIII 1).

4.4.4.

Let us give some examples of diagonalizable groups. One has first

D(ℤ) = G_m    and    D(ℤ^r) = (G_m)^r.

One sets

$${\mu }_n=D(\mathbb{Z}/n\mathbb{Z}),$$

and calls it the group of $n$-th roots of unity. Indeed, one has

μ_n(S) = Hom_{groups}(ℤ/nℤ, Γ(S, OS)^×) = {f ∈ Γ(S, OS) | f^n = 1}.

The $S$-group ${\mu }_{n,S}$ corresponds to the OS-algebra $OS[T]/(T^n-1)$. Suppose in particular that $S$ is the spectrum of a field $k$ of characteristic $p=n$. Setting $T-1=s$, one finds

$$k[T]/(T^p-1)=k[s]/(s^p),$$

which shows that the underlying topological space of ${\mu }_{p,k}$ reduces to a point, the local ring of this point being the artinian $k$-algebra $k[s]/(s^p)$. (In the same vein, let us note that the $S$-schemes $G_{a,S},G_{m,S},OS$ are smooth over $S$, that $D_S(M)$ is

flat over $S$ and that it is formally smooth (resp. smooth) over $S$ if and only if no residue characteristic of $S$ divides the torsion of $M$ (resp. and if moreover $M$ is of finite type), cf. Exp. VIII, 2.1).

4.5. Other examples of groups

The preceding procedure applies to the "classical groups" (linear groups $G{L}_n$, symplectic $S{p}_n$, orthogonal $O_n$, etc.). One defines for example $G{L}_n$ as representing the functor $G{L}_n$ such that:

GL_n(S) = GL(n, Γ(S, OS)) = Aut_{OS}(OS^n).

One can construct it for example as the open subscheme of $\mathrm{Spec}\mathbb{Z}[T_{ij}]$ ($1⩽i,j⩽n$) defined by the function $f=det((T_{ij}{)}_{i,j=1}^n)$, that is, $\mathrm{Spec}\mathbb{Z}[T_{ij},f^{-1}]$.

4.6. Functor-modules in the category of schemes

We propose to associate to every OS-module on the scheme $S$ an OS-module (where OS denotes the functor-ring introduced in 4.3.3). This can be done in two different ways. Precisely:

Definition 4.6.1. Let $S$ be a scheme. For every OS-module $F$ one denotes by $V(F)$ and $W(F)$ the contravariant functors on $(Sch)/S$ defined by:

V(F)(S′) = Hom_{O_{S′}}(F ⊗_{OS} O_{S′}, O_{S′}),
W(F)(S′) = Γ(S′, F ⊗_{OS} O_{S′})

(where $F{\otimes }_{OS}{O}_{S^{\prime }}$ denotes the inverse image on $S^{\prime }$ of the OS-module $F$).

Then $V(F)$ and $W(F)$ are endowed in an obvious manner with a structure of OS-modules (recall that $OS(S^{\prime })=\Gamma (S^{\prime },O_{S^{\prime }})=W(OS)(S^{\prime })$), in such a way that one has in fact defined two functors $V$ and $W$ from the category of OS-modules to the category of OS-modules, with $V$ contravariant and $W$ covariant.

We restrict ourselves in the rest of this paragraph to the case where the OS-modules in question are quasi-coherent, that is, we consider $V$ and $W$ as

functors from the category (OS-Mod.q.c.) of quasi-coherent OS-modules to the category of OS-modules

$$V:(OS-Mod.q.c.{)}^{\circ }\to (OS-Mod.),W:(OS-Mod.q.c.)\to (OS-Mod.).$$

Remark 4.6.1.1.³³ The reader will note that, in what follows, all the propositions involving only the functor $W$ are valid for arbitrary OS-modules, not necessarily quasi-coherent.

Proposition 4.6.2. (i) The functors $V$ and $W$ commute with base extension: if $S^{\prime }$ is above $S$ and if $F$ is a quasi-coherent OS-module, one has

V(F ⊗ O_{S′}) ≃ V(F)_{S′}    and    W(F ⊗ O_{S′}) ≃ W(F)_{S′}.

(ii) The functors $V$ and $W$ are fully faithful: the canonical maps

Hom_{OS}(F, F′) → Hom_{OS}(V(F′), V(F))
Hom_{OS}(F, F′) → Hom_{OS}(W(F), W(F′))

are bijective.

(iii) The functors $V$ and $W$ are additive:

V(F ⊕ F′) ≃ V(F) ×_S V(F′)    and    W(F ⊕ F′) ≃ W(F) ×_S W(F′).

Parts (i) and (iii) are obvious from the definitions. For (ii), one takes for $S^{\prime }$ open subschemes of $S$. We leave the proof to the reader (for $V$, use EGA II, 1.7.14).

Recall (cf. 3.1.4) that if $F,F^{\prime }$ are OS-modules, ${\mathrm{Hom}}_{OS}(F,F^{\prime })$ denotes the $S$-functor (in abelian groups) which to every $S^{\prime }\to S$ associates ${\mathrm{Hom}}_{O_{S^{\prime }}}(F_{S^{\prime }},F_{S^{\prime }}^{\prime })$.