Éléments de Géométrie Algébrique — English

§16. Differential invariants. Differentially smooth morphisms

In this section we present, in global form, certain notions of differential calculus particularly useful in algebraic geometry. We pass over many developments that are classical in differential geometry (connections, infinitesimal transformations associated with a vector field, jets, etc.), although these notions are written in a particularly natural way in the framework of schemes. We likewise pass over here the phenomena special to characteristic $p>0$ (some of which are studied, in the affine setting, in (0, 21)). For certain complements to the differential formalism in preschemes the reader may consult Exposés II and VII of [42], as well as later chapters of this Treatise.

16.1. Normal invariants of an immersion

(16.1.1).

Let $(X,{\mathcal{O}}_X)$, $(Y,{\mathcal{O}}_Y)$ be two ringed spaces, $f=(\psi ,\theta ):Y\to X$ a morphism of ringed spaces $(0_I,\mathrm{4.1.1})$ such that the homomorphism

$${\theta }^{♯}:\psi \ast ({\mathcal{O}}_X)\to {\mathcal{O}}_Y$$

is surjective, so that ${\mathcal{O}}_Y$ is identified with a quotient sheaf of rings $\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f$. One can then endow $\psi \ast ({\mathcal{O}}_X)$ with the ${\mathcal{I}}_f$-preadic filtration.

Definition (16.1.2).

The ${\mathcal{O}}_Y$-augmented sheaf of rings $\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f^{n+1}$ is called the $n$-th normal invariant of $f$; the ringed space $(Y,\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f^{n+1})$ is called the $n$-th infinitesimal neighbourhood of $Y$ for the morphism $f$, and denoted $Y_f^{(n)}$ or simply $Y^{(n)}$. The sheaf of graded rings associated with the sheaf of filtered rings $\psi \ast ({\mathcal{O}}_X)$

  (16.1.2.1)    𝒢ℛ_•(f) = ⨁_{n ⩾ 0} (𝓘_f^n / 𝓘_f^{n+1})

is called the sheaf of graded rings associated with $f$. The sheaf ${\mathcal{G}\mathcal{R}}_1(f)={\mathcal{I}}_f/{\mathcal{I}}_f^2$ is called the conormal sheaf of $f$ (and is also denoted ${\mathcal{N}}_{Y/X}$ when no confusion results).

It is clear that the ${\mathcal{O}}_{Y^{(n)}}=\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f^{n+1}$ (also denoted ${\mathcal{O}}_{Y_f^{(n)}}$) form a

projective system of sheaves of rings on $Y$, the transition homomorphism ${\varphi }_{nm}:{\mathcal{O}}_{Y^{(m)}}\to {\mathcal{O}}_{Y^{(n)}}$ for $n⩽m$ identifying ${\mathcal{O}}_{Y^{(n)}}$ with the quotient of ${\mathcal{O}}_{Y^{(m)}}$ by the power $({\mathcal{I}}_f/{\mathcal{I}}_f^{m+1}{)}^{n+1}$ of the augmentation ideal of ${\mathcal{O}}_{Y^{(m)}}$, kernel of ${\varphi }_{0n}:{\mathcal{O}}_{Y^{(n)}}\to {\mathcal{O}}_Y$. The $Y^{(n)}$ therefore form an inductive system of ringed spaces, all having the space $Y$ as underlying space, and one has canonical morphisms of ringed spaces $h_n:Y^{(n)}\to X$ equal to $(\psi ,{\theta }_n)$, where ${\theta }_n^{♯}$ is the canonical morphism $\psi \ast ({\mathcal{O}}_X)\to \psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f^{n+1}$. It is clear that the sheaf ${\mathcal{G}\mathcal{R}}_{\bullet }(f)$ is a sheaf of graded algebras over the sheaf of rings ${\mathcal{O}}_Y={\mathcal{G}\mathcal{R}}_0(f)$, and the ${\mathcal{G}\mathcal{R}}_k(f)$ are ${\mathcal{O}}_Y$-Modules.

As for any sheaf of filtered rings, one has a canonical surjective homomorphism of graded ${\mathcal{O}}_Y$-algebras

$$(\mathrm{16.1.2.2}){\mathbf{S}}_{{\mathcal{O}}_Y}^{\bullet }({\mathcal{G}\mathcal{R}}_1(f))\to {\mathcal{G}\mathcal{R}}_{\bullet }(f)$$

coinciding in degrees 0 and 1 with the identity homomorphisms.

Examples (16.1.3).

