Exposé VII_A. Infinitesimal study: differential operators and restricted $p$-Lie algebras

by P. Gabriel

In Exposé II we restricted ourselves to the study of first-order differential invariants and did not address certain phenomena specific to characteristic $p>0$ or to characteristic 0. Our object in part A of this Exposé is to fill in this gap.

Moreover, the infinitesimal study of arbitrary order of a group scheme is related to that of the associated formal group; the object of the second part of this Exposé is to present the first definitions and properties concerning formal groups.

A) Differential operators and restricted $p$-Lie algebras1

1. Differential operators

In this section, as well as in sections 2 and 3, $S$ denotes a fixed scheme and the products considered are cartesian products in the category of $S$-schemes.² If $X$ is an $S$-scheme, we write $p_{X/S}$, $p_X$ or simply $p$ for the structural morphism of $X$ into $S$.

1.1.

Let $u:Y\to X$ be a morphism of $S$-schemes and endow the direct image $u_{\ast }(O_Y)$ of the structure sheaf of $Y$ with the O_X-module structure induced by $u$. The sheaf $H={\mathrm{Hom}}_{p_X^{-1}(O_S)}(O_X,u_{\ast }(O_Y))$ of $p_X^{-1}(O_S)$-module homomorphisms from O_X into $u_{\ast }(O_Y)$ is therefore naturally equipped with a structure of O_X-bimodule: if $U$ is an open of $X$, $f$ and $d$ sections of O_X and $H$ on $U$, then fd and df are respectively the morphisms $g\mapsto fd(g)$ and $g\mapsto d(fg)$ from O_X into $u_{\ast }(O_Y)$. We shall henceforth write (ad f)(d) in place of $fd-df$.

Definition 1.1.1. An $S$-deviation of order $⩽n$ is by definition a pair $D=(u,d)$ consisting of a morphism of $S$-schemes $u:Y\to X$ and a morphism of $p_X^{-1}(O_S)$-modules $d:O_X\to u_{\ast }(O_Y)$ such that, for every open $U$ of $X$ and every sequence of $n+1$ sections $f_0,\cdots ,f_n\in O_X(U)$, one has in ${\mathrm{Hom}}_{p_U^{-1}(O_S)}(O_U,u_{\ast }(O_Y){|}_U)$ the equality:

(∗_n)                    (ad f₀)(ad f₁) ··· (ad f_n)(d) = 0.

³

In this case, we shall also say that $d$ is an $S$-deviation of $u$ of order $⩽n$. In particular, an $S$-deviation of $u$ of order $⩽0$ is a morphism of O_X-modules from O_X into $u_{\ast }(O_Y)$, i.e., an element of $\Gamma (Y,O_Y)$.

Definition 1.1.2. A morphism of $p_X^{-1}(O_S)$-modules $d:O_X\to u_{\ast }(O_Y)$ is an $S$-deviation of $u$ if, for every point $y$ of $Y$, there exist an open neighborhood $U$ of $u(y)$ in $X$ and an open neighborhood $V$ of $y$ in $Y$ satisfying the following conditions:

a) $u(V)\subset U$;

b) if $v:V\to U$ is the morphism induced by $u$, there is an integer $n$ such that the morphism $O_U\to v_{\ast }(O_V)$ induced by $d$ is an $S$-deviation of $v$ of order $⩽n$.⁴

If $d$ is an $S$-deviation of $u$, we also say that the pair $D=(u,d)$ is an $S$-deviation and it will happen that we write $Y\stackrel{D}{\to }X$ or $Y\stackrel{d}{\to }X$ (over $u$).

When $d$ is the algebra homomorphism $u^{♮}:O_X\to u_{\ast }(O_Y)$ corresponding to the morphism $u:Y\to X$, we shall also write $u$ in place of $D$.

Remarks 1.1.3.⁵ Let $D\acute{e}v(u)$ (resp. $D\acute{e}{v}_{⩽n}(u)$) be the set of $S$-deviations of $u$ (resp. $S$-deviations of $u$ of order $⩽n$). It is equipped with a natural structure of $O_Y(Y)$-module: if $\lambda \in O_Y(Y)$, $\lambda d$ is the deviation sending $f$ to $\lambda d(f)$, for every section $f$ of O_X on an open $U$.

For every open $V$ of $Y$, set $\mathcal{D}\acute{e}v(u)(V)=D\acute{e}v(u{|}_V)$, i.e., $\mathcal{D}\acute{e}v(u)(V)$ is the set of

d_V ∈ Hom_{p_X^{-1}(O_S)}(O_X, (u|_V)_*(O_V)) ≃ Hom_{p_Y^{-1}(O_S)}((u|_V)^{-1} O_X, O_V)
                                              ≃ Hom_{p_Y^{-1}(O_S)}(u^{-1} O_X, O_Y)(V)

such that, for every open $U$ of $X$, the map $d_V(U):O_X(U)\to O_Y(u^{-1}(U)\cap V)$ satisfies $({\ast }_n)$. This defines a presheaf of O_Y-modules on $Y$, and one sees easily that it is a sheaf (more precisely, a subsheaf of ${\mathrm{Hom}}_{p_Y^{-1}(O_S)}(u^{-1}{O}_X,O_Y)$).

1.2.

Consider now two $S$-deviations $D=(u,d)$ and $E=(v,e)$:

Z ─v,e→ Y ─u,d→ X.

When $U$ ranges over the opens of $X$, the composed maps

Γ(U, O_X) ─d(U)→ Γ(u^{-1} U, O_Y) ─e(u^{-1} U)→ Γ(v^{-1} u^{-1} U, O_Z)

define an $S$-deviation of uv which we shall denote de; when $d$ is of order $⩽m$ and $e$ of order $⩽n$, de is of order $⩽m+n$. We shall also write

(†)                              D ∘ E = (uv, de)

⁶

and we shall say that $D\circ E$ or DE is the composed $S$-deviation. When $d=u^{♮}$ (i.e., $D=u$ with the convention of 1.1), one also says that DE is the image of $E$ by $u$.

The map $(D,E)\mapsto D\circ E$ we have just defined will henceforth allow us to speak of the category of $S$-deviations, whose objects are the $S$-schemes and whose morphisms are the $S$-deviations.⁷

Definition 1.2.0.⁸ Let $w:Z\to X$ be an $S$-morphism. An $S$-derivation of $w$, or $S$-derivation of O_X into $w_{\ast }(O_Z)$, is a morphism of $p^{-1}(O_S)$-modules $d:O_X\to w_{\ast }(O_Z)$ such that, for every open $U$ of $X$ and $f,g\in O_X(U)$,

d(fg) = w^♮(f) d(g) + w^♮(g) d(f).

Then $d$ is a deviation of $w$ of order $⩽1$ which vanishes on the unit section of O_X. We denote by $D\acute{e}{r}_S(w)$ the set of $S$-derivations of $w$; it is an $O(Z)$-module.

With the notations of 1.2, take $Y$ equal to $I_Z=\mathrm{Spec}{O}_Z[t]$, where $t^2=0$, and $v$ equal to the zero section $\tau :Z\to I_Z$, defined by the morphism of O_Z-algebras $O_Z[t]\to O_Z$ sending $t$ to 0, and take $e$ equal to the morphism of O_Z-modules $\sigma :O_Z[t]\to O_Z$ defined by $\sigma (1)=0$ and $\sigma (t)=1$,⁹ which it is convenient to denote ${\partial }_t$.

If $u:I_Z\to X$ is a morphism satisfying $w=u\circ \tau$, then $\sigma \circ u^{♮}$ is an $S$-derivation of O_X into $w_{\ast }(O_Z)$. Conversely, to every $S$-derivation $d$ we associate the morphism $u:I_Z\to X$ such that $u=w$ on the underlying spaces, and

u^♮(f) = w^♮(f) + d(f) t,

for every section $f$ of O_X on an open $U$. One thus obtains:

Lemma 1.2.1. Let $E=(\tau ,{\partial }_t)$ be the deviation of $\tau :Z\to I_Z$ defined above. For every $S$-morphism $w:Z\to X$, the map $u\mapsto u\circ E$ is a bijection between the $S$-morphisms $u:I_Z\to X$ such that $u\circ \tau =w$, and the $S$-derivations of $w$.

1.2.2.

Let $d$ be an $S$-deviation of $u:Y\to X$. On the one hand, $d$ is obviously an $S^{\prime }$-deviation of $u$ for every morphism $s:S\to S^{\prime }$.

