Exposé III. Infinitesimal extensions

by M. Demazure

[^N.D.E-III-0] In this Exposé, we place ourselves in the following general situation. We have a scheme $S$ and a coherent nilpotent ideal $I$ on $S$. We denote by $S_n$ the closed subscheme of $S$ defined by the ideal $I^{n+1}$ ($n\ge 0$). In particular $S_0$ is defined by $I$. Since $I$ is nilpotent, $S_n$ is equal to $S$ for $n$ large enough, and the Sᵢ have the same underlying topological space. A typical example of this situation is the following: $S$ is the spectrum of a local Artinian ring $A$, $I$ is the ideal defined by the radical of $A$, so that $S_0$ is the spectrum of the residue field of $A$.

In the preceding situation, one is given a certain number of data above $S_0$, and one looks above $S$ for data which lift them, that is to say, which give them back by base change from $S$ to $S_0$. This is done step by step, by way of the $S_n$. At each step, we propose to define the obstructions encountered, and to classify, when they exist, the solutions obtained.

The passage from $S_n$ to $S_{n+1}$ can be generalized as follows: one has a scheme $S$, two ideals $I$ and $J$ with $I\supset J$, and $I\cdot J=0$ (in the preceding case $S$, $I$ and $J$ are respectively $S_{n+1}$, $I/I^{n+2}$, $I^{n+1}/I^{n+2}$). We denote by $S_0$ (resp. S_J) the closed subscheme of $S$ defined by $I$ (resp. $J$), and we pose a problem of extension from S_J to $S$.

In (SGA 1, III), problems of extension of morphisms of schemes and of extension of schemes were treated. We pose here the problems of extension of morphisms of groups, of extension of group structures, and of extension of subgroups.

We have gathered in a § 0 the results of (SGA 1, III) which will be useful to us, in order to put them in the most convenient form for our purpose, and to spare the reader from having to refer constantly to (SGA 1, III).[^N.D.E-III-1] § 1 collects some computations of group cohomology which will be useful in what follows and which have nothing to do with scheme theory. §§ 2 and 3 treat respectively of the extension of group morphisms and the extension of group structures. In § 4, we have briefly recalled the proof of a result stated in (TDTE IV) concerning the extension of subschemes, and applied this result to the problem of extension of subgroups. For the rest of the Seminar, only the result of § 2, concerning the extension of morphisms of groups, will be indispensable.[^N.D.E-III-2]

The idea of reducing infinitesimal extension problems to the usual computations of cohomology in extensions of groups was suggested by J. Giraud at the oral exposé (whose computations were notably more complicated and less transparent). Unfortunately, it seems that this method only applies well to the first two problems studied, and we have been unable to escape rather painful computations in the case of extensions of subgroups.

To simplify the language, we shall call a $Y$-functor, resp. $Y$-scheme, etc., a functor, resp. scheme, etc., equipped with a morphism into the functor $Y$, thus extending the definitions of Exposé I (which concerned only the case of a representable $Y$).

0. Reminders from SGA 1 III and various remarks

Let us first state a general definition.

Definition 0.1. Let $C$ be a category, $X$ an object of Ĉ, $G$ a Ĉ-group acting on $X$. We say that $X$ is formally principal homogeneous[^N.D.E-III-3] under $G$ if the following equivalent conditions are satisfied:

(i) for each object $S$ of $C$, the set $X(S)$ is empty or principal homogeneous under $G(S)$;

(ii) the morphism of functors $G\times X\to X\times X$ defined set-theoretically by $(g,x)\mapsto (gx,x)$ is an isomorphism.

This being so, we shall put the results of (SGA 1, III, § 5)[^N.D.E-III-4] in the form which will be most useful to us. We shall employ the following general notations throughout this section. We have a scheme $S$ and on $S$ two quasi-coherent ideals $I$ and $J$ such that

I ⊃ J   and   I · J = 0.

In particular we shall therefore have $J^2=0$. We shall denote by $S_0$ (resp. S_J) the closed subscheme of $S$ defined by the ideal $I$ (resp. $J$). For every $S$-functor $X$, we shall systematically denote by $X_0$ and X_J the functors obtained by base change from $S$ to $S_0$ and S_J. Same notation for a morphism.

Definition 0.1.1.[^N.D.E-III-5] Let $X$ be an $S$-functor. Define a functor $X^{+}$ above $S$ by the formula:

Hom_S(Y, X⁺) = Hom_{S_J}(Y_J, X_J) = Hom_S(Y_J, X)

for a variable $S$-scheme $Y$. In the notations of (Exp. II, 1), one has

$$X^{+}\simeq \prod _{S_J/S}{X}_J.$$

The identity morphism of X_J defines by construction an $S$-morphism

$$p_X:X\to X^{+}.$$

[^N.D.E-III-6] Explicitly, for every $S$-scheme $Y$, the map

p_X(Y) : Hom_S(Y, X) → Hom_S(Y, X⁺) = Hom_S(Y_J, X)

is the map induced by the morphism $Y_J\to Y$.

Remark 0.1.2. Let us now observe that if $X$ is an $S$-group functor, then X_J is an S_J-group functor; then the formula defining $X^{+}$ endows it with a structure of $S$-group functor, and the description of $p_X$ above shows that $p_X:X\to X^{+}$ is a morphism of $S$-group functors.

Remark 0.1.3. On the other hand, for every $S$-group functor $Y$, one has:

Hom_{S-gr.}(Y, X⁺) = Hom_{S_J-gr.}(Y_J, X_J).

Indeed, let $f\in {\mathrm{Hom}}_S(Y,X^{+})$, corresponding to $f_J\in {\mathrm{Hom}}_{S_J}(Y_J,X_J)$; the condition that $f\in {\mathrm{Hom}}_{S-gr.}(Y,X^{+})$ is that, for every $T\to S$ and $y,y^{\prime }\in Y(T)$, one has $f(y\cdot y^{\prime })=f(y)\cdot f(y^{\prime })$, and this is equivalent to

f_J(y_J) · f_J(y_J′) = f_J((y · y′)_J);

since $(y\cdot y^{\prime }{)}_J=y_J\cdot y_J^{\prime }$ (because $Y(T)\to Y(T_J)=Y_J(T_J)$ is a morphism of groups), this is the condition for $f_J$ to be a morphism of groups. Applying this to $Y=X$, we recover that $p_X$, which corresponds to $i{d}_{X_J}$, is a morphism of $S$-group functors.

Let us now return to the general case, but assume that $X$ is an $S$-scheme. Since an $S$-morphism of a variable $S$-scheme $Y$ into $X^{+}$ is by definition an S_J-morphism $g_J$ of Y_J into X_J, we shall define an $X^{+}$-functor in abelian groups L_X as follows.

Scholie 0.1.4.[^N.D.E-III-7] If $\pi :Y\to S$ is a morphism of schemes and $M$ an O_S-module, we denote by $M{\otimes }_{O_S}{O}_Y$ the inverse image $\pi \ast (M)$. If $J$ is an ideal of O_S, we denote by $J\cdot O_Y$ the ideal of O_Y image of the morphism

π*(J) = J ⊗_{O_S} O_Y → O_Y .

Note that, for every morphism of $S$-schemes $f:Z\to Y$, one has an epimorphism of O_Z-modules:

f*(J · O_Y) → J · O_Z ,                              (0.1.4)

as follows from the commutative diagram below:

J ⊗_{O_S} O_Y ⊗_{O_Y} O_Z   ─────►   J ⊗_{O_S} O_Z
            │                                │
            ▼                                ▼
f*(J · O_Y) = (J · O_Y) ⊗_{O_Y} O_Z   ───►   J · O_Z .

Definition 0.1.5.[^N.D.E-III-8] Let $X$ be an $S$-scheme. For every $X^{+}$-scheme $Y$, given by a morphism $g_J:Y_J\to X_J$, we set:

Hom_{X⁺}(Y, L_X) = Hom_{O_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), J · O_Y),

where ${\Omega }_{X_0/S_0}^1$ denotes the module of relative differentials of $X_0$ with respect to $S_0$ (cf. SGA 1, I.1 or EGA IV₄, 16.3), and where $J\cdot O_Y$ is regarded as an $O_{Y_0}$-module thanks to the fact that it is annihilated by $I$.

Then L_X is an $X^{+}$-functor in abelian groups.[^N.D.E-III-9] Indeed, for every $X^{+}$-morphism $f:Z\to Y$, the functor $f_0\ast$ and the morphism $f_0\ast (J\cdot O_Y)\to J\cdot O_Z$ of (0.1.4) induce a natural morphism of abelian groups $L_X(f)$:

Hom_{O_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), J · O_Y)
        → Hom_{O_{Z₀}}(f₀* g₀*(Ω¹_{X₀/S₀}), f₀*(J · O_Y))
        → Hom_{O_{Z₀}}(f₀* g₀*(Ω¹_{X₀/S₀}), J · O_Z).