(i) Suppose that $X$ is a space ringed in local rings, that $Y$ is reduced to a single point $y$ (endowed with a ring ${\mathcal{O}}_y$), and that, if $x=\psi (y)$, ${\theta }^{♯}:{\mathcal{O}}_x\to {\mathcal{O}}_y$ is a surjective homomorphism of rings having as kernel the maximal ideal ${\mathfrak{m}}_x$ of ${\mathcal{O}}_x$. Then the ${\mathcal{O}}_{Y^{(n)}}$ are identified with the rings ${\mathcal{O}}_x/{\mathfrak{m}}_x^{n+1}$, and ${\mathcal{G}\mathcal{R}}_{\bullet }(f)$ with the graded ring associated with the local ring ${\mathcal{O}}_x$ endowed with its ${\mathfrak{m}}_x$-preadic filtration.

(ii) Suppose that $Y$ is a closed subset of an open subspace $U$ of $X$ and that ${\mathcal{O}}_Y$ is induced on $Y$ by a quotient sheaf ${\mathcal{O}}_U/\mathcal{I}$, where $\mathcal{I}$ is an Ideal of ${\mathcal{O}}_U$ such that ${\mathcal{I}}_x={\mathcal{O}}_x$ for every $x\notin Y$; if $X$ is a space ringed in local rings, we suppose in addition that ${\mathcal{I}}_x\ne {\mathcal{O}}_x$ for $x\in Y$, so that $(Y,{\mathcal{O}}_Y)$ is again a space ringed in local rings. Let ${\psi }_0:Y\to U$ be the canonical injection, and denote by ${\theta }_0:{\mathcal{O}}_U\to ({\psi }_0{)}_{\ast }({\mathcal{O}}_Y)$ the homomorphism such that ${\theta }_0^{♯}$ is the canonical homomorphism ${\psi }_0\ast ({\mathcal{O}}_U)={\mathcal{O}}_U|Y\to ({\mathcal{O}}_U/\mathcal{I})|Y$, so that $j_0=({\psi }_0,{\theta }_0):Y\to U$ is a morphism of ringed spaces (and of spaces ringed in local rings if $X$ is a space ringed in local rings); if $i:U\to X$ is the canonical injection (morphism of ringed spaces), $j=i\circ j_0$ is the morphism $(\psi ,\theta )$ of $Y$ into $X$, where $\psi :Y\to X$ is the canonical injection and $\theta :{\mathcal{O}}_X\to {\psi }_{\ast }({\mathcal{O}}_Y)$ is the homomorphism such that ${\theta }^{♯}={\theta }_0^{♯}$. Since ${\theta }^{♯}$ is surjective, one can apply the preceding definitions; ${\mathcal{O}}_{Y^{(n)}}$ is equal to ${\psi }_0\ast ({\mathcal{O}}_U/{\mathcal{I}}^{n+1})$, and one has $({\psi }_0{)}_{\ast }({\mathcal{O}}_{Y^{(n)}})={\mathcal{O}}_U/{\mathcal{I}}^{n+1}$ and ${\mathcal{G}\mathcal{R}}_n(j)={\mathcal{G}\mathcal{R}}_n(j_0)={\psi }_0\ast ({\mathcal{I}}^n/{\mathcal{I}}^{n+1})=j_0\ast ({\mathcal{I}}^n/{\mathcal{I}}^{n+1})$.

(16.1.4).

The example (16.1.3, (ii)) shows that in general the ${\mathcal{O}}_{Y^{(n)}}$ are not canonically endowed with a structure of ${\mathcal{O}}_Y$-Module, still less a fortiori with a structure of ${\mathcal{O}}_Y$-Algebra. To give such a structure amounts to giving a homomorphism of sheaves of rings ${\lambda }_n:{\mathcal{O}}_Y\to {\mathcal{O}}_{Y^{(n)}}$, right inverse of the augmentation homomorphism ${\varphi }_{0n}$; equivalently, to giving a morphism of ringed spaces $(1_Y,{\lambda }_n):Y^{(n)}\to Y$, left inverse of the canonical morphism $(1_Y,{\varphi }_{0n}):Y\to Y^{(n)}$.

Proposition (16.1.5).

Let $f=(\psi ,\theta ):Y\to X$ be an immersion of preschemes. Then:

(i) ${\mathcal{G}\mathcal{R}}_{\bullet }(f)$ is a quasi-coherent graded ${\mathcal{O}}_Y$-Algebra.

(ii) The $Y^{(n)}$ are preschemes, canonically isomorphic to sub-preschemes of $X$.