On the other hand, let $t:T\to S$ be a morphism with target $S$, and let $u_T:Y_T\to X_T$ be the morphism deduced from $u$ by base change, and $t_Y:Y_T\to Y$ and $t_X:X_T\to X$ the canonical projections. Then there exists one and only one $T$-deviation of $u_T$, which we shall denote $d_T$ or $d\times T$, satisfying the equality $t_X{d}_T=d{t}_Y$, in the sense of (†) above, i.e., for every open $U$ of $X$, one has a commutative diagram:¹⁰

                 t_X^♮
   O(U)  ─────────────────→  O(U × T)

  d(U)                              d_T(U × T)

                 t_Y^♮
   O(u^{-1} U)  ────────→  O(u^{-1} U × T).

If one sets $D=(u,d)$, one will also write $D_T=(u_T,d_T)$ and we shall say that $d_T$ and D_T are deduced from $d$ and $D$ by base change.

1.2.3.

For example, let $u:Y\to X$ and $v:Z\to T$ be two $S$-morphisms, $d$ and $e$ $S$-deviations of $u$ and $v$. One has a commutative diagram

            u_T
X × T  ←─────────  Y × T
  ↑                  ↑
v_X │  u×v        v_Y │
  │                  │
            u_Z
X × Z  ←─────────  Y × Z

and we shall denote by $d\times e$ (the product of $d$ and $e$) the $S$-deviation of $u\times v$ equal to $d_T{e}_Y=e_X{d}_Z$ (with the convention (†) above), i.e., for every open $U$ of $X\times T$, if we denote

                       d_T(U)
       O(U)  ──────────────────→  O(u_T^{-1} U)
         ╲                              ↑
   e_X(U) ╲    (d × e)(U)               │ e_Y(u_T^{-1} U)
           ╲                            │
                d_Z(v_X^{-1} U)
       O(v_X^{-1} U)  ────────────→  O(W).

If one sets $D=(u,d)$ and $E=(v,e)$, we shall also write $D\times E=(u\times v,d\times e)$.

1.3.

¹¹ Let $u:Y\to X$ be a morphism of $S$-schemes. Recall that the adjunction isomorphism

Hom_{p_X^{-1}(O_S)}(O_X, u_*(O_Y)) ⥲ Hom_{p_Y^{-1}(O_S)}(u^{-1}(O_X), O_Y)

associates with every morphism of $p^{-1}(O_S)$-modules $d:O_X\to u_{\ast }(O_Y)$ the morphism $d^{\prime }=ϵ\circ u^{-1}(d)$, where $ϵ$ is the canonical morphism $u^{-1}{u}_{\ast }(O_Y)\to O_Y$.

Let us write $J_u$ (resp. $I_u$) for the kernel of the algebra homomorphism $u^{♮}:O_X\to u_{\ast }(O_Y)$ (resp. $u^{{♮}^{\prime }}:u^{-1}(O_X)\to O_Y$), and let $d:O_X\to u_{\ast }(O_Y)$ be a morphism of $p^{-1}(O_S)$-modules. If $U$ is an open of $X$ and $f_0,\cdots ,f_n,g\in O_X(U)$, one sees easily by induction on $n$ that the condition $({\ast }_n)$ is equivalent to the following equality (cf. EGA IV₄, 16.8.8.2):

(∗∗_n)                     0 = ∑_{I ⊂ ⟦0,n⟧} (−1)^{|I|} u^♮(f_{⟦0,n⟧ − I}) d(f_I g),

where $f_I$ denotes the product of the $f_i$, for $i\in I$. It follows that if $d$ satisfies $({\ast }_n)$, then $d$ vanishes on the ideal $J_u^{n+1}$.

Suppose now $Y$ equal to $S$; then $u:S\to X$ is a section of $p:X\to S$, hence is an immersion (cf. EGA I, 5.3.13). Then, on the one hand, $ϵ:u^{-1}{u}_{\ast }{O}_S\to O_S$ is an isomorphism, so that $u^{-1}(J_u)=I_u$. On the other hand, one has an isomorphism:

$$(\star )u^{-1}(O_X)\simeq O_S\oplus I_u.$$

Suppose that $d$ vanishes on $J_u^{n+1}$. Then $d^{\prime }=ϵ\circ u^{-1}(d)$ vanishes on $I_u^{n+1}$ and hence $d^{\prime }$ satisfies the analogues $(\ast {\ast }_n^{\prime })$ and $({\ast }_n^{\prime })$ of $(\ast {\ast }_n)$ and $({\ast }_n)$, when $f_0,\cdots ,f_n\in I_u(u^{-1}(U))$. Moreover, since $(ada)(\varphi )=0$ for every $a\in O_S(u^{-1}(U))$ and every morphism of $O_{u^{-1}(U)}$-modules $\varphi :u^{-1}(O_U)\to O_{u^{-1}(U)}$, one deduces from $(\star )$ that $d^{\prime }$ satisfies the analogue $({\ast }_n^{\prime })$ of $({\ast }_n)$. It follows that $d$ satisfies $({\ast }_n)$. Consequently, one has obtained:

Lemma. If $u:S\to X$ is a section of $p:X\to S$, then $d$ is an $S$-deviation of $u$ of order $⩽n$ if and only if $d^{\prime }$ vanishes on $I_u^{n+1}$.

This interpretation may be generalized as follows. Let $u:Y\to X$ be an arbitrary $S$-morphism and ${\Gamma }_u$ the graph of $u$, i.e., the morphism $Y\to Y\times X$ with components $i{d}_Y$ and $u$. For every $S$-deviation $d$ of $u$ of order $⩽n$, one obtains by composition

Y ──diag.→ Y × Y ──d_Y→ Y × X (over u_Y)

a $Y$-deviation of ${\Gamma }_u$ of order $⩽n$ which we shall denote ${\Gamma }_d$ (the graph of $d$).

Conversely, to every $Y$-deviation $e$ of ${\Gamma }_u$ one associates the composed $S$-deviation $e_X=p{r}_2\circ e$:

Y ──e→ Y × X ──pr₂→ X (over Γ_u).

One sees at once that $({\Gamma }_d{)}_X=d$, and the equality ${\Gamma }_{e_X}=e$ follows from the fact that $e$ is O_Y- linear.¹² One thus obtains an isomorphism of $O_Y(Y)$-modules:

{ S-deviations of u of order ⩽ n }  ⥲  { Y-deviations of Γ_u of order ⩽ n }
                                 d ↦ Γ_d.

Moreover, one sees easily that $d$ is an $S$-derivation of $u$ if and only if ${\Gamma }_d$ is a $Y$-derivation of ${\Gamma }_u$.

Let us call $I_{{\Gamma }_u}$ the kernel of the algebra homomorphism $({\Gamma }_u{)}^{-1}(O_{Y\times X})\to O_Y$ corresponding to ${\Gamma }_u$. Taking into account the preceding lemma, one has obtained:

Proposition. Let $u:Y\to X$ be an $S$-morphism and ${\Gamma }_u:Y\to Y\times X$ its graph. The $S$-deviations of $u$ of order $⩽n$ are identified with the $Y$-deviations of ${\Gamma }_u$ of order $⩽n$, which are in bijection with

Hom_{O_Y}((Γ_u)^{-1}(O_{Y × X}) / I_{Γ_u}^{n+1}, O_Y).

1.3.1.

¹³ Let us return to the case where $u:S\to X$ is a section of $p:X\to S$. Then the homomorphism $\varphi :u^{-1}(O_X)\to O_S$ admits a section, which we shall denote simply $g\mapsto g\cdot 1$, so that, with the notations of 1.3, one has an isomorphism of O_S-modules:

$$(\star )u^{-1}(O_X)\simeq O_S\oplus I_u,$$

and for every section $f$ of $u^{-1}(O_X)$, $f-\varphi (f)\cdot 1$ is a section of $I_u$.

Let $d$ be an $S$-deviation of $u$ of order $⩽1$, and $d^{\prime }$ the O_S-morphism $u^{-1}(O_X)\to O_S$ corresponding to $d$. If a, b are sections of $u^{-1}(O_X)$, one has:

0 = d'((a − φ(a) · 1)(b − φ(b) · 1)) = d'(ab) − φ(a) d'(b) − φ(b) d'(a) + φ(ab) d'(1).

Consequently, one sees that $d$ is an $S$-derivation of $u$ (cf. 1.2.1 and N.D.E. (2)) if and only if $d^{\prime }(1)=0$. One thus obtains:

Lemma. The $S$-derivations of $u$ are exactly the $S$-deviations of $u$ of order 1 which vanish on the unit section of O_X; they correspond to the $O_S(S)$-module

$${\mathrm{Hom}}_{O_S}(I_u/I_u^2,O_S),$$

and one has an isomorphism of $O_S(S)$-modules $D\acute{e}{v}_{⩽1}(u)\simeq O_S(S)\oplus D\acute{e}{r}_S(u)$.

Returning to the general case, one deduces, with the notations of 1.3,

Corollary. Let $u:Y\to X$ be an $S$-morphism and ${\Gamma }_u:Y\to Y\times X$ its graph. One has a canonical isomorphism of $O_Y(Y)$-modules

Dér_S(u) ≃ Dér_Y(Γ_u) ≃ Hom_{O_Y}(I_{Γ_u} / I_{Γ_u}², O_Y).