Let us finally note that $L_X(f)$ is described locally as follows. Note first that $Y$ and Y_J have the same underlying topological space, and similarly for $V$ and V_J if $V$ is an open subset of $Y$. Let then $U=\mathrm{Spec}(A)$ be an affine open of $X$ above an affine open $\mathrm{Spec}(\Lambda )$ of $S$, $V=\mathrm{Spec}(B)$ an affine open of $Y$ such that $g_J(V_J)\subset U$, and $W=\mathrm{Spec}(C)$ an affine open of $f^{-1}(V)$. Let $J$ and $I$ be the ideals of $\Lambda$ corresponding to $J$ and $I$. Then $f$ (resp. $g_J$) induces a morphism of $\Lambda$-algebras $\theta :B\to C$ (resp. $\varphi :A\to B/JB$), and one obviously has $\theta (JB)\subset JC$. On the other hand, $m\in {\mathrm{Hom}}_{O_{Y_0}}(g_0\ast ({\Omega }_{X/S}^1),J\cdot O_Y)$ induces an element $D$ of

Hom_{O_{V₀}}(g₀*(Ω¹_{U/S}), J · O_V) = Hom_{B/IB}(Ω¹_{A/Λ} ⊗_A B/IB, JB) = Der_Λ(A, JB),

and the image of $L_X(f)(m)$ in

Hom_{O_{W₀}}(f₀* g₀*(Ω¹_{U/S}), J · O_Z) = Hom_{C/IC}(Ω¹_{A/Λ} ⊗_A C/IC, JC) = Der_Λ(A, JC)

is none other than $\theta \circ D$.

Remark 0.1.6.[^N.D.E-III-10] Let $f:X\to W$ be an $S$-morphism. It induces an $S$-morphism $f^{+}:X^{+}\to W^{+}$ defined as follows: if $g$ is an element of ${\mathrm{Hom}}_S(Y,X^{+})$, corresponding to an $S$-morphism $g_J:Y_J\to X$, then $f^{+}(g)$ is the element $f\circ g_J$ of ${\mathrm{Hom}}_S(Y,W^{+})$. It is clear that the following diagram is commutative:

        f
   X ─────► W
   │         │
p_X│         │p_W
   ▼   f⁺    ▼
   X⁺ ────► W⁺ .

Reminders 0.1.7.[^N.D.E-III-11] In this paragraph, given an $S$-scheme $X$, we "recall" certain functorial properties of the module of differentials ${\Omega }_{X/S}^1$ and of the first infinitesimal neighborhood of the diagonal ${\Delta }_{X/S}^{(1)}$, properties which follow easily from (EGA IV₄, §§ 16.1–16.4), but which do not figure there explicitly.

a) Let us begin by recalling the following facts (cf. EGA II, §§ 1.2–1.5). Let $g:Y\to X$ be a morphism of schemes, $\pi :X^{\prime }\to X$ an affine $X$-scheme, $B$ the quasi-coherent O_X-algebra ${\pi }_{\ast }(O_{X^{\prime }})$; then the $Y$-scheme $Y{\times }_X{X}^{\prime }$ is affine and corresponds to the quasi-coherent O_Y-algebra $g\ast (B)$, and one has a commutative diagram of bijections:

Hom_X(Y, X′)            ──∼──►    Hom_Y(Y, Y ×_X X′)
    │                                       │
    ≀                                       ≀
    ▼                                       ▼
Hom_{O_X-alg.}(B, g_*(O_Y))   ──∼──►   Hom_{O_Y-alg.}(g*(B), O_Y).

Moreover, these bijections are functorial in the pair $(X,X^{\prime })$, i.e., if $W^{\prime }$ is an affine $W$-scheme, corresponding to the quasi-coherent O_W-algebra $A$, if one has a commutative diagram of morphisms of schemes

            X′ ──f′──► W′
          ↗ g′         │
         /             │
        ↗              │
      Y ──g──► X ──f──► W ,

and if we denote by $\varphi :A\to f_{\ast }(B)$ and $\varphi ♯:f\ast (A)\to B$ (resp. $\theta :B\to g_{\ast }(O_Y)$ and $\theta ♯:g\ast (B)\to O_Y$) the algebra morphisms associated to $f^{\prime }$ (resp. to a variable $X$-morphism $g^{\prime }:Y\to X^{\prime }$), then one has the following commutative diagram (where $Y$ is viewed as a $W$-scheme via $f\circ g$):

                         g′ ↦ f′ ∘ g′
Hom_X(Y, X′) ──────────────────────► Hom_W(Y, W′)
                                          │
                                          ≃
                                          ▼
                                Hom_{O_W-alg.}(A, f_* g_*(O_Y))
                                          │
                                          ≃
       θ ↦ θ ∘ φ♯                         ▼
Hom_{O_X-alg.}(B, g_*(O_Y)) ──────► Hom_{O_X-alg.}(f*(A), g_*(O_Y))
       │                                  │
       ≀                                  ≀
       ▼   ψ ↦ ψ ∘ g*(φ♯)                 ▼
Hom_{O_Y-alg.}(g*(B), O_Y) ──────► Hom_{O_Y-alg.}(g* f*(A), O_Y).

b) Let now $X$ be an $S$-scheme. Let ${\Omega }_{X/S}^1$ be the module of differentials of $X$ over $S$, and ${\Delta }_{X/S}^{(1)}$ the first infinitesimal neighborhood of the diagonal immersion ${\delta }_X:X\to X{\times }_{S}X$; it is a subscheme of $X{\times }_{S}X$, of which the diagonal ${\Delta }_{X/S}$ is a closed subscheme. We denote by $p{r}_X^i$ ($i=1,2$) the two projections $X{\times }_{S}X\to X$, and by ${\pi }_X$ the restriction of $p{r}_X^1$ to ${\Delta }_{X/S}^{(1)}$.

On the one hand, every morphism $f:X\to W$ of $S$-schemes induces an $S$-morphism ${\Delta }^{(1)}f:{\Delta }_{X/S}^{(1)}\to {\Delta }_{W/S}^{(1)}$ such that the following diagram is commutative:

       δ_X            pr_X
   X ─────► Δ⁽¹⁾_{X/S} ─────► X ×_S X ─────► X
   │              │               │           │
 f │       Δ⁽¹⁾f │            f×f│         f │
   ▼       δ_W   ▼                ▼   pr_W   ▼
   W ─────► Δ⁽¹⁾_{W/S} ─────► W ×_S W ─────► W .

On the other hand, ${\Delta }_{X/S}^{(1)}$ is, via the projection ${\pi }_X$, an affine $X$-scheme, spectrum of the augmented quasi-coherent O_X-algebra

$$P_{X/S}^1=O_X\oplus {\Omega }_{X/S}^1,$$

where ${\Omega }_{X/S}^1$ is an ideal of square zero; the augmentation is the morphism of O_X-algebras ${ϵ}_X:P_{X/S}^1\to O_X$ which vanishes on ${\Omega }_{X/S}^1$ and which corresponds to the closed immersion ${\delta }_X:X\hookrightarrow {\Delta }_{X/S}^{(1)}$. Then, every morphism of $S$-schemes $f:X\to W$ induces a morphism of augmented O_X-algebras

f*(P¹_{W/S}) = O_X ⊕ f*(Ω¹_{W/S}) → P¹_{X/S} = O_X ⊕ Ω¹_{X/S}

that is, equivalently, a morphism of O_X-modules

$$f_{X/W/S}:f\ast ({\Omega }_{W/S}^1)\to {\Omega }_{X/S}^1,$$

cf. (EGA IV₄, 16.4.3.6) (and (16.4.18.2) for the notation $f_{X/W/S}$).

Since ${\pi }_X:{\Delta }_{X/S}^{(1)}\to X$ is affine, then, by a), ${\Delta }^{(1)}f$ is entirely determined by $f_{X/W/S}$ and, for every $X$-scheme $g:Y\to X$, the set

Hom_X(Y, Δ⁽¹⁾_{X/S}) ≃ Hom_{O_Y-alg.}(O_Y ⊕ g*(Ω¹_{X/S}), O_Y)

is identified with a subset of ${\mathrm{Hom}}_{O_Y}(g\ast ({\Omega }_{X/S}^1),O_Y)$, namely the subset

$${\widetilde{\mathrm{Hom}}}_{O_Y}(g\ast ({\Omega }_{X/S}^1),O_Y)$$

formed by the O_Y-morphisms $\psi :g\ast ({\Omega }_{X/S}^1)\to O_Y$ such that $Im(\psi )$ is an ideal of O_Y of square zero.[^N.D.E-III-12]

Consequently, applying a) to the diagram

              Δ⁽¹⁾ f
   Δ⁽¹⁾_{X/S} ────► Δ⁽¹⁾_{W/S}
       ↗
    g′│
       │
    Y ──g──► X ──f──► W

and taking into account that ${\Delta }^{(1)}f$ is the restriction to ${\Delta }_{X/S}^{(1)}$ of $f\times f$, one obtains the following commutative diagram, functorial in the $X$-scheme Y ──g──► X:

                              (g, g′) ↦ (f∘g, f∘g′)
Hom_X(Y, X ×_S X) ──────────────────────────────► Hom_W(Y, Δ⁽¹⁾_{W/S})
       ↑                                                  ↑
       │                                                  │
       │                              g′ ↦ Δ⁽¹⁾f ∘ g′
Hom_X(Y, Δ⁽¹⁾_{X/S}) ────────────────────────► Hom_W(Y, Δ⁽¹⁾_{W/S})       (0.1.7 (∗))
       │                                                  │
       ≃                                                  ≃
       ▼            ψ ↦ ψ ∘ g*(f_{X/W/S})                  ▼
Hom̃_{O_Y}(g*(Ω¹_{X/S}), O_Y) ──────────────► Hom̃_{O_Y}(g* f*(Ω¹_{W/S}), O_Y).