(iii) Every homomorphism of sheaves of rings ${\lambda }_n:{\mathcal{O}}_Y\to {\mathcal{O}}_{Y^{(n)}}$, right inverse of the augmentation homomorphism ${\varphi }_{0n}$, makes the ${\mathcal{O}}_{Y^{(n)}}$ and the ${\mathcal{O}}_{Y^{(k)}}$ for $k⩽n$ into quasi-coherent ${\mathcal{O}}_Y$-Algebras; the ${\mathcal{O}}_Y$-Module structures deduced from the preceding structures on the ${\mathcal{G}\mathcal{R}}_k(f)$ for $k⩽n$ coincide with those defined in (16.1.2).

(i) The question being local on $X$ and on $Y$, one can restrict oneself to the case where $Y$ is a closed sub-prescheme of $X$ defined by a quasi-coherent Ideal $\mathcal{I}$ of ${\mathcal{O}}_X$; since ${\mathcal{O}}_Y$ is the restriction to $Y$ of ${\mathcal{O}}_X/\mathcal{I}$, assertion (i) is evident, and $Y^{(n)}$ is the closed sub-prescheme of $X$ defined by the quasi-coherent Ideal ${\mathcal{I}}^{n+1}$ of ${\mathcal{O}}_X$. Finally, to prove (iii), note that the datum of ${\lambda }_n$ makes the Ideal $\mathcal{I}/{\mathcal{I}}^{n+1}$ of the augmentation ${\varphi }_{0n}$ and its quotients ${\mathcal{I}}^k/{\mathcal{I}}^{n+1}$ ($1⩽k⩽n$) into ${\mathcal{O}}_Y$-Modules, and it suffices to prove by induction on $k$ that the ${\mathcal{I}}^k/{\mathcal{I}}^{n+1}$ are quasi-coherent ${\mathcal{O}}_Y$-Modules and that the quotient ${\mathcal{O}}_Y$-Module structure deduced on ${\mathcal{I}}^k/{\mathcal{I}}^{k+1}$ is the same as that defined in (16.1.2). The second assertion is immediate, ${\mathcal{I}}^k/{\mathcal{I}}^{k+1}$ being annihilated by $\mathcal{I}/{\mathcal{I}}^{n+1}$; the first follows by induction on $k$: it is trivial for $k=1$, and ${\mathcal{I}}^k/{\mathcal{I}}^{n+1}$ is an extension of ${\mathcal{I}}^k/{\mathcal{I}}^{k+1}$ by ${\mathcal{I}}^{k+1}/{\mathcal{I}}^{n+1}$ (III, 1.4.17).

Corollary (16.1.6).

Under the general hypotheses of (16.1.5), if the immersion $f$ is locally of finite presentation, the ${\mathcal{G}\mathcal{R}}_n(f)$ are quasi-coherent ${\mathcal{O}}_Y$-Modules of finite type.

Indeed, with the notation of the proof of (16.1.5), $\mathcal{I}$ is an Ideal of finite type of ${\mathcal{O}}_X$ (1.4.7), so the ${\mathcal{I}}^n/{\mathcal{I}}^{n+1}$ are ${\mathcal{O}}_Y$-Modules of finite type, whence the conclusion.

Corollary (16.1.7).

Under the general hypotheses of (16.1.5), let $g:X\to Y$ be a morphism of preschemes left inverse to $f$. Then, for every $n$, the composite morphism $(1_Y,{\lambda }_n):Y^{(n)}\to X\to Y$ defines a homomorphism of sheaves of rings ${\lambda }_n:{\mathcal{O}}_Y\to {\mathcal{O}}_{Y^{(n)}}$ right inverse of the augmentation ${\varphi }_{0n}$, making ${\mathcal{O}}_{Y^{(n)}}$ into a quasi-coherent ${\mathcal{O}}_Y$-Algebra; for these homomorphisms, the transition homomorphisms ${\varphi }_{nm}:{\mathcal{O}}_{Y^{(m)}}\to {\mathcal{O}}_{Y^{(n)}}$ ($n⩽m$) are homomorphisms of ${\mathcal{O}}_Y$-Algebras. In addition, if $g$ is locally of finite type, the ${\mathcal{O}}_{Y^{(n)}}$ are quasi-coherent ${\mathcal{O}}_Y$-Modules of finite type.

The first assertion follows at once from the definitions and from (16.1.5). On the other hand, if $g$ is locally of finite type, then $f$ is locally of finite presentation (1.4.3, (v)); the ${\mathcal{G}\mathcal{R}}_n(f)$ are then quasi-coherent ${\mathcal{O}}_Y$-Modules of finite type by (16.1.6), so the same holds for the ${\mathcal{O}}_Y$-Modules $\mathcal{I}/{\mathcal{I}}^{n+1}$, which are extensions of finitely many of the ${\mathcal{G}\mathcal{R}}_k(f)$ (III, 1.4.17).