Definition 1.4.

Let $X$ be an $S$-scheme. We call $S$-differential operator (resp. $S$-differential operator of order $⩽n$) on $X$ any $S$-deviation (resp. any $S$-deviation of order $⩽n$) of the identity morphism of $X$.

According to 1.1, an $S$-differential operator of order $⩽n$ is therefore an endomorphism of $p^{-1}(O_S)$-module of O_X which satisfies the equalities $({\ast }_n)$ of 1.1. We shall denote by $Di{f}_{X/S}^n$ the $\Gamma (O_S)$-module ¹⁴ formed by the $S$-differential operators of order $⩽n$, and by $Di{f}_{X/S}$ that formed by all the $S$-differential operators.

As we saw in 1.2, one can compose $S$-deviations of $i{d}_X$, which equips $Di{f}_{X/S}$ with a structure of $\Gamma (O_S)$-algebra; we shall say that this is the algebra of differential operators of $X/S$.

Similarly, for every open $V$ of $X$, set $\mathcal{D}i{f}_{X/S}(V)=Di{f}_{V/S}=D\acute{e}v(i{d}_V)$; according to 1.1.3, this defines a sheaf of O_X-modules, called the sheaf of $S$-differential operators on $X$.¹⁵

1.4.1.

As we saw in 1.3, one can interpret the differential operators of $X/S$ by means of the graph of the identity morphism of $X$, i.e., of the diagonal morphism ∆ = ∆_{X/S} of $X$ into $X\times X$. Let us translate into the present context the statements of 1.3.

Endow $O_{X\times X}$ with the $p{r}_1^{-1}(O_X)$-algebra structure defined by $p{r}_1$, so that ∆^{-1}(O_{X × X}) is equipped with a structure of algebra over O_X = ∆^{-1} pr₁^{-1}(O_X). Let $I_{X/S}$ be the kernel of the homomorphism

∆^{-1}(O_{X × X}) ──m→ O_X

adjoint to the homomorphism O_{X × X} → ∆_*(O_X), and let $P_{X/S}^m$ be the O_X-algebra

∆^{-1}(O_{X × X}) / I_{X/S}^{m+1}.

If $V$ is an affine open of $S$ and $U$ an affine open of $X$ above $V$, and if one sets

Γ(U, P^m_{X/S}) = (A ⊗_k A) / I^{m+1},

where $I$ is the ideal generated by the elements $a\otimes 1-1\otimes a$, for $a\in A$. This being so, one has according to 1.3 an isomorphism of $O_X(X)$-modules:

j_X : Dif^m_{X/S} ⥲ Hom_{O_X}(P^m_{X/S}, O_X)

which one can define as follows: if $d$ belongs to $Di{f}_{X/S}^m$ and if $c$ is a section of $P_{X/S}^m$ on $U$ of the form $a\otimes b+I^{m+1}$, one has $j_X(d)(c)=a\cdot d(b)$.¹⁶

1.4.2.

Let $d$ be a differential operator and $u$ a section of $X$ over $S$. We call value of $d$ at $u$ the composed $S$-deviation

S ──u→ X ──d→ X (over id_X).

According to 1.3 and 1.4.1, if $d$ is a differential operator of order $⩽m$, then du (resp. $d$) is canonically associated with a morphism of O_S-modules $d^{\prime }:u^{-1}(O_X)/I_u^{m+1}\to O_S$ (resp. a morphism of O_X-modules $d^{\prime \prime }:P_{X/S}^m\to O_X$).

It is clear that one can construct $d^{\prime }$ from d'' as follows: the square

                 u × X
   X ≃ S ×_X X  ──────→  X × X
       │                   │
       p                   pr₁
       │           u       │
       ▼                   ▼
       S          ───────→ X

is cartesian, which allows us to identify $X$ with $S{\times }_X(X\times X)$, $u$ with S ×_X ∆, hence $u^{\ast }(P_{X/S}^m)$ with $u^{-1}(O_X)/I_u^{m+1}$. One thus identifies $u^{\ast }(d^{\prime \prime })$ with a morphism $u^{-1}(O_X)/I_u^{m+1}\to O_S$, which is none other than $d^{\prime }$.

1.5.

Set as usual $I_S=\mathrm{Spec}{O}_S[T]/(T^2)$. Let $\tau :S\to I_S$ be the zero section and $\sigma$ the canonical deviation of $\tau$ which we defined in 1.2.0, i.e. the homomorphism of O_S-modules which vanishes on the unit section of $O_S[T]/(T^2)$ and which sends the class $t$ of $T$ modulo $T^2$ to the unit section of O_S.

Let $X$ be an $S$-scheme. To every I_S-automorphism $u$ of $I_S\times X$ inducing the identity on $X$ there is associated by composition a differential operator $D_u$ of $X$:

X ≃ S × X ──σ × X→ I_S × X ──u→ I_S × X ──pr₂→ X.

According to II, 3.14, the map $u\mapsto D_u$ is an isomorphism of the $\Gamma (O_S)$-Lie algebra

Lie(Aut X) := Lie(Aut X)(S)

onto the $\Gamma (O_S)$-Lie algebra of $p^{-1}(O_S)$-derivations of O_X. The inverse isomorphism associates with every derivation $D$ the automorphism of $I_S\times X$ corresponding to the automorphism $a+bt\mapsto a+(Da+b)t$ of $O_X[T]/(T^2)$.

2. Invariant differential operators on group schemes

2.1.

Let $G$ be an $S$-group scheme; we denote by $ϵ$ or ${ϵ}_G:S\to G$ the unit section of $G$.

Definition. Let $U(G)$ be the $\Gamma (O_S)$-module of $S$-deviations of ${ϵ}_G$ (or $S$-deviations of the origin) (cf. 1.1).

If $d$ and $e$ are two elements of $U(G)$, $d\times e$ is an $S$-deviation of $ϵ\times ϵ:S\simeq S\times S\to G\times G$. The image of $d\times e$ by the multiplication morphism $m:G\times G\to G$ (cf. 1.2) will be called the product of $d$ and $e$ and will be denoted $d\cdot e$.

The $\Gamma (O_S)$-module $U(G)$ is thus equipped with a structure of associative $\Gamma (O_S)$-algebra having ${ϵ}_G$ as unit element (1.1). We shall say that $U(G)$ is the infinitesimal algebra of $G$.¹⁷

When $T$ ranges over the schemes above $S$, the infinitesimal algebra $U(G_T)$ of the $T$-group $G\times T$ obviously varies contravariantly in $T$, so that we may speak of the infinitesimal algebra functor.

When $T$ ranges over the opens of $S$, one therefore obtains a presheaf $T\mapsto U(G_T)$ of O_S-algebras; moreover, according to 1.1.3, this is a sheaf. We shall denote it $\mathcal{U}(G)$ and we shall call it the sheaf of infinitesimal algebras of $G$.

The algebra $U(G)$ is also a covariant functor in $G$. Indeed, if $u:G\to H$ is a homomorphism of $S$-groups and $d$ an $S$-deviation of ${ϵ}_G$, the image of $d$ by $u$ is an element $U(u)(d)=ud$ of $U(H)$. The map $U(u):U(G)\to U(H)$ thus defined is obviously a homomorphism of $\Gamma (O_S)$-algebras. One defines similarly a homomorphism $\mathcal{U}(u)$ from $\mathcal{U}(G)$ to $\mathcal{U}(H)$.

2.2.

Let $d$ be an element of $U(G)$, i.e., an $S$-deviation of the origin of $G$. Consider the $S$-deviation $d\times G$ of $ϵ\times G:G\simeq S\times G\to G\times G$ obtained from $d$ by base change (1.2.2); the image of $d\times G$ by the multiplication morphism $m:G\times G\to G$ is an $S$-deviation of $m\circ (ϵ\times i{d}_G)=i{d}_G$, i.e., an element of $Di{f}_{G/S}$, which we shall denote $d_G$.

The map $d\mapsto d_G$ is obviously $\Gamma (O_S)$-linear and the "commutative" diagram below shows that one has $(e\cdot d{)}_G=d_G\cdot e_G$:¹⁸

                                 m × G
   G × G × G  ─────────────────────────────→  G × G
       △                                       △
       │ G × d × G                              │ G × m
       │ ε × G × G                              │

  G × G                              G × G                     m
   △  △                              △  △
   │  │ e × G   m                   │  │ d × G   m
   │  │ ε × G                       │  │ ε × G
   │                                │
   │           e_G                  │           d_G
   G  ─────────────→  G  ─────────────→  G       (over id_G everywhere).