Remark 0.1.7.1. Let us end this paragraph with the following remark, which will be useful later (cf. 0.1.10). If we denote by $L_X$ the $X$-functor which to every $X$-scheme $g:Y\to X$ associates ${\mathrm{Hom}}_{O_Y}(g\ast ({\Omega }_{X/S}^1),O_Y)$, and $L_f:L_X\to L_W$ the morphism of functors defined above (which to every $\psi \in {\tilde{L}}_X(Y)$ associates $\psi \circ g\ast (f_{X/W/S})$), what precedes shows that we have a commutative diagram of functors:

X ×̃ X L̃_X  ◄──∼──  Δ⁽¹⁾_{X/S}  ─────► X ×_S X
    │                   │                 │
f ×̃_f│                Δ⁽¹⁾f│           f×f│
    ▼                   ▼                 ▼
W ×̃ W L̃_W  ◄──∼──  Δ⁽¹⁾_{W/S}  ─────► W ×_S W.

Theorem 0.1.8. (SGA 1, III 5.1)[^N.D.E-III-13] Let $Y$, $X$ be two $S$-schemes, $J$ a quasi-coherent ideal of O_Y of square zero, Y_J the closed subscheme of $Y$ defined by $J$, and $g_J:Y_J\to X$ an $S$-morphism.

a) The set $P(g_J)$ of $S$-morphisms $g:Y\to X$ which extend $g_J$ is either empty, or principal homogeneous under the abelian group

$${\mathrm{Hom}}_{O_{Y_J}}(g_J\ast ({\Omega }_{X/S}^1),J).$$

b) If $i:Y_0\hookrightarrow Y_J$ is the closed immersion defined by a quasi-coherent ideal $I\supset J$ such that $I\cdot J=0$, and if $g_0=g_J\circ i$, the preceding abelian group is isomorphic to

$${\mathrm{Hom}}_{O_{Y_0}}(g_0\ast ({\Omega }_{X/S}^1),J).$$

Proof. (b) follows at once from (a). Indeed, $J$, being annihilated by $I$, can be considered as an $O_{Y_0}$-module, whence, by adjunction:

Hom_{O_{Y_J}}(g_J*(Ω¹_{X/S}), J) = Hom_{O_{Y₀}}(i* g_J*(Ω¹_{X/S}), J).

To prove (a), we may assume $P(g_J)\ne \varnothing$, i.e. that there exists an $S$-morphism $g:Y\to X$ extending $g_J$. Let us denote by $j$ the immersion $Y_J\hookrightarrow Y$. Then $P(g_J)$ is the set of $S$-morphisms $g^{\prime }:Y\to X$ such that $g^{\prime }\circ j=g_J$. The datum of such a $g^{\prime }$ is equivalent to the datum of an $S$-morphism

h : Y → X ×_S X

such that $p{r}_1\circ h=g$ and $h_J=\delta \circ g_J$, where $h_J=h\circ j$ and $\delta$ is the diagonal immersion $X\hookrightarrow X{\times }_{S}X$:

                h_J = δ ∘ g_J
   X ×_S X  ◄──────────────── Y_J
    │  ▲ h                    │
pr₁│   ╲                    j │
    ▼    ╲                     ▼
    X  ◄─g───                  Y .

Since $h_J$ factors through $\delta$ and $Y$ is in the first infinitesimal neighborhood of the immersion $j:Y_J\to Y$ (since $J^2=0$), then, by functoriality (cf. EGA IV₄, 16.2.2 (i)), the $h$'s sought factor uniquely through ${\Delta }_{X/S}^{(1)}$ (cf. 0.1.7). Set

Y′ = Δ⁽¹⁾_{X/S} ×_X Y    and    Y_J′ = Y′ ×_Y Y_J = Δ⁽¹⁾_{X/S} ×_X Y_J .

Then the $h$'s sought are in bijection with the sections $u$ of $Y^{\prime }\to Y$ which extend the section $u_J=(\delta \circ g_J,id)$ of $Y_J^{\prime }\to Y_J$. On the other hand, $Y^{\prime }$ (resp. $Y_J^{\prime }$) is an affine scheme over $Y$ (resp. Y_J), corresponding to the quasi-coherent algebra

A = O_Y ⊕ g*(Ω¹_{X/S}),    resp.    A_J = A ⊗_{O_Y} O_{Y_J} = O_{Y_J} ⊕ g_J*(Ω¹_{X/S}).

Let us denote by $ϵ:A\to O_Y$ the canonical augmentation of $A$ (i.e. the morphism of O_Y-algebras $A\to O_Y$ which vanishes on $g\ast ({\Omega }_{X/S}^1)$), and likewise define ${ϵ}_J:A_J\to O_{Y_J}$. Then,

Γ(Y′/Y) ≃ Hom_{O_Y-alg.}(A, O_Y),    Γ(Y_J′/Y_J) ≃ Hom_{O_{Y_J}-alg.}(A_J, O_{Y_J})

and, via these isomorphisms, the section $u=(\delta \circ g,id)$ (resp. $u_J$) corresponds to $ϵ$ (resp. ${ϵ}_J$). Consequently, $P(g_J)$ is in bijection with the set of algebra morphisms $A\to O_Y$ which reduce to ${ϵ}_J$, and via this bijection, $g$ corresponds to $ϵ$.

Set $M=g\ast ({\Omega }_{X/S}^1)$. Then ${\mathrm{Hom}}_{O_Y-alg.}(A,O_Y)$ is identified with the set of O_Y-morphisms $\psi :M\to O_Y$ such that $Im(\psi )$ is an ideal of square zero, and we are interested in those which induce the null morphism $M\to O_{Y_J}=O_Y/J$, i.e. which send $M$ into $J$. Conversely, since $J^2=0$, every O_Y-morphism $\varphi :M\to J$ comes from a (unique) algebra morphism $A\to O_Y$, reducing to ${ϵ}_J$. Finally, we have ${\mathrm{Hom}}_{O_Y}(g\ast ({\Omega }_{X/S}^1),J)={\mathrm{Hom}}_{O_{Y_J}}(g_J\ast ({\Omega }_{X/S}^1),J)$ since $J^2=0$ (cf. the proof of (b) already seen). One thus obtains a bijection

$$P(g_J)\simeq {\mathrm{Hom}}_{O_{Y_J}}(g_J\ast ({\Omega }_{X/S}^1),J)$$

by which $g$ corresponds to the null morphism.

For every $m\in {\mathrm{Hom}}_{O_{Y_J}}(g_J\ast ({\Omega }_{X/S}^1),J)$, denote by $m\cdot g$ the element of $P(g_J)$ associated to $g$ and $m$ by the preceding bijection. We have already seen that $0\cdot g=g$; it remains to see that

m′ · (m · g) = (m + m′) · g.                            (0.1.8 (∗))

This is verified locally.[^N.D.E-III-14] Indeed, the two preceding morphisms $Y\to X$ induce the same continuous map as $g$ between the underlying topological spaces; it therefore suffices to verify that for every affine open $U=\mathrm{Spec}(A)$ of $X$ above an affine open $\mathrm{Spec}(\Lambda )$ of $S$, and every affine open $V=\mathrm{Spec}(B)$ of $g^{-1}(U)$, they induce the same morphism of $\Lambda$-algebras $A\to B$.

Let $J=\Gamma (V,J)$ and let $\varphi ,\psi$ and $\eta$ be the morphisms $A\to B$ induced by $g$, $m\cdot g$ and $m^{\prime }\cdot (m\cdot g)$ respectively; they coincide modulo $J$. One can uniquely write $\psi =\varphi +D$ (resp. $\eta =\psi +D^{\prime }$), where $D$ (resp. $D^{\prime }$) is an element of

Der_φ(A, J) = {δ ∈ Hom_Λ(A, J) | δ(ab) = φ(a) δ(b) + φ(b) δ(a)}

(resp. ${\mathrm{Der}}_{\psi }(A,J)$). But ${\mathrm{Der}}_{\varphi }(A,J)={\mathrm{Der}}_{\psi }(A,J)$ since $J^2=0$, and both are identified with

Hom_{B/J}(Ω¹_{A/Λ} ⊗_A B/J, J),

and via this identification $D$ corresponds to $m$ and $D^{\prime }$ to $m^{\prime }$. Then $\eta =\varphi +D+D^{\prime }$ and $D+D^{\prime }$ corresponds to $m+m^{\prime }$, whence the equality (∗).