Proposition (16.1.8).

Let $X$ be a locally Noetherian prescheme, $j:Y\to X$ an immersion. Then the $Y^{(n)}$ are locally Noetherian preschemes, the ${\mathcal{G}\mathcal{R}}_n(j)$ are coherent ${\mathcal{O}}_Y$-Modules, and ${\mathcal{G}\mathcal{R}}_{\bullet }(j)$ is a coherent sheaf of rings on the space $Y$.

Everything being local on $X$ and $Y$, one is reduced to the case where $X$ is affine and $j$ is a closed immersion; then all the assertions are evident except the last, which follows from the fact that if $A$ is a Noetherian ring and $\mathfrak{J}$ is an ideal of $A$, then $g{r}_{\mathfrak{J}}^{\bullet }(A)$ is a Noetherian ring, taking into account the exactness of the functor $\psi \ast$ and $(0_I,\mathrm{5.3.7})$.

Proposition (16.1.9).

Let $X$ be a prescheme, $j:Y\to X$ an immersion locally of finite presentation, $y$ a point of $Y$. The following conditions are equivalent:

a) There exists an open neighbourhood $U$ of $y$ in $Y$ such that $j|U$ is a homeomorphism of $U$ onto an open subset of $X$.

b) There exists an integer $n>0$ such that the canonical homomorphism

  (φ_{n-1, n})_y : 𝒪_{Y^{(n)}, y} → 𝒪_{Y^{(n-1)}, y}

is bijective.

c) There exists an integer $n>0$ such that $({\mathcal{G}\mathcal{R}}_n(j){)}_y=0$.

Furthermore, if the integer $n$ satisfies b) or c), then there exists a neighbourhood $V$ of $y$ in $Y$ such that ${\mathcal{G}\mathcal{R}}_m(j)|V=0$ for $m⩾n$ and ${\varphi }_{nm}|V:{\mathcal{O}}_{Y^{(m)}}|V\to {\mathcal{O}}_{Y^{(n)}}|V$ is bijective for $m⩾n$.

The question being local on $Y$, one can restrict to the case where $j$ is a closed immersion, $Y$ being defined by a quasi-coherent Ideal of finite type $\mathcal{I}$ of ${\mathcal{O}}_X$. The equivalence of b) and c), for a given $n$, is then immediate; furthermore, since ${\mathcal{I}}^n/{\mathcal{I}}^{n+1}$ is an ${\mathcal{O}}_X$-Module of finite type, there exists an open neighbourhood $U$ of $y$ in $X$ such that ${\mathcal{I}}^n|U={\mathcal{I}}^{n+1}|U$ $(0_I,\mathrm{5.2.2})$, hence also ${\mathcal{I}}^n|U={\mathcal{I}}^m|U$ for $m⩾n$, which proves the last assertions. To prove that a) implies b), one can restrict to the case where the space underlying $Y$ is equal to the space underlying $X$, and where $\mathcal{I}$ is generated by a finite number of its sections over $X$: as $\mathcal{I}$ is then contained in the nilradical $\mathcal{N}$ of ${\mathcal{O}}_X$ (I, 5.1.2), it is nilpotent, which proves b). Finally, to prove that b) implies a), one can also restrict to the case where ${\mathcal{I}}^n={\mathcal{I}}^{n+1}$; then, for every $y\in Y$, since ${\mathcal{I}}_y\subset {\mathfrak{m}}_y$, the maximal ideal of ${\mathcal{O}}_{X,y}$, one necessarily has ${\mathcal{I}}_y^n=0$ by Nakayama's lemma, since ${\mathcal{I}}_y$ is an ideal of finite type. The set of $x\in X$ such that ${\mathcal{I}}_x^n=0$ is therefore an open subset $U$ of $X$ containing $Y$ $(0_I,\mathrm{5.2.2})$; since on the other hand ${\mathcal{I}}_x\ne 0$ for $x\notin Y$, one necessarily has $U=Y$.

Corollary (16.1.10).

For the restriction of the immersion $j$ to a neighbourhood of $y$ in $Y$ to be an open immersion (in other words, for $j$ to be a local isomorphism at the point $y$), it is necessary and sufficient that $({\mathcal{G}\mathcal{R}}_1(j){)}_y=({\mathcal{N}}_{Y/X}{)}_y=0$.