The commutativity of the two bottom triangles follows from the definition of $d_G$ and $e_G$; on the other hand, the composed $S$-deviation of $e\times G$ and $G\times d\times G$ is $(e\times d)\times G$ (cf. 1.2.2), its image by $m\times G$ is $(e\cdot d)\times G$, and the image of the latter by $m$ is therefore equal to $(e\cdot d{)}_G$.

One thus obtains an anti-homomorphism $U(G)\to Di{f}_{G/S}$ of $\Gamma (O_S)$-algebras, called right translation.¹⁹

If $\mathcal{D}i{f}_{G/S}$ denotes the sheaf of $S$-differential operators on $G$ (cf. 1.4) and $p$ the structural morphism $G\to S$, one defines similarly a "right translation": $\mathcal{U}(G)\to p_{\ast }(\mathcal{D}i{f}_{G/S})$.

2.3.

We shall now characterize the differential operators of $G$ over $S$ of the form $d_G$. Let $g:S\to G$ be a section of the structural morphism of $G$ and $g_G$ the right translation of $G$ by $g$, i.e., the composed morphism:

g_G : G ≃ G × S ──G × g→ G × G ──m→ G.

For every differential operator $D$ of $G$ over $S$, the composition $g_G^{-1}D{g}_G$ (cf. 1.2) is again an $S$-deviation of $i{d}_G$, i.e., an element of $Di{f}_{X/S}$; we shall denote:

D^g = g_G^{-1} D g_G.

We shall say that $D$ is right-invariant if, for every base change $t:T\to S$ and every section $g:T\to G\times T$, one has $(D_T{)}^g=D_T$.

Lemma. For every differential operator $D$ of $G$ over $S$, the following assertions are equivalent (where $m$ is the multiplication morphism of $G$):

(i) $D$ is right-invariant.

(ii) The two following deviations of $m$ are equal: $Dm=m(D\times G)$.

(ii) ⇒ (i): since the condition (ii) is stable under base change, it suffices to show that (ii) entails the equality $D^g=D$ for every section $g:S\to G$. Let $h$ be the morphism $G\times g:G\simeq G\times S\to G\times G$, so that $m\circ h$ is the right translation $g_G$. The equality $D^g=D$ is equivalent to the equality $g_G\circ D=D\circ g_G$, and this follows from the commutative diagram:

                    m                     h
       G  ←──────────  G × G  ←──────────  G

   D, id_G        D × G, id_{G × G}      D, id_G

                    m                     h
       G  ←──────────  G × G  ←──────────  G.

(i) ⇒ (ii): take indeed for $t:T\to S$ the structural morphism $p:G\to S$, for section $g:T\to G\times T$ the diagonal morphism ∆ : G → G × G. The right translation

∆_{G × G} : G × G  ⟶  G × G

is then the morphism from $G\times G$ into $G\times G$ with components $m$ and $p{r}_2$. The equality (D_G)^∆ = D_G is then equivalent to the commutativity of the first square of the following diagram:

                  ∆_{G × G}                  pr₁
       G × G  ──────────────→  G × G  ──────────→  G

   D_G, id_{G × G}          D_G, id_{G × G}    D, id_G

                  ∆_{G × G}                  pr₁
       G × G  ──────────────→  G × G  ──────────→  G.

The equality (ii) thus follows from the fact that m = pr₁ ∘ ∆_{G × G}.

Consider for example an element $d$ of the infinitesimal algebra $U(G)$. The squares of the diagram

                          d × G × G                            m × G
       G × G  ←────  S × G × G  ──────────────→  G × G × G  ──────────→  G × G
                          ε × G × G

         m                    S × m                  G × m                  m

                          d × G                                m
         G  ←────  S × G  ──────────────→  G × G  ──────────→  G
                          ε × G

are then commutative. Since one has

m ∘ (d × G) = d_G    and    (m × G) ∘ (d × G × G) = d_G × G,

one also has $d_G\circ m=m\circ (d_G\times G)$. Therefore: for every $S$-deviation $d$ of the origin, $d_G$ is a right-invariant differential operator.

2.4. Theorem.

(i) The map $d\mapsto d_G$ is an anti-isomorphism²⁰ of the infinitesimal algebra $U(G)$ onto the subalgebra $Di{f}_{G/S}^G$ of $Di{f}_{G/S}$ formed by the right-invariant differential operators.

(ii) Similarly, the map $d\mapsto {}_{G}d$ is an isomorphism of $U(G)$ onto the subalgebra of $Di{f}_{G/S}$ formed by the left-invariant differential operators.

Let in fact $D$ be an arbitrary differential operator of $G$ over $S$ and let us denote by D_0 its value at the origin, i.e., the composed deviation $S\stackrel{ϵ}{\to }G\stackrel{D}{\to }G$ (over $i{d}_G$). The right-invariant differential operator $(D_0{)}_G$ is then obtained by composition:

G ≃ S × G ──ε × G→ G × G ──D × G→ G × G ──m→ G (over id_{G × G}).

If $D$ is right-invariant, one has $Dm=m(D\times G)$, whence

D = D m(ε × G) = m(D × G)(ε × G) = (D_0)_G.

In particular, the map $d\mapsto d_G$ is surjective.

Conversely, let $d$ be an $S$-deviation of the origin. One then has a commutative square

                          d × G
       G × G  ←──────────  G
         △                  △
   G × ε │                  │ ε
                       d
       G × S ≃ G  ←──────  S

whence it follows that $d=m(G\times ϵ)d=m(d\times G)ϵ=(d_G{)}_0$. A fortiori, the map $d\mapsto d_G$ is injective. This proves the theorem.

When $S$ varies, Theorem 2.4 obviously implies that the right translation $\mathcal{U}(G)\to p_{\ast }(\mathcal{D}i{f}_{G/S})$ is an anti-isomorphism of O_S-algebras of $\mathcal{U}(G)$ onto the sheaf of O_S-algebras $p_{\ast }(\mathcal{D}i{f}_{G/S}{)}^G$, which to every open $U$ of $S$ associates $Di{f}_{G_U/U}^{G_U}$.

2.4.1. Remark.

Consider the commutative diagram

                       η
        G  ←──────────  G × G
        △               △    △
     p  │   ε        pr₁ │    │ ∆
        │       p        │
        S  ←──────────  G,

where $\eta$ denotes the morphism "$(x,y)\mapsto y{x}^{-1}$".²¹ The latter induces morphisms

η' : η^{-1}(O_G) ⟶ O_{G × G}    and    ∆^{-1}(η') : p^{-1} ε^{-1}(O_G) ⟶ ∆^{-1}(O_{G × G}).

For every integer $n⩾1$, set $p_{G/S}^n={ϵ}^{-1}(O_G)/I_{ϵ}^{n+1}$ (cf. 1.3 and 1.4 for the notations).²² Since the square formed by the morphisms $ϵ$, $\eta$, ∆ and $p$ is cartesian, ∆^{-1}(η') induces an isomorphism of O_G-modules:

$$p^{\ast }(p_{G/S}^n)\stackrel{\sim }{\to }{P}_{G/S}^n.$$

The differential operators of $G$ over $S$ of order $⩽n$ therefore correspond bijectively to the morphisms of O_G-modules $p^{\ast }(p_{G/S}^n)⟶O_G$, i.e., to the morphisms of O_S-modules

$$p_{G/S}^n⟶p_{\ast }(O_G).$$

In this bijection, the right-invariant differential operators correspond to the composed arrows

p^n_{G/S} ──→ O_S ──can.→ p_*(O_G).

One thus recovers the isomorphism of Theorem 2.4.

2.5.

²³ Let $Lie(G)$ be the Lie algebra of $G$;²⁴ we shall define a morphism of $\Gamma (O_S)$-Lie algebras $\alpha :Lie(G)\to U(G)$.

Let $s:S\to I_S$ be the zero section of $I_S\to S$ and $\sigma$ the deviation of $s$ defined in 1.2.0. Recall (cf. II, 4.1) that $Lie(G)$ is the set of morphisms $x:I_S\to G$ such that $x\circ s={ϵ}_G$. Then the composition

S ──σ→ I_S ──x→ G (over s)

is an $S$-deviation of ${ϵ}_G$, i.e., an element of $U(G)$; with the notations of 1.2 (†), it is denoted $\sigma x$. Moreover, according to 1.2.1, the map $\alpha :x\mapsto \sigma x$ is an isomorphism of $O_S(S)$-modules from $Lie(G)$ onto the submodule $D\acute{e}r({ϵ}_G)$ of $U(G)$ formed by the $S$-derivations of ${ϵ}_G$. We shall see that $\alpha$ is a morphism of Lie algebras.²⁵ Let

$${\rho }^{\prime }:U(G)⟶Di{f}_{G/S}$$

be the algebra morphism which to an $S$-deviation $d$ of ${ϵ}_G$ associates the left-invariant differential operator ${}_{G}d\in Di{f}_{G/S}$, cf. 2.2, N.D.E. (17).