Corollary 0.1.9.[^N.D.E-III-15] Let $X$ be an $S$-scheme; resume the notations of 0.1.5. Then $X$ is endowed with a (left) action of the $X^{+}$-abelian group L_X, which makes $X$ into a formally principal homogeneous object under L_X above $X^{+}$, i.e. one has an isomorphism of $X^{+}$-functors:

L_X × X ──∼──► X × X
       X⁺          X⁺

(defined set-theoretically by $(m,x)\mapsto (x,m\cdot x)$).

Proof. Let $i_0$ be the immersion $X_0\hookrightarrow X$. Note first that, since $X_0=X{\times }_S{S}_0$, one has $i_0\ast ({\Omega }_{X/S}^1)\simeq {\Omega }_{X_0/S_0}^1$ (cf. EGA IV, 16.4.5).

Let $Y$ be an $X^{+}$-scheme, given by an $S$-morphism $g_J:Y_J\to X$, and let $g_0:Y_0\to X_0$ be the morphism obtained by base change. By 0.1.8, if ${\mathrm{Hom}}_{X^{+}}(Y,X)$ is non-empty, it is a principal homogeneous set under the group

Hom_{O_{Y₀}}(g₀* i₀*(Ω¹_{X/S}), J · O_Y),

which is identified with ${\mathrm{Hom}}_{O_{Y_0}}(g_0\ast ({\Omega }_{X_0/S_0}^1),J\cdot O_Y)=L_X(Y)$. One therefore has a bijection

L_X(Y) × Hom_{X⁺}(Y, X)  ──∼──►  Hom_{X⁺}(Y, X ×_{X⁺} X)

given by $(m,g)\mapsto (g,m\cdot g)$. Let us show that this is "functorial in $Y$".

Let $f:Z\to Y$ be a morphism of $S$-schemes. It is a question of showing that the diagram below is commutative:

L_X(Y) × Hom_{X⁺}(Y, X) ──∼──► Hom_{X⁺}(Y, X ×_{X⁺} X)
   │                                       │
L_X(f) × f                             f × f
   ▼                                       ▼
L_X(Z) × Hom_{X⁺}(Z, X) ──∼──► Hom_{X⁺}(Z, X ×_{X⁺} X).

If ${\mathrm{Hom}}_{X^{+}}(Y,X)=\varnothing$, there is nothing to show. It therefore suffices to see that, for every $S$-morphism $g:Y\to X$ extending $g_J$ and every $m\in L_X(Y)$, one has:

(m · g) ∘ f = L_X(f)(m) · (g ∘ f).                       (0.1.9 (∗))

These two $S$-morphisms $Z\to X$ coincide on Z_J with $g_J\circ f_J$; in particular, they induce the same continuous map as $g\circ f$ between the underlying topological spaces. Consequently, it suffices to see that, if $z\in Z$, $y=f(z)$, $x=g(y)$, and if $A$, $B$, $C$ denote respectively the local rings $O_{X,x}$, $O_{Y,y}$, $O_{Z,z}$, then the morphisms $A\to C$ induced by $(m\cdot g)\circ f$ and $L_X(f)(m)\cdot (g\circ f)$ coincide. Denote by $s$ the image of $x$ in $S$, $\Lambda =O_{S,s}$, $J$ and $I$ the ideals of $\Lambda$ corresponding to $J$ and $I$, and let $\varphi ,\psi :A\to B$ and $\theta :B\to C$ be the morphisms of $\Lambda$-algebras induced by $g$, $m\cdot g$, and $f$. Then $m$ induces an element $D$ of

Hom_{B/IB}(Ω¹_{A/Λ} ⊗_A B/IB, JB) = Der_Λ(A, JB)

and one has $\psi =\varphi +D$; hence $(m\cdot g)\circ f$ corresponds to $\theta \circ \psi =\theta \circ \varphi +\theta \circ D$. Now, we have seen in 0.1.5 that $\theta \circ D$ is the image of $L_X(f)(m)$ in

Hom_{C/IC}(Ω¹_{A/Λ} ⊗_A C/IC, JC) = Der_Λ(A, JC);

consequently, $\theta \circ \varphi +\theta \circ D$ is the image of $L_X(f)(m)\cdot (g\circ f)$. This proves the equality (0.1.9 (∗)).

Corollary 0.1.10.[^N.D.E-III-16] a) L_X depends functorially on $X$: for every $S$-morphism $f:X\to W$, there exists an $S$-morphism $L_f:L_X\to L_W$ which is a morphism of abelian groups "above $f^{+}$", i.e., the diagram

        L_f
   L_X ────► L_W
   │           │
   ▼   f⁺      ▼
   X⁺ ──────► W⁺

is commutative, and for every $Y\to X^{+}$,

L_f(Y) : Hom_{X⁺}(Y, L_X) → Hom_{W⁺}(Y, L_W)

(where $Y$ is above $W^{+}$ via $f^{+}$) is a morphism of abelian groups.

b) Moreover, the following diagram is commutative:

L_X × X ─∼─► X ×_{X⁺} X
   │              │
L_f × f        f × f
   ▼              ▼
L_W × W ─∼─► W ×_{W⁺} W .

Proof. a) $L_f$ is induced by the morphism of $O_{X_0}$-modules $f_{X_0/W_0/S_0}:f_0\ast ({\Omega }_{W_0/S_0}^1)\to {\Omega }_{X_0/S_0}^1$ (cf. 0.1.7 b)): for every $X^{+}$-scheme $Y$, given by an $S$-morphism $g_J:Y_J\to X$, one has a commutative diagram, functorial in $Y$:

                          L_f(Y)
Hom_{O_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), J · O_Y) ───► Hom_{O_{Y₀}}(g₀* f₀*(Ω¹_{W₀/S₀}), J · O_Y)
       │                                                       │
       ▼                                                       ▼
     {g_J}                                                {f_J ∘ g_J}

where $L_f(Y)$ is the map $\psi \mapsto \psi \circ g_0\ast (f_{X_0/W_0/S_0})$, which is indeed a morphism of abelian groups.[^N.D.E-III-17]

Let us prove (b). Let $Y$ be an $X^{+}$-scheme; if ${\mathrm{Hom}}_{X^{+}}(Y,X)=\varnothing$ there is nothing to show. So let $g\in {\mathrm{Hom}}_{X^{+}}(Y,X)$; it must be seen that for every $m\in L_X(Y)$, one has:

f ∘ (m · g) = L_f(Y)(m) · (f ∘ g).                       (0.1.10 (∗))

Now, $g$ being fixed, ${\mathrm{Hom}}_X(Y,X{\times }_{X^{+}}X)$ is a subset of ${\mathrm{Hom}}_X(Y,{\Delta }_{X/S}^{(1)})$ and

L_X(Y) = Hom_{O_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), J · O_Y) = Hom_{O_Y}(g*(Ω¹_{X/S}), J · O_Y)

a subset of $L_X(Y)$ (cf. 0.1.7); finally, $L_f(Y)$ is the restriction to $L_X(Y)$ of the map $L_f(Y)$. Moreover, the bijection

L_X(Y) ──∼──► Hom_X(Y, X ×_{X⁺} X),    m ↦ (g, m · g)

is (the inverse of) the restriction to $L_X(Y)\subset L_X(Y)$ of the bijection Hom_X(Y, Δ⁽¹⁾_{X/S}) ──∼──► {g} × L̃_X(Y) considered in 0.1.7.1. Consequently, the equality (0.1.10 (∗)) results from (0.1.7 (∗)); indeed, if we denote by $g^{\prime }$ the $X$-morphism $Y\to {\Delta }_{X/S}^{(1)}$ defined by $(g,m\cdot g)$, then the element of $L_W(Y)$ corresponding to $(f\circ g,f\circ g^{\prime })$ is ${\tilde{L}}_f(Y)(m)=L_f(Y)(m)$, i.e., one indeed has

L_f(m) · (f ∘ g) = f ∘ (m · g).

Lemma 0.1.11. Let $X$, $X^{\prime }$ be two $S$-schemes. One has a commutative diagram:

L_X ×_S L_{X′} ──∼──► L_{X ×_S X′}
       │                  │
       ▼                  ▼
X⁺ ×_S X′⁺  ──∼──► (X ×_S X′)⁺ .

[^N.D.E-III-18] Proof. First, for every $S$-scheme $Y$, ${\mathrm{Hom}}_S(Y,X^{+}{\times }_S{X}^{\prime +})$ equals ${\mathrm{Hom}}_S(Y,X^{+})\times {\mathrm{Hom}}_S(Y,X^{\prime +})$ and this is isomorphic to

Hom_S(Y_J, X) × Hom_S(Y_J, X′) = Hom_S(Y, (X ×_S X′)⁺);

this proves that $X^{+}{\times }_S{X}^{\prime +}\simeq (X{\times }_S{X}^{\prime }{)}^{+}$.