The condition is obviously necessary, and the preceding reasoning, applied for $n=1$, proves that it is sufficient.

Remarks (16.1.11).

(i) Under the conditions of Definition (16.1.1), the projective limit of the projective system $({\mathcal{O}}_{Y^{(n)}},{\varphi }_{nm})$ of sheaves of rings on $Y$ is called the normal invariant of infinite order of $f$, and sometimes denoted ${\mathcal{O}}_{Y^{(\infty )}}$. When $X$ is a locally Noetherian prescheme and $j:Y\to X$ is a closed immersion, so that $Y$ is a closed sub-prescheme of $X$ defined by a coherent Ideal $\mathcal{I}$, ${\mathcal{O}}_{Y^{(\infty )}}$ is none other than the formal completion of ${\mathcal{O}}_X$ along $Y$ (I, 10.8.4), and $Y^{(\infty )}=(Y,{\mathcal{O}}_{Y^{(\infty )}})$ is the formal prescheme that is the completion of $X$ along $Y$ (I, 10.8.5). In all cases, one may say that $Y^{(\infty )}$ is the formal neighbourhood of $Y$ in $X$ (for the morphism $f$). In the particular case just considered, it is therefore the formal prescheme that is the inductive limit of the infinitesimal neighbourhoods of order $n$.

(ii) Note that for a morphism of preschemes $f=(\psi ,\theta ):Y\to X$, it can happen that the homomorphism ${\theta }^{♯}:\psi \ast ({\mathcal{O}}_X)\to {\mathcal{O}}_Y$ is surjective without $f$ being a local

immersion, and without $\psi$ being injective. One has an example by taking for $Y$ a sum of preschemes $Y_{\lambda }$ all isomorphic to $\mathrm{Spec}({\mathcal{O}}_x)$, where $x\in X$, and for $f$ the morphism equal to the canonical morphism on each of the $Y_{\lambda }$.

16.2. Functorial properties of normal invariants of an immersion

(16.2.1).

Let $f=(\psi ,\theta ):Y\to X$ and $f^{\prime }=({\psi }^{\prime },{\theta }^{\prime }):Y^{\prime }\to X^{\prime }$ be two morphisms of ringed spaces such that the homomorphisms ${\theta }^{♯}$ and ${\theta }^{\prime ♯}$ are surjective; consider a commutative diagram of morphisms of ringed spaces

  (16.2.1.1)
                  Y  ──f──>  X
                  ↑          ↑
                  u          v
                  │          │
                  Y' ──f'─>  X'

Set $u=(\rho ,\lambda )$, $v=(\sigma ,\mu )$. One has $\rho \ast (\psi \ast ({\mathcal{O}}_X))={\psi }^{\prime }\ast (\sigma \ast ({\mathcal{O}}_X))$, and consequently a commutative diagram of homomorphisms of sheaves of rings on $Y^{\prime }$

  ρ*(ψ*(𝒪_X)) = ψ'*(σ*(𝒪_X)) ──ψ'*(μ^#)──> ψ'*(𝒪_{X'})
              │                                │
              ρ*(θ^#)                          θ'^#
              ↓                                ↓
            ρ*(𝒪_Y) ────────λ^#──────────> 𝒪_{Y'}

from which one concludes, if $\mathcal{I}$ and ${\mathcal{I}}^{\prime }$ are the kernels of ${\theta }^{♯}$ and ${\theta }^{\prime ♯}$, that one has ${\psi }^{\prime }\ast ({\mu }^{♯})(\rho \ast (\mathcal{I}))\subset {\mathcal{I}}^{\prime }$, by exactness of the functor $\rho \ast$. One deduces at once that for every integer $n$, ${\psi }^{\prime }\ast ({\mu }^{♯})(\rho \ast ({\mathcal{I}}^n))\subset {\mathcal{I}}^{\prime n}$, which shows that ${\psi }^{\prime }\ast ({\mu }^{♯})$ defines, by passage to the quotients, a homomorphism of sheaves of rings

$$(\mathrm{16.2.1.2}){\nu }_n:\rho \ast (\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}^{n+1})\to {\psi }^{\prime }\ast ({\mathcal{O}}_{X^{\prime }})/{\mathcal{I}}^{\prime n+1}$$

and consequently a morphism of ringed spaces $w_n=(\rho ,{\nu }_n):Y^{\prime (n)}\to Y^{(n)}$ (which, for $n=0$, is none other than $u$). It follows at once from this definition that the diagrams

  Y^{(n)}  ──h_{mn}──>  Y^{(m)}  ──h_m──>  X
     ↑                    ↑                ↑
     w_n                  w_m              v        (n ⩽ m)
     │                    │                │
  Y'^{(n)} ──h'_{mn}──> Y'^{(m)} ──h'_m──> X'

(where the horizontal arrows are the canonical morphisms (16.1.2)) are commutative.