Let $\rho :G\to {\mathrm{Aut}}_S(G)$ be the homomorphism of group functors which to an $S$-morphism $g:T\to G$ associates the right translation of G_T by $g$, i.e. the morphism:

G_T ≃ T ×_T G_T ──G_T × g→ G_T ×_T G_T ──m_T→ G_T.

Recall also (cf. 1.5 and II, 3.14) that Lie(Aut G) = Lie(Aut_S(G)/S)(S) is identified with the infinitesimal automorphisms of $G$, i.e., with the automorphisms of $I_S\times G$ inducing the identity on $G$.

Since $\rho$ is a monomorphism, the same holds for the morphism Lie(ρ) : Lie(G/S) → Lie(Aut_S(G)/S) (see, for example, Exp. II, N.D.E. (50)), hence Lie(ρ) : Lie(G) → Lie(Aut G) is injective.

On the other hand, according to 1.5, the map $\beta$ which to every infinitesimal automorphism $u$ of $G$ associates the differential operator $D_u$ of $G$:

G ≃ S × G ──σ × G→ I_S × G ──u→ I_S × G ──pr₂→ G

is an isomorphism of $Lie(\mathrm{Aut}G)$ onto the Lie subalgebra of $Di{f}_{G/S}$ formed by the $p^{-1}(O_S)$-derivations of O_G.

For every $x\in Lie(G)$, one has the commutative square below which determines the image of $x$ by $Lie(\rho )$:

                       Lie(ρ)(x)
       I_S × G  ───────────────→  I_S × G

   x × G                                pr₂
                          m
       G × G  ──────────────────→  G.

Taking this diagram into account, the image of $Lie(\rho )(x)$ by $\beta$ is the composed deviation

G ≃ S × G ──σ × G→ I_S × G ──G × x→ G × G ──m→ G (over s × G)

which, according to 2.2 N.D.E. (17), is none other than ${}_G(\sigma x)={\rho }^{\prime }(\alpha (x))$. One thus obtains a commutative diagram:

                       Lie(ρ)
       Lie(G)  ─────────────────→  Lie(Aut G)
          │                              │
        α │                              │ β
          │              ρ'              │
          ▼                              ▼
       U(G)  ──────────────────────────→  Dif_{G/S}

where $Lie(\rho )$, $\beta$ and ${\rho }^{\prime }$ are morphisms of Lie algebras. Since ${\rho }^{\prime }$ is injective, it follows that $\alpha$ is also a morphism of Lie algebras. Consequently, one has obtained:

Proposition. $\alpha$ is an isomorphism of $O_S(S)$-Lie algebras, from $Lie(G)$ into the Lie algebra of $S$-derivations of ${ϵ}_G$, itself isomorphic via $Lie(\rho )$ to the Lie algebra of $S$-derivations of $G$ invariant under left translation.²⁶

3. Coalgebras and Cartier duality

3.1.

Let $S$ be a scheme (or, more generally, a ringed space). An O_S-coalgebra²⁷ is a pair (𝒰, ∆_𝒰) consisting of an O_S-module $\mathcal{U}$ and a morphism of O_S-modules ∆_𝒰 : 𝒰 → 𝒰 ⊗_{O_S} 𝒰 (called the diagonal morphism or comultiplication) such that:

(i) σ ∘ ∆_𝒰 = ∆_𝒰, where $\sigma (a\otimes b)=b\otimes a$.

(ii) The square

                              ∆_𝒰
       𝒰  ──────────────────────→  𝒰 ⊗_{O_S} 𝒰

   ∆_𝒰                                    id_𝒰 ⊗ ∆_𝒰

                          ∆_𝒰 ⊗ id_𝒰
       𝒰 ⊗_{O_S} 𝒰  ──────────────────→  𝒰 ⊗_{O_S} 𝒰 ⊗_{O_S} 𝒰

is commutative.

(iii) There exists a morphism of O_S-modules ${ϵ}_{\mathcal{U}}:\mathcal{U}\to O_S$, called the augmentation, such that the composed morphisms

𝒰 ──∆_𝒰→ 𝒰 ⊗_{O_S} 𝒰 ──id_𝒰 ⊗ ε_𝒰→ 𝒰 ⊗_{O_S} O_S ≃ 𝒰
𝒰 ──∆_𝒰→ 𝒰 ⊗_{O_S} 𝒰 ──ε_𝒰 ⊗ id_𝒰→ O_S ⊗_{O_S} 𝒰 ≃ 𝒰

are the identity morphism of $\mathcal{U}$.

If ${ϵ}_{\mathcal{U}}$ and ${ϵ}_{\mathcal{U}}^{\prime }$ are two augmentations, one has ε_𝒰 = (ε_𝒰 ⊗ ε'_𝒰) ∘ ∆_𝒰 = ε'_𝒰; the augmentation is therefore uniquely determined by (iii).

If (𝒰, ∆_𝒰) and (𝒱, ∆_𝒱) are two O_S-coalgebras, a morphism from the first into the second is a morphism of O_S-modules $f:\mathcal{U}\to \mathcal{V}$ such that the diagrams

        f                              f
   𝒰  ────→  𝒱                    𝒰  ────→  𝒱
   │          │                    │          │
 ∆_𝒰         ∆_𝒱      and        ε_𝒰        ε_𝒱
   ▼          ▼                    │          │
                                   ▼          ▼
   𝒰 ⊗ 𝒰  ─f ⊗ f→  𝒱 ⊗ 𝒱                  O_S

are commutative. Morphisms of coalgebras compose like morphisms of O_S-modules, so that we shall be able to speak of the category of O_S-coalgebras.

3.1.0.

²⁸ This category has finite products: the final object is the O_S-module O_S, the comultiplication being the identity; the product of two coalgebras (𝒰, ∆_𝒰) and (𝒱, ∆_𝒱) is the tensor product $\mathcal{U}{\otimes }_{O_S}\mathcal{V}$, the comultiplication being the composed morphism

𝒰 ⊗ 𝒱 ──∆_𝒰 ⊗ ∆_𝒱→ 𝒰 ⊗ 𝒰 ⊗ 𝒱 ⊗ 𝒱 ──id_𝒰 ⊗ σ ⊗ id_𝒱→ 𝒰 ⊗ 𝒱 ⊗ 𝒰 ⊗ 𝒱

where $\sigma (a\otimes b)=b\otimes a$; the canonical projections of $\mathcal{U}\otimes \mathcal{V}$ onto the factors $\mathcal{U}$ and $\mathcal{V}$ are the morphisms $i{d}_{\mathcal{U}}\otimes {ϵ}_{\mathcal{V}}$ and ${ϵ}_{\mathcal{U}}\otimes i{d}_{\mathcal{V}}$,²⁹ and the "diagonal morphism" $\mathcal{U}\to \mathcal{U}\otimes \mathcal{U}$ (corresponding to the pair of morphisms $(i{d}_{\mathcal{U}},i{d}_{\mathcal{U}})$) is none other than the comultiplication ∆_𝒰.

3.1.1.

Let $\mathcal{A}$ be a commutative O_S-algebra, locally free and of finite type as an O_S-module. If we set

$${\mathcal{A}}^{\ast }={\mathrm{Hom}}_{O_S-Mod.}(\mathcal{A},O_S),$$

the canonical morphism $\varphi$ from ${\mathcal{A}}^{\ast }{\otimes }_{O_S}{\mathcal{A}}^{\ast }$ into $(\mathcal{A}{\otimes }_{O_S}\mathcal{A}{)}^{\ast }$ is invertible. If $m:\mathcal{A}\otimes \mathcal{A}\to \mathcal{A}$ is the morphism defining the multiplication of $\mathcal{A}$, one obtains by composition a diagonal morphism

∆_{𝒜^*} : 𝒜^* ──m^*→ (𝒜 ⊗ 𝒜)^* ──φ^{-1}→ 𝒜^* ⊗ 𝒜^*.

This diagonal morphism obviously makes ${\mathcal{A}}^{\ast }$ an O_S-coalgebra whose augmentation is the

it is clear that:

The functor $\mathcal{A}\mapsto {\mathcal{A}}^{\ast }$ is an anti-equivalence of the category of O_S-algebras which are locally free and of finite type as O_S-modules, onto the category of O_S-coalgebras locally free and of finite type as O_S-modules.

3.1.2.

To every O_S-coalgebra $\mathcal{U}$ is canonically associated an $S$-functor

Spec^* 𝒰 : (Sch/S)° ⟶ (Ens).