Next, let $Y$ be a scheme above $X^{+}{\times }_S{X}^{\prime +}$ via a morphism $h:Y_J\to X{\times }_S{X}^{\prime }$; set $f=p\circ h$ and $g=q\circ h$, where we have denoted by p, q the projections of $X{\times }_S{X}^{\prime }$ to $X$ and $X^{\prime }$. Since ${\Omega }_{(X_0{\times }_{S_0}{X}_0^{\prime })/S_0}^1\cong p_0\ast ({\Omega }_{X_0/S_0}^1)\oplus q_0\ast ({\Omega }_{X_0^{\prime }/S_0}^1)$ (cf. EGA IV₄, 16.4.23), one obtains a natural isomorphism:

Hom_{O_{Y₀}}(f₀*(Ω¹_{X₀/S₀}), J · O_Y) × Hom_{O_{Y₀}}(g₀*(Ω¹_{X′₀/S₀}), J · O_Y)
    ≃ Hom_{O_{Y₀}}(h₀*(Ω¹_{(X₀ ×_{S₀} X′₀)/S₀}), J · O_Y)

i.e., $L_X(Y)\times L_{X^{\prime }}(Y)\simeq L_{X{\times }_S{X}^{\prime }}(Y)$.

Remark 0.1.12.[^N.D.E-III-19] Let $C$ be a category stable under fibered products, $S$ an object of $C$, $T_1$, $T_2$ two objects above $S$ and, for $i=1,2$, $L_i$ and $X_i$ two objects above $T_i$:

L_1 ──► T_1 ◄── X_1        L_2 ──► T_2 ◄── X_2
              ╲                    ╱
                ╲                ╱
                  ▼            ▼
                       S                  .

Then one has a natural isomorphism:

(L_1 ×_{T_1} X_1) ×_S (L_2 ×_{T_2} X_2) ≃ (L_1 ×_S L_2) ×_{T_1 ×_S T_2} (X_1 ×_S X_2).

Consequently, from the preceding lemma one deduces the:

Corollary 0.1.13. Let X_1, X_2 be two $S$-schemes. One has a commutative diagram of isomorphisms:

        L_{X_1 ×_S X_2} ×_{(X_1 ×_S X_2)⁺} (X_1 ×_S X_2)
              │                                ╲
   (0.1.11) ≃ ▼                                  ≃
        (L_{X_1} ×_S L_{X_2}) ×_{(X_1⁺ ×_S X_2⁺)} (X_1 ×_S X_2)
                              ──∼──►
        (L_{X_1} ×_{X_1⁺} X_1) ×_S (L_{X_2} ×_{X_2⁺} X_2) .
                                                (0.1.12)

We can now state:

Proposition 0.2. For every $S$-scheme $X$, one can define a (left) action of the $X^{+}$-abelian group L_X on the $X^{+}$-object $X$, such that:

(i) this action makes $X$ into a formally principal homogeneous object under L_X above $X^{+}$, i.e. the morphism

L_X × X ──► X × X
       X⁺          X⁺

is an isomorphism of $X^{+}$-functors;

(ii) this action is functorial in the $S$-scheme $X$, i.e., for every $S$-morphism $f:X\to W$, the following diagram is commutative:

L_X ×_{X⁺} X ─────► X
     │              │
 L_f × f          f │
     ▼              ▼
L_W ×_{W⁺} W ─────► W ;

(iii) this action "commutes with fibered product", i.e. for all $S$-schemes X_1 and X_2, the following diagram is commutative:

L_{X_1 ×_S X_2} ×_{(X_1 ×_S X_2)⁺} (X_1 ×_S X_2)  ──►  X_1 ×_S X_2
        │                                                    ▲
        ≃                                                    │
        ▼                                                    │
(L_{X_1} ×_S L_{X_2}) ×_{(X_1⁺ ×_S X_2⁺)} (X_1 ×_S X_2) ──∼──► (L_{X_1} ×_{X_1⁺} X_1) ×_S (L_{X_2} ×_{X_2⁺} X_2).

Proof.[^N.D.E-III-20] (i) and (ii) follow respectively from Corollaries 0.1.9 and 0.1.10. To prove (iii), denote $P(X)=L_X{\times }_{X^{+}}X$, for every $S$-scheme $X$. Then, by (ii) applied to the projections $p_i:X_1{\times }_S{X}_2\to X_i$, one obtains commutative squares

P(X_1 ×_S X_2)  ─────►  X_1 ×_S X_2
       │                     │
  L_{p_i} × p_i             p_i
       ▼                     ▼
   P(X_i)         ─────►   X_i

for $i=1,2$, and hence a commutative square:

P(X_1 ×_S X_2)         ─────►   X_1 ×_S X_2
       │                                ║
       ▼                                ▼
P(X_1) ×_S P(X_2)      ─────►   X_1 ×_S X_2 .

Combining this with Corollary 0.1.13, one obtains that the vertical arrow is an isomorphism, and that one has the commutative diagram announced in (iii).

Remark 0.3. Suppose the $X^{+}$-scheme $Y$ flat over $S$ (cf. SGA 1, IV). Then one can write

Hom_{X⁺}(Y, L_X) = Hom_{O_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), J ⊗_{O_{S₀}} O_{Y₀}).

Remark 0.4. Denote by ${\pi }_0:X_0\to S_0$ the structural morphism and suppose there exists an $O_{S_0}$-module ${\omega }_{X_0/S_0}^1$ such that ${\Omega }_{X_0/S_0}^1\simeq {\pi }_0\ast ({\omega }_{X_0/S_0}^1)$ (the case will arise in particular when $X_0$ is an $S_0$-group, cf. II, 4.11). If one defines a functor $L_X^{\prime }$ above $S$ by the formula

Hom_S(Y, L′_X) = Hom_{O_{Y₀}}(ω¹_{X₀/S₀} ⊗_{O_{S₀}} O_{Y₀}, J · O_Y),     (0.4.1)

one then has ${\mathrm{Hom}}_{X^{+}}(Y,L_X)={\mathrm{Hom}}_S(Y,L_X^{\prime })$ for every $X^{+}$-scheme $Y$, that is to say

L_X = L′_X × X⁺ .
              S

[^N.D.E-III-21] Then, since $L_X{\times }_{X^{+}}X=L_X^{\prime }{\times }_{S}X$, the action of L_X on $X$ induces an action of $L_X^{\prime }$ on $X$, and this action respects the morphism $p_X:X\to X^{+}$; indeed, if $Y$ is an $S$-scheme, $h:Y\to X$ an $S$-morphism and $m$ an element of $L_X^{\prime }(Y)$, then $h$ and $m\cdot h$ have the same restriction to Y_J, i.e. $p_X(m\cdot h)=p_X(h)$.

Remark 0.5. Keep the hypotheses and notations of 0.4 and suppose moreover that $Y$ is an $S$-scheme flat over $S$. Then we have

Hom_{X⁺}(Y, L_X) = Hom_S(Y, L′_X) = Hom_{S₀}(Y₀, L⁰_X),

where the $S_0$-functor in abelian groups $L_X^0$ is defined by the following identity (with respect to the variable $S_0$-scheme $T$):

Hom_{S₀}(T, L⁰_X) = Hom_{O_T}(ω¹_{X₀/S₀} ⊗_{O_{S₀}} O_T, J ⊗_{O_{S₀}} O_T).    (0.5.1)

In the notations of (II, 1), we have therefore shown that the functors $L_X^{\prime }$ and ${\prod }_{S_0/S}{L}_X^0$ have the same restriction to the full subcategory of $(Sch)/S$ whose objects are the $S$-schemes $Y$ flat over $S$.

Remark 0.6. Keep the hypotheses and notations of 0.5[^N.D.E-III-22] and suppose moreover that there exists a section ${ϵ}_0$ of ${\pi }_0:X_0\to S_0$; one then has ${\omega }_{X_0/S_0}^1\simeq {ϵ}_0\ast ({\Omega }_{X_0/S_0}^1)$.

First, one has (independently of the preceding hypothesis):

Hom_{S₀}(T, L⁰_X) = Γ(T, Hom_{O_T}(ω¹_{X₀/S₀} ⊗_{O_{S₀}} O_T, J ⊗_{O_{S₀}} O_T)).

Now suppose that ${\omega }_{X_0/S_0}^1$ admits a finite presentation (cf. EGA 0_I, 5.2.5), which will in particular be the case if $X_0$ is locally of finite presentation over $S_0$ (cf. EGA IV₄, 16.4.22). Then, if $T$ is flat over $S_0$, it follows from (EGA 0_I, 6.7.6) that

Hom_{O_T}(ω¹_{X₀/S₀} ⊗_{O_{S₀}} O_T, J ⊗_{O_{S₀}} O_T) ≃ Hom_{O_{S₀}}(ω¹_{X₀/S₀}, J) ⊗_{O_{S₀}} O_T ,

whence

Hom_{S₀}(T, L⁰_X) = Γ(T, Hom_{O_{S₀}}(ω¹_{X₀/S₀}, J) ⊗_{O_{S₀}} O_T).

Introducing the notation $W(\cdot )$ of (I, 4.6.1), we have therefore proved that for every $S_0$-scheme $T$ flat over $S_0$, one has

Hom_{S₀}(T, L⁰_X) = Hom_{S₀}(T, W(Hom_{O_{S₀}}(ω¹_{X₀/S₀}, J))).