By passage to the quotients from the homomorphisms (16.2.1.2), and taking into

account the exactness of the functor $\rho \ast$, one obtains a di-homomorphism of graded Algebras (relative to the homomorphism ${\lambda }^{♯}:\rho \ast ({\mathcal{O}}_Y)\to {\mathcal{O}}_{Y^{\prime }}$)

$$(\mathrm{16.2.1.3})gr(u):\rho \ast ({\mathcal{G}\mathcal{R}}_{\bullet }(f))\to {\mathcal{G}\mathcal{R}}_{\bullet }(f^{\prime })$$

(or, if one prefers, a $\rho$-morphism $(0_I,\mathrm{3.5.1})$ ${\mathcal{G}\mathcal{R}}_{\bullet }(f)\to {\mathcal{G}\mathcal{R}}_{\bullet }(f^{\prime })$), and in particular a di-homomorphism of conormal sheaves

$$g{r}_1(u):\rho \ast ({\mathcal{G}\mathcal{R}}_1(f))\to {\mathcal{G}\mathcal{R}}_1(f^{\prime }).$$

It is immediate, moreover, that these homomorphisms give rise to a commutative diagram

  (16.2.1.4)
       ρ*(𝐒_{𝒪_Y}^•(𝒢ℛ_1(f))) ────────> ρ*(𝒢ℛ_•(f))
              │                              │
              𝐒(gr_1(u))                     gr(u)
              ↓                              ↓
       𝐒_{𝒪_{Y'}}^•(𝒢ℛ_1(f'))  ────────>  𝒢ℛ_•(f')

where the horizontal arrows are the canonical homomorphisms (16.1.2.2).

Finally, if one has a commutative diagram of morphisms of ringed spaces

       Y   ──f──>  X
       ↑           ↑
       u           v
       │           │
       Y'  ──f'─>  X'
       ↑           ↑
       u'          v'
       │           │
       Y'' ──f''─> X''

where $f^{\prime \prime }=({\psi }^{\prime \prime },{\theta }^{\prime \prime })$ is such that ${\theta }^{\prime \prime ♯}$ is surjective, and if $w_n$ and $w_n^{\prime }$ are defined from $u$, $v$ on the one hand, and from $u^{\prime \prime }=u\circ u^{\prime }$, $v^{\prime \prime }=v\circ v^{\prime }$ on the other, then one has $w_n^{\prime \prime }=w_n\circ w_n^{\prime }$, as follows at once from the definitions and from $(0_I,\mathrm{3.5.5})$; likewise $gr(u^{\prime \prime })=gr(u^{\prime })\circ {\rho }^{\prime }\ast (gr(u))$ if $u^{\prime }=({\rho }^{\prime },{\lambda }^{\prime })$. One can therefore say that the $Y^{(n)}$ and the ${\mathcal{G}\mathcal{R}}_{\bullet }(f)$ depend functorially on $f$.

Proposition (16.2.2).

With the notation and hypotheses of (16.2.1), suppose moreover that $f$, $f^{\prime }$, $u$ and $v$ are morphisms of preschemes. Then:

(i) The morphisms $w_n:Y^{\prime (n)}\to Y^{(n)}$ are morphisms of preschemes.

(ii) If $Y^{\prime }=Y{\times }_X{X}^{\prime }$, with $u$ and $f^{\prime }$ the canonical projections, and if $f$ is an immersion or $v$ is flat, one has $Y^{\prime (n)}=Y^{(n)}{\times }_X{X}^{\prime }$.

(iii) If $Y^{\prime }=Y{\times }_X{X}^{\prime }$ and if $v$ is flat (resp. if $f$ is an immersion), the homomorphism

  Gr(u) = gr(u) ⊗ 1 : 𝒢ℛ_•(f) ⊗_{𝒪_Y} 𝒪_{Y'} → 𝒢ℛ_•(f')

is bijective (resp. surjective).

(i) The hypotheses at once imply that for every $y^{\prime }\in Y^{\prime }$, ${\rho }_{y^{\prime }}\ast ({\theta }_{{\psi }^{\prime }(y^{\prime })}^{♯})$ is a local homomorphism (I, 1.6.2), so $w_n$ is a morphism of preschemes (I, 2.2.1).