Note indeed that, for every $S$-scheme $q:T\to S$, $q^{\ast }(\mathcal{U}{\otimes }_{O_S}\mathcal{U})$ is identified with $q^{\ast }(\mathcal{U}){\otimes }_{O_T}{q}^{\ast }(\mathcal{U})$, so that q^*(∆_𝒰) makes ${\mathcal{U}}_T=q^{\ast }(\mathcal{U})$ into an O_T-coalgebra; we can therefore set by definition and with an obvious abuse of notation:³⁰

(Spec^* 𝒰)(T) = { x ∈ Γ(T, 𝒰_T) | ε_{𝒰_T}(x) = 1 and ∆_{𝒰_T}(x) = x ⊗ x }.

The sections $x$ of ${\mathcal{U}}_T$ obviously correspond to the morphisms of O_T-modules $\xi :O_T\to {\mathcal{U}}_T$; the conditions $ϵ(x)=1$ and ∆(x) = x ⊗ x simply express that $\xi$ is a morphism of coalgebras. One therefore also has:

(Spec^* 𝒰)(T) = Hom_{O_T-coalg.}(O_T, 𝒰_T).

In particular, one has the following proposition:³¹

Proposition 3.1.2.1. Let $\mathcal{A}$ be a commutative O_S-algebra which is locally free of finite type as an O_S-module. Then the $S$-functor ${\mathrm{Spec}}^{\ast }{\mathcal{A}}^{\ast }$ is represented by $\mathrm{Spec}\mathcal{A}$.

Indeed, for every $S$-scheme $T$, one has canonical isomorphisms:

(Spec^* 𝒜^*)(T) = Hom_{O_T-coalg.}(O_T, 𝒜_T^*) ≃ Hom_{O_T-alg.}(𝒜_T, O_T) ≃ (Spec 𝒜)(T).

3.2.

An O_S-coalgebra in groups (i.e., a group in the category of O_S-coalgebras) consists of the data of an O_S-coalgebra (𝒰, ∆_𝒰) and three morphisms of O_S-coalgebras $m_{\mathcal{U}}:\mathcal{U}\otimes \mathcal{U}\to \mathcal{U}$, ${\eta }_{\mathcal{U}}:O_S\to \mathcal{U}$ and $c_{\mathcal{U}}:\mathcal{U}\to \mathcal{U}$ satisfying the conditions (ii), (iii) and (vi) below; on the other hand, the fact that $m_{\mathcal{U}}$ is a morphism of cogebras translates into the commutativity of diagrams (iv) and (v)

                              m_𝒰
       𝒰 ⊗ 𝒰  ────────────────────→  𝒰

   ∆_𝒰 ⊗ ∆_𝒰
       │
(iv)   ▼                                  ∆_𝒰
   𝒰 ⊗ 𝒰 ⊗ 𝒰 ⊗ 𝒰

   id_𝒰 ⊗ σ ⊗ id_𝒰
                              m_𝒰 ⊗ m_𝒰
   𝒰 ⊗ 𝒰 ⊗ 𝒰 ⊗ 𝒰  ──────────────────→  𝒰 ⊗ 𝒰

                              m_𝒰
       𝒰 ⊗ 𝒰  ────────────────────→  𝒰
              ╲                    ╱
(v)            ╲                  ╱
        ε_𝒰 ⊗ ε_𝒰              ε_𝒰
                  ╲          ╱
                       O_S

(ii)* The square

                              id_𝒰 ⊗ m_𝒰
       𝒰 ⊗ 𝒰 ⊗ 𝒰  ──────────────────→  𝒰 ⊗ 𝒰

   m_𝒰 ⊗ id_𝒰                                 m_𝒰

                              m_𝒰
       𝒰 ⊗ 𝒰  ────────────────────→  𝒰

is commutative.

(iii)* The two compositions below equal the identity morphism of $\mathcal{U}$:

𝒰 ≃ 𝒰 ⊗ O_S ──id_𝒰 ⊗ η_𝒰→ 𝒰 ⊗ 𝒰 ──m_𝒰→ 𝒰
𝒰 ≃ O_S ⊗ 𝒰 ──η_𝒰 ⊗ id_𝒰→ 𝒰 ⊗ 𝒰 ──m_𝒰→ 𝒰.

(vi) The composed morphism below is equal to ${\eta }_{\mathcal{U}}\circ {ϵ}_{\mathcal{U}}$:

𝒰 ──∆_𝒰→ 𝒰 ⊗ 𝒰 ──c_𝒰 ⊗ id_𝒰→ 𝒰 ⊗ 𝒰 ──m_𝒰→ 𝒰.

3.2.1.

The morphisms ${\eta }_{\mathcal{U}}$ and $c_{\mathcal{U}}$ are uniquely determined by $m_{\mathcal{U}}$. On the other hand, conditions (ii)* and (iii)* simply express that $m_{\mathcal{U}}$ makes $\mathcal{U}$ an O_S-algebra having as unit section the image by ${\eta }_{\mathcal{U}}$ of the unit section of O_S. Condition (iv) also expresses that the diagonal morphism ∆_𝒰 is compatible with multiplication; and indeed, ∆_𝒰 : 𝒰 → 𝒰 ⊗ 𝒰 must be a homomorphism of group coalgebras, which also entails the commutativity of the triangle

                       O_S
                  ╱         ╲
             η_𝒰              η_𝒰 ⊗ η_𝒰
(v)*            ╱               ╲
              ╱                   ╲
       𝒰  ───────────────────→  𝒰 ⊗ 𝒰.
                       ∆_𝒰

On the other hand, as in every category, the antipode $c_{\mathcal{U}}$ is an isomorphism of $\mathcal{U}$ onto the "opposite" group object;³² in particular, $c_{\mathcal{U}}$ induces an algebra isomorphism of $\mathcal{U}$ onto the opposite algebra ${\mathcal{U}}^{\circ }$.

3.2.2.

Since the functor $\mathcal{U}\mapsto {\mathrm{Spec}}^{\ast }\mathcal{U}$ commutes with finite products, it transforms a group coalgebra into a group $S$-functor; and indeed, for every $S$-scheme $T$, the elements $x\in \Gamma (T,{\mathcal{U}}_T)$ belonging to $({\mathrm{Spec}}^{\ast }\mathcal{U})(T)$ form a group under the multiplication of the algebra $\Gamma (T,{\mathcal{U}}_T)$; the inverse of $x$ is none other than $c_{\mathcal{U}}(x)$. According to 3.1.2.1, one has:

Scholium 3.2.2.1.³³ Let $\mathcal{U}$ be an O_S-coalgebra in groups, finite and locally free as an O_S-module. Then the group $S$-functor ${\mathrm{Spec}}^{\ast }\mathcal{U}$ is represented by the $S$-group, finite and locally free, $\mathrm{Spec}{\mathcal{U}}^{\ast }$.

Remark 3.2.2.2. Let $\mathcal{L}$ be an O_S-Lie algebra and $\mathcal{U}(\mathcal{L})$ the enveloping algebra of $\mathcal{L}$, i.e., the sheaf on $S$ associated with the presheaf which to every open $V$ assigns the enveloping algebra $U(\Gamma (V,\mathcal{L}))$ of the Lie algebra $\Gamma (V,\mathcal{L})$.

Every homomorphism from $\mathcal{L}$ into the Lie algebra underlying an associative O_S-algebra factors in one and only one way through the canonical morphism from $\mathcal{L}$ into $\mathcal{U}(\mathcal{L})$; moreover, this universal property entails, besides the functoriality of $\mathcal{U}(\mathcal{L})$ in $\mathcal{L}$, that the enveloping algebra of a product of Lie algebras is identified with the tensor product of the enveloping algebras.

In particular, the diagonal morphism $\delta :\mathcal{L}\to \mathcal{L}\times \mathcal{L}$ induces an algebra homomorphism ∆ : 𝒰(ℒ) → 𝒰(ℒ × ℒ) ≃ 𝒰(ℒ) ⊗ 𝒰(ℒ). The zero morphism $\mathcal{L}\to 0$ induces a homomorphism $ϵ:\mathcal{U}(\mathcal{L})\to \mathcal{U}(0)\simeq O_S$. The isomorphism $x\mapsto -x$ of $\mathcal{L}$ onto the opposite Lie algebra ${\mathcal{L}}^{\circ }$ induces an anti-isomorphism $c$ of the algebra $\mathcal{U}(\mathcal{L})$. One then verifies easily that the multiplication $m$ of the algebra $\mathcal{U}(\mathcal{L})$ makes (𝒰(ℒ), ∆) an O_S-coalgebra in groups with $ϵ$ as augmentation and $c$ as antipode.³⁴

3.2.3.

³⁵ Let $\mathcal{U}$ be an O_S-coalgebra in groups. We shall see that the group $S$-functor $G={\mathrm{Spec}}^{\ast }\mathcal{U}$ is very good, in the sense of II, 4.6 and 4.10.