In summary, if ${\omega }_{X_0/S_0}^1$ admits a finite presentation, and if one restricts to the category of $S$-schemes flat over $S$, one has

L′_X = ∏_{S₀/S} W(Hom_{O_{S₀}}(ω¹_{X₀/S₀}, J)),                  (0.6.1)

and ${\mathrm{Hom}}_{O_{S_0}}({\omega }_{X_0/S_0}^1,J)$ is a quasi-coherent $O_{S_0}$-module, by EGA I, 9.1.1.

Note finally that if ${\omega }_{X_0/S_0}^1$ is moreover locally free (of finite rank), for example if $X_0$ is smooth over $S_0$ (in which case it is automatically locally of finite presentation over $S_0$), one has

Hom_{O_{S₀}}(ω¹_{X₀/S₀}, J) ≃ Lie(X₀/S₀) ⊗_{O_{S₀}} J ,           (0.6.2)

where, by abuse of language ($X_0$ not being necessarily an $S_0$-group), we denote by $Lie(X_0/S_0)$ the dual of the $O_{S_0}$-module ${\omega }_{X_0/S_0}^1$.[^N.D.E-III-23]

Proposition 0.2 (and its proof) has two important corollaries.[^N.D.E-III-24]

Corollary 0.7. Let $X$ be an $S$-scheme.

a) Every $S$-endomorphism of $X$ inducing the identity on X_J is an automorphism.

b) One has an exact sequence of groups:

0 ──► Hom_{O_{X₀}}(Ω¹_{X₀/S₀}, J · O_X) ──i──► Aut_S(X) ──► Aut_{S_J}(X_J) .

c) Moreover, if one makes ${\mathrm{Aut}}_S(X)$ act on the first group by transport of structure, one has, for all $u\in {\mathrm{Aut}}_S(X)$ and $m\in {\mathrm{Hom}}_{O_{X_0}}({\Omega }_{X_0/S_0}^1,J\cdot O_X)$:

i(u · m) = u · i(m) · u⁻¹.

Proof. By 0.2 (i), ${\mathrm{Hom}}_{X^{+}}(X,X)$ is a principal homogeneous set under ${\mathrm{Hom}}_{X^{+}}(X,L_X)$, since it is certainly non-empty: it contains a marked point, namely the identity automorphism of $X$.[^N.D.E-III-25] Consequently, the map $m\mapsto m\cdot i{d}_X$ induces a bijection

Hom_{O_{X₀}}(Ω¹_{X₀/S₀}, J · O_X) = L_X(X) ──∼──► Hom_{X⁺}(X, X).

Let $m\in L_X(X)$ and let $f=m^{\prime }\cdot i{d}_X$ be an element of ${\mathrm{Hom}}_{X^{+}}(X,X)$. Applying 0.2 (ii) to $f$, one obtains:

f ∘ (m · id_X) = L_f(X)(m) · f = L_f(X)(m) · (m′ · id_X).

On the other hand, since $f$ is an $X^{+}$-endomorphism of $X$, one has $f_J=i{d}_{X_J}$ and therefore $f_0=i{d}_{X_0}$; since $L_f$ depends only on $f_0$ (cf. N.D.E. (17) in 0.1.10), one therefore has $L_f(X)(m)=m$. Consequently, the equality above rewrites as:

(m′ · id_X) ∘ (m · id_X) = m · (m′ · id_X) = (m + m′) · id_X .

This shows that the bijection $m\mapsto m\cdot i{d}_X$ transforms the group law of ${\mathrm{Hom}}_{X^{+}}(X,L_X)$ into the composition law of $X^{+}$-endomorphisms of $X$.

It follows first that every element of ${\mathrm{Hom}}_{X^{+}}(X,X)$ is invertible, which is the first assertion of the statement, and then that one has an exact sequence

0 ──► Hom_{X⁺}(X, L_X) ──i──► Aut_S(X) ──► Aut_{S_J}(X_J) ,

which is the second.

Let us now note that the morphism $i$ defined above is functorial in $X$ for isomorphisms, because it is defined in structural terms from the action of L_X on $X$ above $X^{+}$, itself functorial in $X$ by assertion (ii) of Proposition 0.2.[^N.D.E-III-26] Hence every automorphism $u$ of $X$ above $S$ induces by transport of structure isomorphisms

h : Hom_{X⁺}(X, L_X) ──∼──► Hom_{X⁺}(X, L_X)

and f : Aut_S(X) ──∼──► Aut_S(X) such that the following diagram is commutative:

            i
Hom_{X⁺}(X, L_X) ────► Aut_S(X)
      │                    │
      h                    f
      ▼     i              ▼
Hom_{X⁺}(X, L_X) ────► Aut_S(X)

i.e. such that $f\circ i=i\circ h$. On the other hand, $f$ is given by the commutative diagram

        a
   X ─────► X
   │         │
 u │         │ u
   ▼  f(a)   ▼
   X ─────► X ,

that is, $f(a)=u\circ a\circ u^{-1}$ for every $a\in {\mathrm{Aut}}_S(X)$. Writing $i(h(m))=f(i(m))$, one finds the desired formula.

Corollary 0.7.bis. Let $X$ be an $S$-scheme such that X_J is an S_J-monoid. Then L_X is endowed with a structure of $S$-monoid, one has a split exact sequence of $S$-monoids:

                i           p
1 ──► L′_X ────► L_X ⇄ ──► X⁺ ──► 1
                     s

and the monoid law induced on $L_X^{\prime }$ coincides with its abelian group structure. In particular, if X_J is an S_J-group, then L_X is an $S$-group and is the semidirect product of $X^{+}$ and $L_X^{\prime }$.

Proof. Indeed, since X_J is an S_J-monoid, then $X^{+}={\prod }_{S_J/S}{X}_J$ is an $S$-monoid (indeed, one has $X^{+}(Y)=X_J(Y_J)$ for every $Y\to S$). For every $S$-scheme $Y$, denote by $Y_J$ the Y_J-affine scheme corresponding to the quasi-coherent $O_{Y_J}$-algebra $O_{Y_J}\oplus J\cdot O_Y$ (i.e. the graded algebra associated to the filtration $O_Y\supset J\cdot O_Y$). Then $L_X(Y)$ is identified with $X_J(Y_J)$ and $L_X^{\prime }(Y)$ with the kernel of the morphism $p:X_J(Y_J)\to X_J(Y_J)$ induced by the "zero section" $Y_J\to Y_J$ (i.e. by the morphism of $O_{Y_J}$-algebras $O_{{\tilde{Y}}_J}\to O_{Y_J}$ vanishing on the ideal $J\cdot O_Y$). One has therefore, for every $Y\to S$, a split exact sequence of monoids, functorial in $Y$:

                       i              p
1 ──► L′_X(Y) ──────► L_X(Y) ⇄ ──► X⁺(Y) ──► 1 .
                            s

It remains to see that the monoid law induced on $L_X^{\prime }$ coincides with its abelian group structure. Denote by µ the monoid law of L_X and $e$ its unit section; one must show that for all $m,m^{\prime }\in L_X^{\prime }(Y)$, one has

µ(m · e, m′ · e) = (m + m′) · e .

This can be seen in either of the following ways. On the one hand, one can revisit the proof of equality (0.1.10 (∗)) by replacing the morphism $f:X\to W$ appearing there with the morphism µ : L_X ×_S L_X → L_X. Identifying $X^{+}(Y)=X_J(Y_J)$ with its image by $s$ in $L_X(Y)=X_J({\tilde{Y}}_J)$, one obtains that, for all $g,g^{\prime }\in X_J(Y_J)$ and $m,m^{\prime }\in L_X^{\prime }(Y)$, one has

µ(m · g, m′ · g′) = L_µ^{(g, g′)}(m, m′) · µ(g, g′),                      (⋆)

where L_µ^{(g, g′)} denotes the morphism derived from µ at the point $(g,g^{\prime })$ (i.e. ${\tilde{Y}}_J$ is above $L_X{\times }_S{L}_X$ via $(g,g^{\prime })$). In particular, one has µ(m · e, m′ · e) = L_µ^{(e, e)}(m, m′) · e; now L_µ^{(e, e)}(m, m′) = L_{ℓ_e}(m′) + L_{r_e}(m), where ${\ell }_e$ (resp. $r_e$) denotes left (resp. right) translation by $e$, which is the identity map of X_J, whence L_µ^{(e, e)}(m, m′) = m + m′.

Alternatively, one can proceed as follows (cf. the proof of [DG70], § II.4, Th. 3.5). By Lemma 0.1.11, the formation of $X^{+}$ and of L_X "commutes with products", and hence the same holds for $L_X^{\prime }$; it follows that the morphism µ′ : L′_X × L′_X → L′_X induced by µ is a homomorphism for the abelian-group structure. One then deduces from Lemma 3.10 of Exp. II that µ′ coincides with the abelian-group law.