(ii) and (iii). If $f$ is an immersion, one can restrict to the case where $f$ is a closed immersion, $Y$ being defined by the quasi-coherent Ideal $\mathcal{I}$ of ${\mathcal{O}}_X$, and $Y^{(n)}$ by the Ideal ${\mathcal{I}}^{n+1}$; the assertions then follow from (I, 4.4.5).

Suppose next that $v$ is flat; one can restrict to the case where $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(B)$, $X^{\prime }=\mathrm{Spec}(A^{\prime })$ are affine, $A^{\prime }$ being a flat $A$-module; then $Y^{\prime }=\mathrm{Spec}(B^{\prime })$ with $B^{\prime }=B{\otimes }_A{A}^{\prime }$. Moreover, if $\mathfrak{J}$ is the kernel of the homomorphism $A\to B$, the kernel ${\mathfrak{J}}^{\prime }$ of $A^{\prime }\to B^{\prime }$ is identified with $\mathfrak{J}{\otimes }_A{A}^{\prime }$ by flatness, and ${\mathfrak{J}}^{\prime n}/{\mathfrak{J}}^{\prime n+1}=({\mathfrak{J}}^n/{\mathfrak{J}}^{n+1}){\otimes }_A{A}^{\prime }$. One deduces at once, taking $(0_I,\mathrm{4.3.3})$ into account, that the ${\mathcal{O}}_{Y^{\prime }}$-Module ${\mathcal{I}}^{\prime n}/{\mathcal{I}}^{\prime n+1}$ is equal to $({\mathcal{I}}^n/{\mathcal{I}}^{n+1}){\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y^{\prime }}$, and in particular for $n=0$ one has

  𝒪_{Y'} = ρ*(𝒪_Y) ⊗_{ρ*(ψ*(𝒪_X))} ψ'*(𝒪_{X'}),

which proves (iii). Set now $C_n=\Gamma (Y,{\mathcal{O}}_{Y^{(n)}})$, $C_n^{\prime }=\Gamma (Y^{\prime },{\mathcal{O}}_{Y^{\prime (n)}})$. Since $Y^{(n)}$ and $Y^{\prime (n)}$ are affine schemes (16.1.5), the kernel ${\mathfrak{K}}_n$ (resp. ${\mathfrak{K}}_n^{\prime }$) of the homomorphism $C_n\to C_{n-1}$ (resp. $C_n^{\prime }\to C_{n-1}^{\prime }$) is $\Gamma (Y,{\mathcal{I}}^n/{\mathcal{I}}^{n+1})$ (resp. $\Gamma (Y^{\prime },{\mathcal{I}}^{\prime n}/{\mathcal{I}}^{\prime n+1})$), so one deduces from the foregoing that ${\mathfrak{K}}_n^{\prime }={\mathfrak{K}}_n{\otimes }_A{A}^{\prime }$. One has a commutative diagram

  0 ──> 𝔎_n ⊗_A A' ──> C_n ⊗_A A' ──> C_{n-1} ⊗_A A' ──> 0
         │                │                │
         r                s_n              s_{n-1}
         ↓                ↓                ↓
  0 ──>  𝔎'_n   ──────> C'_n  ─────────> C'_{n-1}  ─────> 0

where the left vertical arrow is bijective and the two rows are exact ($A^{\prime }$ being a flat $A$-module). One deduces by induction that $s_n$ is bijective for all $n$, since it is so by hypothesis for $n=0$, and the induction step follows from the five lemma. This proves the second assertion of (ii).

Corollary (16.2.3).

Let $g:X\to Y$, $u:Y^{\prime }\to Y$ be two morphisms of preschemes, $X^{\prime }=X{\times }_Y{Y}^{\prime }$, and let $g^{\prime }:X^{\prime }\to Y^{\prime }$ and $v:X^{\prime }\to X$ be the canonical projections. Let $f:Y\to X$ be a $Y$-section of $X$ (hence an immersion), $f^{\prime }=f_{(Y^{\prime })}:Y^{\prime }\to X^{\prime }$ the $Y^{\prime }$-section of $X^{\prime }$ deduced from $f$ by the base change $u$. Then:

(i) The morphism $w_n:Y_{f^{\prime }}^{\prime (n)}\to Y_f^{(n)}$ corresponding to $f$, $f^{\prime }$, $u$, $v$ (16.2.1) and the canonical morphism $h_n^{\prime }:Y_{f^{\prime }}^{\prime (n)}\to X^{\prime }$ identify $Y_{f^{\prime }}^{\prime (n)}$ with the product $Y_f^{(n)}{\times }_X{X}^{\prime }$.