Let $M$ be a free O_S-module of rank $r$, and let $T\to S$ be an $S$-scheme. Since $I_T(M)=\mathrm{Spec}(O_T\oplus M_T)$, so that $\pi :I_T(M)\to T$ is affine, one has

π_*(𝒰_{I_T(M)}) = 𝒰_T ⊗_{O_T} π_*(O_{I_T(M)}) = 𝒰_T ⊗_{O_T} (O_T ⊕ M_T),

and so

(1)    Γ(I_T(M), 𝒰_{I_T(M)}) ≃ Γ(T, 𝒰_T) ⊗_{O(T)} (O(T) ⊕ Γ(T, M_T)).

Let $(d_1,\cdots ,d_r)$ be a basis of $M$. Then an element $u_0+{\sum }_i{u}_i{d}_i$ of $\Gamma (I_T(M),{\mathcal{U}}_{I_T(M)})$ belongs to $G(I_T(M))$ if and only if one has:

1 = ε(u_0 + ∑_i u_i d_i) = ε(u_0) + ∑_i ε(u_i) d_i,

and

(u_0 + ∑_i u_i d_i) ⊗ (u_0 + ∑_i u_i d_i) = ∆(u_0 + ∑_i u_i d_i) = ∆(u_0) + ∑_i ∆(u_i) d_i,

that is to say:

(2)    { ε(u_0) = 1,  ∆ u_0 = u_0 ⊗ u_0,    (i.e. u_0 ∈ G(T))
       { ε(u_i) = 0,  ∆(u_i) = u_i ⊗ u_0 + u_0 ⊗ u_i,  for i = 1, …, r.

Moreover, the morphism $G(I_T(M))\to G(T)$ corresponding to the zero section of $I_T(M)\to T$ is given by $u_0+{\sum }_i{u}_i{d}_i\mapsto u_0$. From this, combined with (1) and (2), one deduces that, if $N$ is a second free O_S-module of finite rank, the diagram of sets

       G(I_T(M ⊕ N))          ─────→  G(I_T(N))
           │                              │
           ▼                              ▼
       G(I_T(M))               ─────→  G(T)

is cartesian, i.e. $G$ satisfies condition (E) of II, 3.5.

Let us denote by $Prim\Gamma (T,{\mathcal{U}}_T)$ the sub-$O(T)$-module of $\Gamma (T,{\mathcal{U}}_T)$ formed by the primitive elements, i.e., the elements $u$ which satisfy (with the abuse of notation signaled in 3.1.2):

∆u = u ⊗ 1 + 1 ⊗ u,    ε(u) = 0.

³⁶

Since (Lie G)(T) is the set of elements $u_0+ud\in G(I_T)$ above the unit element $u_0=1$ of $G(T)$, one obtains an isomorphism of $O(T)$-modules, functorial in $T$:³⁷

(Lie G)(T) ≃ Prim Γ(T, 𝒰_T).

On the other hand, one deduces from (1) that

Prim Γ(I_T(M), 𝒰_{I_T(M)}) ≃ Prim Γ(T, 𝒰_T) ⊗_{O(T)} O(I_T(M)),

and it follows that the natural morphism of $O(I_T(M))$-modules:

(Lie G)(T) ⊗_{O(T)} O(I_T(M)) ⟶ (Lie G)(I_T(M))

is an isomorphism, i.e. Lie G is a good O_S-module (cf. II, Déf. 4.4).

So $G$ is a good group $S$-functor (cf. II, Déf. 4.6), and according to II, 4.7.2, Lie G is equipped with an O_S-bilinear "Lie bracket" satisfying the Jacobi identity. It remains to show that $G$ is very good, i.e., that the "bracket" on (Lie G)(T) satisfies $[u,u]=0$ for every $u\in (LieG)(T)$ (cf. II, 4.10).

Let u, v be two elements of (Lie G)(T), i.e., two primitive elements of $\Gamma (T,{\mathcal{U}}_T)$. Set $I=\mathrm{Spec}{O}_S[d]/(d^2)$ and $I^{\prime }=\mathrm{Spec}{O}_S[d^{\prime }]/(d^{\prime 2})$. Since the composition law of $G(I\times I^{\prime })$ is induced by the multiplication of the algebra ${\mathcal{U}}_{I\times I^{\prime }}$, one has in $G(I\times I^{\prime })$ the equality:

(1 + ud)(1 + vd')(1 + ud)^{-1}(1 + vd')^{-1} = (1 + ud)(1 + vd')(1 − ud)(1 − vd')
                                              = 1 + (uv − vu) dd'

According to the description of the bracket [u, v] given before Prop. 4.8 of Exp. II, one obtains that

[u, v] = uv − vu,

where the right-hand term is the commutator of $u$ and $v$ in the algebra $\Gamma (T,{\mathcal{U}}_T)$, whence $[u,u]=0$. One has thus obtained the following proposition:³⁸

Proposition. Let $\mathcal{U}$ be an O_S-coalgebra in groups. The group $S$-functor $G={\mathrm{Spec}}^{\ast }\mathcal{U}$ is very good, and one has an isomorphism $LieG\simeq Prim\mathcal{W}(\mathcal{U})$ of O_S-Lie algebras, where $Prim\mathcal{W}(\mathcal{U})$ denotes the functor which to every $T\to S$ associates the $O(T)$-Lie algebra formed by the primitive elements of $\mathcal{W}(\mathcal{U})(T)=\Gamma (T,{\mathcal{U}}_T)$.

3.3.

Suppose finally that $\mathcal{U}$ is a commutative group coalgebra, i.e., the triangle

                              σ
       𝒰 ⊗ 𝒰  ────────────────────→  𝒰 ⊗ 𝒰
              ╲                    ╱
(i)*           ╲                  ╱
              m_𝒰              m_𝒰
                  ╲          ╱
                       𝒰

is commutative, or in other words that $m_{\mathcal{U}}$ makes $\mathcal{U}$ a commutative O_S-algebra. Conditions (i), (ii), (iii), (iv), (v), (vi), (i), (ii), (iii)* and (v)* then also signify that $\mathcal{U}$ is a cogroup in the category of commutative O_S-algebras. Therefore, if moreover $\mathcal{U}$ is a quasi-coherent O_S-module, then the affine $S$-scheme $\mathrm{Spec}\mathcal{U}$ is a commutative $S$-group scheme.

In this case, since the diagonal morphism ∆' of $O_S[T,T^{-1}]$ sends $T$ to $T\otimes T$, the morphisms of $S$-groups from $\mathrm{Spec}\mathcal{U}$ into ${\mathbb{G}}_{m,S}$ (I 4.3.2) correspond bijectively to the morphisms of unital O_S-algebras

φ : O_S[T, T^{-1}] ⟶ 𝒰

such that (φ ⊗ φ) ∘ ∆' = ∆_𝒰 ∘ φ (in this case, ${ϵ}_{\mathcal{U}}\circ \varphi$ is the neutral element of ${\mathbb{G}}_{m,S}(S)$, i.e., the augmentation ${ϵ}^{\prime }$). Such a morphism $\varphi$ is determined by the image $\varphi (T)$, which must be an invertible element $x$ of $\mathcal{U}$ satisfying ∆_𝒰 x = x ⊗ x and ${ϵ}_{\mathcal{U}}(x)={ϵ}^{\prime }(T)=1$. One therefore has:

Hom_{S-gr.}(Spec 𝒰, 𝔾_{m, S}) ≃ (Spec^* 𝒰)(S)

and since this formula remains valid after any base change, this gives:

Spec^* 𝒰 ≃ Hom_{S-gr.}(Spec 𝒰, 𝔾_{m, S}).

One has therefore obtained the

Proposition 3.3.0. If $\mathcal{U}$ is an O_S-coalgebra in commutative groups, quasi-coherent as an O_S-module, then the affine $S$-scheme $G=\mathrm{Spec}\mathcal{U}$ is a commutative $S$-group scheme, and one has an isomorphism of group $S$-functors

Spec^* 𝒰 ≃ Hom_{S-gr.}(G, 𝔾_{m, S}).

If one supposes moreover that $\mathcal{U}$ is a locally free O_S-module of finite type, then, according to 3.1.2.1, the group $S$-functor ${\mathrm{Spec}}^{\ast }\mathcal{U}$ is represented by $\mathrm{Spec}{\mathcal{U}}^{\ast }$. One thus obtains the

Proposition 3.3.1 (Cartier duality). The functor

𝒜(G) ↦ 𝒜(G)^* = Hom_{O_S-Mod.}(𝒜(G), O_S)

induces a duality $(\ast )$ of the category of commutative, finite and locally free $S$-group schemes; it associates with $G$ the $S$-group ${\mathrm{Hom}}_{S-gr.}(G,{\mathbb{G}}_{m,S})$.