0.8.[^N.D.E-III-27] Let now $X$ be an $S$-scheme such that X_J is an S_J-group. Suppose there exists an $S$-morphism

P : X × X ──► X
        S

such that the morphism obtained by base change

P_J : X_J × X_J ──► X_J
            S_J

is the group law of X_J. (An important particular case of the preceding situation will be the case where $X$ is an $S$-group and one takes for $P$ its group law.) From this one deduces a morphism

L_P : L_X × L_X ≃ L_{X ×_S X} ──► L_X
              S

which, in fact, does not depend on $P$, because it is computed by means of the group law P_J of X_J, as we shall now see.[^N.D.E-III-28] Indeed, by (ii) and (iii) of 0.2, for every $Y\to S$ and $x,x^{\prime }\in X(Y)$, $m,m^{\prime }\in L_X^{\prime }(Y)$, one has

P(m · x, m′ · x′) = L_P^{(x, x′)}(m, m′) · (x, x′) = L_P^{(x, x′)}(m, m′) · µ(g, g′)

where $g$ (resp. $g^{\prime }$) is the image of $x$ (resp. $x^{\prime }$) in $X^{+}(Y)$. Moreover (cf. the proof of 0.10), $L_P^{(x,x^{\prime })}$ equals L_µ^{(g, g′)} and, by 0.7.bis (⋆), this is the element of $L_X^{\prime }(Y)$ defined by the following equality in $L_X(Y)$:

L_µ^{(g, g′)}(m, m′) · µ(g, g′) = µ(m · g, m′ · g′),

that is, if we denote by $\times$ (instead of µ) the group law of L_X and Ad the "adjoint action" of $X^{+}$ on $L_X^{\prime }$ (which factors through $X_0$ and is induced by the adjoint action of $X_0$ on ${\omega }_{X_0/S_0}^1$), one obtains that

L_µ^{(g, g′)}(m, m′) × g × g′ = m × g × m′ × g′ = (m × Ad(g)(m′)) × g × g′

hence finally $L_P^{(x,x^{\prime })}(m,m^{\prime })=m\times Ad(g)(m^{\prime })$. We therefore obtain:

Proposition 0.8. Let $P:X{\times }_{S}X\to X$ be an $S$-morphism such that P_J endows X_J with a structure of S_J-group. Denote by $\times$ the group law of $L_X^{\prime }$ and by $(m,x)\mapsto m\cdot x$ the morphism $L_X^{\prime }{\times }_{S}X\to X$ defining the action of $L_X^{\prime }$ on $X$, and let $Ad:X^{+}\to {\mathrm{Aut}}_{S-gr.}(L_X^{\prime })$ be the "adjoint action" of $X^{+}$ on $L_X^{\prime }$ (which is induced by the adjoint action of $X_0$ on ${\omega }_{X_0/S_0}^1$). Then, for every $S^{\prime }\to S$ and $x,x^{\prime }\in X(S^{\prime })$, $m,m^{\prime }\in L_X^{\prime }(S^{\prime })$, one has:

P(m · x, m′ · x′) = (m × Ad(p_X(x))(m′)) · P(x, x′).        (0.8.1)

If $X$ is an $S$-group, we shall denote by $\ast$ its law, $e$ its unit section, and $i$ the $S$-morphism defined by:

$$i(m)=m\cdot e,$$

for every $S^{\prime }\to S$ and $m\in L_X^{\prime }(S^{\prime })$.

Corollary 0.9. Let $X$ be an $S$-group. Then $X^{+}$ is naturally endowed with a structure of $S$-group, and $p_X$ is a morphism of $S$-groups. Moreover, the $S$-morphism

i : L′_X ──► X,    m ↦ m · e

is an isomorphism of $S$-groups from $L_X^{\prime }$ onto $Ker(p_X)$, and one has, for all $S^{\prime }\to S$, $x^{\prime }\in X(S^{\prime })$, $m\in L_X^{\prime }(S^{\prime })$:

m · x′ = (m · e) ∗ x′ = i(m) ∗ x′ .                  (0.9.1)

The first two assertions have already been proved in 0.1.2. Since $X$ is formally principal homogeneous over $X^{+}$ under $L_X=L_X^{\prime }{\times }_S{X}^{+}$, the morphism $i$ is indeed an isomorphism of $S$-functors from $L_X^{\prime }$ onto the kernel of $p_X$. The fact that $i$ is a morphism of groups and the formula (0.9.1) follow from formula (0.8.1) applied respectively to $x=x^{\prime }=e$, and to $x=e$, $m^{\prime }=1$.

Corollary 0.10. Let $X$ be an $S$-group. With the preceding notations, for every $S^{\prime }\to S$ and all $x\in X(S^{\prime })$ and $m^{\prime }\in L_X^{\prime }(S^{\prime })$, one has

x ∗ i(m′) ∗ x⁻¹ = i(Ad(p_X(x))(m′)) .                 (0.10.1)

This follows from the equality $i(m^{\prime })\ast x^{-1}=m^{\prime }\cdot x^{-1}$ and from (0.8.1) applied to $m=1$ and $x^{\prime }=x^{-1}$.

When $X$ is an $S$-group, we have therefore explicitly determined the kernel of $X\to X^{+}$ and the action of the inner automorphisms of $X$ on this kernel. We shall now see that one can do the same for certain $S$-group functors not necessarily representable. One case will be useful to us, namely that of the Aut functors (I, 1.7). Let us state at once:

Proposition 0.11. Let $E$ be an $S$-scheme. Denote $X={\mathrm{Aut}}_S(E)$. The kernel of the morphism of $S$-group functors

p_X : X ──► X⁺

is canonically identified with the $S$-functor in commutative groups $L_X^{\prime }$ defined by

Hom_S(Y, L′_X) = Hom_{O_{E₀ ×_{S₀} Y₀}}(Ω¹_{E₀/S₀} ⊗_{O_{S₀}} O_{Y₀}, J · O_{E ×_S Y}),

where $Y$ denotes a variable $S$-scheme.

Indeed, if $Y$ is a variable $S$-scheme, one has ${\mathrm{Hom}}_S(Y,X)={\mathrm{Aut}}_Y(E{\times }_{S}Y)$, and

Hom_S(Y, X⁺) = Hom_S(Y_J, X) = Aut_{Y_J}(E ×_S Y_J) = Aut_{Y_J}((E ×_S Y) ×_Y Y_J).

Applying 0.7 b) to the $Y$-scheme $E{\times }_{S}Y$, one obtains an isomorphism of groups:

Hom_S(Y, L′_X) ≃ Ker(Hom_S(Y, X) ──► Hom_S(Y, X⁺)),

an isomorphism that one verifies easily to be functorial in the $S$-scheme $Y$. One thus obtains an isomorphism of $S$-groups

L′_X ≃ Ker(X ──► X⁺),

which completes the proof of Proposition 0.11.

Corollary 0.12.[^N.D.E-III-29] We keep the notations of 0.11: $E$ is an $S$-scheme and $X={\mathrm{Aut}}_S(E)$. One has a natural action $f$ of $X$ on $L_X^{\prime }$ defined as follows. For every $S$-scheme $Y$, one has

Hom_S(Y, X) = Aut_Y(E ×_S Y)
and    Hom_S(Y, L′_X) = Hom_{O_{E₀ ×_{S₀} Y₀}}(Ω¹_{E₀ ×_{S₀} Y₀ / Y₀}, J · O_{E ×_S Y})

(N. B. ${\Omega }_{E_0/S_0}^1{\otimes }_{S_0}{O}_{Y_0}\simeq {\Omega }_{E_0{\times }_{S_0}{Y}_0/Y_0}^1$, cf. EGA IV₄, 16.4.5); the first group acts on the second by transport of structure and this action is indeed functorial in $Y$. One then has the formula:

x · i(m) · x⁻¹ = i(f(x) m),                              (0.12.1)

for every $Y\to S$ and all $x\in {\mathrm{Hom}}_S(Y,X)$, $m\in {\mathrm{Hom}}_S(Y,L_X^{\prime })$.

Indeed, this follows from 0.7 c) applied to the $Y$-scheme $E{\times }_{S}Y$.

Reminder 0.13. The direct image of a quasi-coherent module by a morphism of finite presentation is quasi-coherent. Under the same conditions, the formation of the direct image commutes with flat base change: in the situation

T ◄────g′──── T′ = T ×_S S′
│                  │
f                  f′
▼      g           ▼
S ◄──────────── S′ ,

if one supposes $f$ (and therefore $f^{\prime }$) of finite presentation and $g$ (and therefore $g^{\prime }$) flat, one has for every quasi-coherent O_T-module $F$

f_*(F) ⊗_{O_S} O_{S′} = f′_*(F ⊗_{O_S} O_{S′}),

where, more elegantly,

$$g\ast (f_{\ast }(F))=f_{\ast }^{\prime }(g^{\prime }\ast (F)).$$

These two facts hold more generally for a quasi-compact and quasi-separated morphism $f$, cf. (EGA I, 9.2.1) and (EGA III₁, 1.4.15) in the quasi-compact separated case (taking into account EGA III₂, Err_III 25) and (EGA IV₁, 1.7.4 and 1.7.21).