4. "Frobeniuseries"

Let $p$ be a fixed prime number and $(Sch/{\mathbb{F}}_p)$ the category of schemes of characteristic $p$, i.e., of schemes above the prime field ${\mathbb{F}}_p$. Following the general conventions of this Seminar, we identify $(Sch/{\mathbb{F}}_p)$ with a subcategory of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$ by means of the functor $h$ of I 1.1. We likewise take advantage of the isomorphism from $\mathrm{Hom}(h_X,F)$ onto $F(X)$ defined in I 1.1 to identify these two sets whenever $X$ is an ${\mathbb{F}}_p$-scheme and $F$ an object of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$.

Notations 4.0.³⁹ If $T$ is an ${\mathbb{F}}_p$-scheme, a $T$-functor is a morphism $q:F\to T$ of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$ having $T$ as target; for every $T$-scheme $r:X\to T$, the set of $T$-morphisms $X\to F$, i.e., of ${\mathbb{F}}_p$-morphisms $s:X\to F$ such that $q\circ s=r$, will then be denoted $q(r)$, $q(X/T)$, $F(r)$ or $F(X/T)$ (or even $F(X)$ when no confusion will be possible with $\mathrm{Hom}(h_X,F)$).

4.1.

For every scheme $S$ of characteristic $p$, we denote by $fr(S)$, or simply fr, the endomorphism of $S$ which induces the identity on the underlying topological space of $S$ and which associates $x^p$ with a section $x$ of O_S on an open $U$.

Then the map $fr:S\mapsto fr(S)$ is an endomorphism of the identity functor of $(Sch/{\mathbb{F}}_p)$,⁴⁰ which implies the following results. Let $E$ be an ${\mathbb{F}}_p$-functor, i.e., an object of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$; the map which to every ${\mathbb{F}}_p$-scheme $S$ associates the endomorphism $E(fr(S))$ of $E(S)$ is a functorial endomorphism of $E$ which we shall denote $fr(E)$ or fr; this notation is compatible with the identification of $(Sch/{\mathbb{F}}_p)$ with a subcategory of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$. Moreover, the map $E\mapsto fr(E)$ is an endomorphism of the identity functor of $(Sch/{\mathbb{F}}_p{)}^{\wedge }$ (which we shall again denote fr).⁴¹

For every ${\mathbb{F}}_p$-scheme $S$ and every $S$-functor $q:X\to S$, we denote by $X^{(p/S)}$ or $X^{(p)}$ the inverse image of $X$ by the base change $fr(S)$:

                  pr_X
       X^{(p/S)}  ──────→  X
           │                  │ q
           ▼                  ▼
                fr(S)
            S  ──────────→  S.

The commutative square

                       fr(X)
              X  ──────────→  X
              │                  │
            q │                  │ q
              ▼       fr(S)       ▼
              S  ──────────→  S

induces an $S$-morphism, denoted $Fr(X/S)$ (or simply Fr), from $X$ into $X^{(p/S)}$ such that $fr(X)=p{r}_X\circ Fr(X/S)$:

              X
              │ ╲
              │  ╲  Fr(X/S)
              │   ╲
              │    ╲             fr(X)
              │     X^{(p/S)}  ──pr_X→  X
              │       │                  │
            q │       │ q^{(p/S)}        │ q
              ▼       ▼       fr(S)       ▼
              S       S       ─────→     S.

We shall say that $Fr(X/S)$ is the Frobenius morphism of $X$ relative to $S$; it is clear that the map $Fr:X\mapsto Fr(X/S)$ is a functorial homomorphism.

⁴² Let $r:T\to S$ be an $S$-scheme. For every $\varphi \in X(r)={\mathrm{Hom}}_S(T,X)$ (cf. 4.0), one has a commutative diagram:

                  Fr(X/S)              pr_X
       X  ──────────────→  X^{(p/S)}  ──────→  X
        ╲                         │                  │
         ╲ φ                      │ q^{(p/S)}        │ q
       r  ╲                       │                  │
            ╲                     ▼      fr(S)       ▼
              T  ──────────────→  S  ──────────→  S.

According to the definition of $X^{(p/S)}$ as fibered product, $p{r}_X$ induces a bijection:

X^{(p/S)}(r) = Hom_S(T, X^{(p/S)})  ⥲  Hom_S(T, X) = X(fr(S) ∘ r).

On the other hand, $r\circ fr(T)=fr(S)\circ r$, since fr is an endomorphism of the identity functor. It follows that the map $Fr(X/S)(r):X(r)\to X^{(p/S)}(r)$ can be characterized by the commutativity of the following square:

                  Fr(X/S)(r)
       X(r)  ────────────────→  X^{(p/S)}(r)

(†)   X(fr(T))                       ≀
                                          
       X(r ∘ fr(T))           X(fr(S) ∘ r).

For example, if $X$ is the subscheme of $S$ defined by a quasi-coherent ideal $\mathcal{I}$, then $X^{(p)}$ is the subscheme of $S$ defined by the ideal ${\mathcal{I}}^p$ generated by the $p$-th powers of the sections of $\mathcal{I}$; moreover, $Fr(X/S)$ is then the canonical immersion of $\mathrm{Spec}(O_X/\mathcal{I})$ into $\mathrm{Spec}(O/{\mathcal{I}}^p)$.

4.1.1.

⁴³ Let $t:T\to S$ be a base change and $X_T=X{\times }_{q,t}T$. Consider the inverse image of X_T by $fr(T)$:

       (X_T)^{(p/T)}  ────→  X_T  ─────→  X
                                              │ q
                  fr(T)              t
              T  ──────→  T  ──→  S.

Since $t\circ fr(T)=fr(S)\circ t$, then $(X_T{)}^{(p/T)}$ is identified with the inverse image of $X^{(p/S)}$ by $t$; in other words, one has a canonical isomorphism:

$$X_T^{(p/T)}\stackrel{\sim }{\to }(X^{(p/S)}{)}_T.$$

It is clear that, in this identification, $Fr(X_T/T)$ is identified with the inverse image $Fr(X/S{)}_T$ of $Fr(X/S)$.

4.1.1.1.

In particular, if $S$ is the spectrum of the prime field ${\mathbb{F}}_p$, $X^{(p/S)}$ is equal to $X$ and $Fr(X/S)$ to $fr(X)$. Consequently, $X_T^{(p/T)}$ is identified with X_T and $Fr(X_T/T)$ with $fr(X{)}_T$.

For example, if $E$ is a set and E_T the constant $T$-scheme of type $E$, one has $E_T^{(p/T)}\simeq E_T$ and $Fr(E_T/T)\simeq i{d}_{E_T}$.

4.1.2.

The functor $X\mapsto X^{(p/S)}$ obviously commutes with products; it therefore transforms an $S$-group $G$ into an $S$-group $G^{(p/S)}$; moreover, since Fr is a functorial homomorphism, then

$$Fr(G/S):G⟶G^{(p/S)}$$

is a homomorphism of $S$-groups. We shall denote Fr G its kernel.

If $r:T\to S$ is a scheme above $S$, it follows from the diagram (†) of 4.1 that the value of Fr G at $r$ is the kernel of the homomorphism

G(fr(T)) : G(r) ⟶ G(r ∘ fr(T)).

Now, when $T$ is the scheme I_R of dual numbers over an $S$-scheme $R$, $fr(I_R)$ factors as follows:

I_R  ──can.→  R  ──fr(R)→  R  ──s→  I_R,

where $s$ is the zero section. It follows that $(FrG)(I_R)$ contains the kernel $Lie(G/S)(R)$ of the morphism $G(s):G(I_R)\to G(R)$, and that one therefore has: $Lie(G/S)=Lie(FrG/S)$.

4.1.3.

More generally, for every $S$-functor $X$, we define the $S$-functor $X^{(p^n)}$ by recursion on $n$ by means of the formulas:

X^{(p)} = X^{(p/S)}    and    X^{(p^n)} = (X^{(p^{n-1})})^{(p)}.

Similarly, $F{r}^n(X/S)$ or $F{r}^n$ denote the composed functorial homomorphism

X ──Fr(X/S)→ X^{(p)} ──Fr(X^{(p)}/S)→ X^{(p²)} ──→ ··· ──→ X^{(p^{n-1})} ──Fr(X^{(p^{n-1})}/S)→ X^{(p^n)}.

One will note that, according to 4.1.1, $Fr(X^{(p)}/S)$ coincides with $Fr(X/S{)}^{(p)}$, i.e., the following diagram is commutative:

       X^{(p)}  ────→  X

   Fr(X^{(p)}/S)         Fr(X/S)

       X^{(p²)}  ────→  X^{(p)}.

If $G$ is a group $S$-functor, $G^{(p^n)}$ is also one and $F{r}^n(G/S)$ is a homomorphism of group $S$-functors.

Definition. We shall denote by $F{r}^{n}G$ the kernel of $F{r}^n(G/S)$ and we shall say that $G$ is of height $⩽n$ if $F{r}^n(G/S)$ is zero, i.e., if $F{r}^{n}G=G$.