Remark 0.14.[^N.D.E-III-30] Resume the notations of 0.11: let $E$ be an $S$-scheme, $X={\mathrm{Aut}}_S(E)$ and $L_X^{\prime }$ the $S$-functor in commutative groups defined by:

L′_X(Y) = Hom_{O_{E₀ ×_{S₀} Y₀}}(Ω¹_{E₀/S₀} ⊗_{O_{S₀}} O_{Y₀}, J · O_{E ×_S Y})
         = Hom_{O_{E ×_S Y}}(Ω¹_{E/S} ⊗_{O_S} O_Y, J · O_{E ×_S Y})
         = Γ(E ×_S Y, Hom_{O_{E ×_S Y}}(Ω¹_{E/S} ⊗_{O_S} O_Y, J · O_{E ×_S Y})).

Suppose $Y$ flat over $S$; then one has isomorphisms:

J · O_{E ×_S Y} ←─∼─ (J · O_E) ⊗_{O_S} O_Y ─∼→ (J · O_E) ⊗_{O_{S₀}} O_{Y₀} .

Suppose moreover $E$ of finite presentation over $S$; then ${\Omega }_{E/S}^1$ is an O_E-module of finite presentation (cf. EGA IV₄, 16.4.22), and hence, by (EGA 0_I, 6.7.6),

Hom_{O_{E ×_S Y}}(Ω¹_{E/S} ⊗_{O_S} O_Y, (J · O_E) ⊗_{O_S} O_Y) ≃ Hom_{O_E}(Ω¹_{E/S}, J · O_E) ⊗_{O_S} O_Y .

Denote by $\pi :E\to S$ and $g:Y\to S$ the structural morphisms; applying 0.13 to the diagram

E ◄────g′──── E ×_S Y
│                 │
π                 π′
▼      g          ▼
S ◄────────────── Y ,

and to the O_E-module $F={\mathrm{Hom}}_{O_E}({\Omega }_{E/S}^1,J\cdot O_E)$, one obtains

Γ(E ×_S Y, g′*F) = Γ(Y, π′_* g′* F) = Γ(Y, g* π_* F) = W(π_* F)(Y).

We have therefore shown that, if $E$ is of finite presentation over $S$, one has

L′_X = W(π_* Hom_{O_E}(Ω¹_{E/S}, J · O_E))                        (0.14.1)

on the category of $S$-schemes flat over $S$. Note moreover that the module of which we take the $W$ is quasi-coherent, by (EGA I, 9.1.1 and 9.2.1).

[^N.D.E-III-31] Denote by $L_0$ the $S_0$-functor

W(π_{0*} Hom_{O_{E₀}}(Ω¹_{E₀/S₀}, J · O_E)).

Then, returning to the definition of $L_X^{\prime }(Y)$ and taking into account the isomorphism

J · O_{E ×_S Y} ≃ (J · O_E) ⊗_{O_{S₀}} O_{Y₀},

one obtains, by arguing as above, that

L′_X(Y) = L₀(Y₀) = L₀(Y ×_S S₀) = ∏_{S₀/S} L₀(Y).

Hence, on the category of $S$-schemes flat over $S$, one has:

L′_X = ∏_{S₀/S} W(π_{0*} Hom_{O_{E₀}}(Ω¹_{E₀/S₀}, J · O_E)).

It is not obvious that the action of $X$ on $L_X^{\prime }$ defined in 0.12 comes from an action of $X_0={\mathrm{Aut}}_{S_0}(E_0)$ on $L_0$; this is however the case when, moreover, $E$ is flat over $S$.

Indeed, in this case one has canonical isomorphisms:

J · O_E ≃ J ⊗_{O_S} O_E ≃ J ⊗_{O_{S₀}} O_{E₀}.

L₀ ≃ W(π_{0*} Hom_{O_{E₀}}(Ω¹_{E₀/S₀}, J ⊗_{O_{S₀}} O_{E₀})),

L′_X = ∏_{S₀/S} W(π_{0*} Hom_{O_{E₀}}(Ω¹_{E₀/S₀}, J ⊗_{O_{S₀}} O_{E₀})).        (0.14.2)

Then, for every $S_0$-scheme $T$, one has

L₀(T) ≃ Hom_{O_{E₀ ×_{S₀} T}}(Ω¹_{E₀ ×_{S₀} T / T}, J ⊗_{O_{S₀}} O_{E₀ ×_{S₀} T})

and ${\mathrm{Hom}}_{S_0}(T,X_0)={\mathrm{Aut}}_T(E_0{\times }_{S_0}T)$ acts by transport of structure on $L_0(T)$, functorially in $T$; finally, for every $S$-scheme $Y$ flat over $S$, the action by transport of structure of ${\mathrm{Hom}}_S(Y,X)={\mathrm{Aut}}_Y(E{\times }_{S}Y)$ on $L_X^{\prime }(Y)=L_0(Y_0)$ factors through ${\mathrm{Aut}}_{Y_0}(E_0{\times }_{S_0}{Y}_0)$.

Let us finally extract from (SGA 1, III) the following two propositions.

Proposition 0.15. (SGA 1, III, 6.8)[^N.D.E-III-32] For every S_J-scheme $Y$ smooth over S_J and affine, there exists an $S$-scheme $X$ smooth over $S$ such that $X{\times }_S{S}_J\simeq Y$, and such an $X$ is unique up to (non-unique) isomorphism.

Proposition 0.16. (SGA 1, III, 5.5)[^N.D.E-III-33] Let $X$ be an $S$-scheme smooth over $S$. For every affine $S$-scheme $Y$, the canonical map

p_X(Y) : Hom_S(Y, X) ──► Hom_S(Y, X⁺) = Hom_{S_J}(Y_J, X_J)

is surjective.

Corollary 0.17. Let $E$ be an $S$-scheme smooth over $S$ and affine; denote $X={\mathrm{Aut}}_S(E)$. For every affine $S$-scheme $Y$, the canonical map

Aut_Y(E ×_S Y) = Hom_S(Y, X) ──► Hom_S(Y, X⁺) = Aut_{Y_J}(E_J ×_{S_J} Y_J)

is surjective.

Indeed, $Y{\times }_{S}E$ is affine over $Y$, itself affine, so affine. Applying 0.16, one deduces that every S_J-morphism $Y_J{\times }_{S_J}{E}_J\to E_J$ extends to an $S$-morphism $Y{\times }_{S}E\to E$. [^N.D.E-III-34] In other words, every Y_J-endomorphism of $Y_J{\times }_{S_J}{E}_J$ lifts to a $Y$-endomorphism of $Y{\times }_{S}E$. Then, 0.7 a) shows that every Y_J-automorphism of $Y_J{\times }_{S_J}{E}_J$ lifts to a $Y$-automorphism of $Y{\times }_{S}E$, which is the announced property.

1. Extensions and cohomology

1.1.

Let $C$ be a category stable under fibered products.[^N.D.E-III-35] Let $S$ be an object of $C$, $G$ an (representable) $S$-group, and $F$ an $S$-functor in commutative groups on which $G$ acts. We defined in (I, 5.1) the cohomology groups $H^n(G,F)$. Recall that these are the homology groups of a complex denoted $C\ast (G,F)$ where, denoting $(G/S{)}^n=G{\times }_S\cdot \cdot \cdot {\times }_{S}G$ ($n$ factors),

Cⁿ(G, F) = Hom_S((G/S)ⁿ, F).

Since $G$, and hence the $(G/S{)}^n$, are representable, one has also

$$C^n(G,F)=F((G/S{)}^n);$$

from this, and from the definition of the boundary operator, one sees that the complex $C\ast (G,F)$ depends only on the restriction of $F$ to the full subcategory of $C/S$ whose objects are the cartesian powers of $G$ over $S$. Consequently, one has the:

Lemma 1.1.1. Let $C$ be a category stable under fibered products,[^N.D.E-III-35] $S$ an object of $C$, $G$ a representable $S$-group. Denote by $C(G)$ the full subcategory of $C/S$ whose objects are the cartesian powers of $G$ over $S$. Let $F$ and $F^{\prime }$ be two $S$-functors in commutative groups on which $G$ acts. If $F$ and $F^{\prime }$ have the same restriction to $C(G)$, one has a canonical isomorphism

H∗(G, F) ──∼──► H∗(G, F′).

1.1.2. Cohomology and restriction of scalars.

[^N.D.E-III-36] Let us state another comparison result. Let now $T\to S$ be a morphism in $C$. If $F$ is a $T$-functor in commutative groups, then the functor obtained by "restriction of scalars" (cf. Exp. II, 1)

$$F_1=\prod _{T/S}F$$

is an $S$-functor in commutative groups and one has a morphism of $S$-group functors

u : ∏_{T/S} Aut_{T-gr.}(F) ──► Aut_{S-gr.}(F₁) .[^N.D.E-III-37]

Let now $G$ be an $S$-group functor and let

G_T ──► Aut_{T-gr.}(F)

be an action of G_T on $F$. By definition of the functor ${\prod }_{T/S}$, one deduces a morphism of $S$-group functors