(ii) If one endows ${\mathcal{O}}_{Y_f^{(n)}}$ (resp. ${\mathcal{O}}_{Y_{f^{\prime }}^{\prime (n)}}$) with the structure of ${\mathcal{O}}_Y$-Algebra defined by $g$ (resp. with the structure of ${\mathcal{O}}_{Y^{\prime }}$-Algebra defined by $g^{\prime }$) (16.1.7), the homomorphism of ${\mathcal{O}}_{Y^{\prime }}$-Algebras

  (16.2.3.1)    ρ*(𝒪_{Y_f^{(n)}}) ⊗_{𝒪_Y} 𝒪_{Y'} → 𝒪_{Y'_{f'}^{(n)}}

deduced from the homomorphism ${\nu }_n$ (16.2.1.2) is bijective. Furthermore, the homomorphism of ${\mathcal{O}}_{Y^{\prime }}$-Modules

  (16.2.3.2)    Gr_1(u) : 𝒢ℛ_1(f) ⊗_{𝒪_Y} 𝒪_{Y'} → 𝒢ℛ_1(f')

is bijective.

(i) Note first that the morphisms $f^{\prime }:Y^{\prime }\to X^{\prime }$ and $u:Y^{\prime }\to Y$ identify $Y^{\prime }$ with the product $Y{\times }_X{X}^{\prime }$ (for the structure morphisms $f:Y\to X$ and $v:X^{\prime }\to X$) (14.5.12.1). The conclusion of (i) then follows from (16.2.2, (ii)), the morphism $g$ being an immersion.

(ii) The commutative diagram

                w_n
  Y_f^{(n)} <─────── Y'_{f'}^{(n)}
     │ h_n               │ h'_n
     ↓                   ↓
     X      <─── v ───   X'
     │ g                 │ g'
     ↓                   ↓
     Y      <─── u ───   Y'

identifies $Y_{f^{\prime }}^{\prime (n)}$ with the product $Y_f^{(n)}{\times }_X{X}^{\prime }$, so (I, 3.3.9) identifies (for the morphisms $g^{\prime }\circ h_n^{\prime }$ and $w_n$) $Y_{f^{\prime }}^{\prime (n)}$ with the product $Y_f^{(n)}{\times }_Y{Y}^{\prime }$. Since $Y_f^{(n)}$ (resp. $Y_{f^{\prime }}^{\prime (n)}$) is the affine prescheme over $Y$ (resp. $Y^{\prime }$) associated with the ${\mathcal{O}}_Y$-Algebra ${\mathcal{O}}_{Y_f^{(n)}}$ (resp. with the ${\mathcal{O}}_{Y^{\prime }}$-Algebra ${\mathcal{O}}_{Y_{f^{\prime }}^{\prime (n)}}$), the fact that the canonical homomorphism (16.2.3.1) is bijective follows from (II, 1.5.2). Finally, the canonical homomorphism (16.2.3.1) is compatible with the augmentations ${\mathcal{O}}_{Y_f^{(n)}}\to {\mathcal{O}}_Y$ and ${\mathcal{O}}_{Y_{f^{\prime }}^{\prime (n)}}\to {\mathcal{O}}_{Y^{\prime }}$; as ${\mathcal{O}}_{Y_f^{(n)}}$ is the direct sum (as an ${\mathcal{O}}_Y$-Module) of ${\mathcal{O}}_Y$ and of the augmentation ideal $\mathcal{I}/{\mathcal{I}}^{n+1}$, one sees that the canonical homomorphism (16.2.3.1), restricted to $(\mathcal{I}/{\mathcal{I}}^{n+1}){\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y^{\prime }}$, is a bijection of the latter onto ${\mathcal{I}}^{\prime }/{\mathcal{I}}^{\prime n+1}$. For $n=1$, this shows that $G{r}_1(u)$ is bijective.

One will note that, under the hypotheses of (16.2.3), the homomorphisms $G{r}_n(u)$ are surjective by virtue of the foregoing, but are not bijective in general for $n⩾2$. However:

Corollary (16.2.4).

Under the hypotheses of (16.2.3), suppose that $u:Y^{\prime }\to Y$ is a flat morphism (resp. that the ${\mathcal{G}\mathcal{R}}_n(f)$ are flat ${\mathcal{O}}_Y$-Modules for $n⩽m$). Then the homomorphism

  Gr_n(u) : 𝒢ℛ_n(f) ⊗_{𝒪_Y} 𝒪_{Y'} → 𝒢ℛ_n(f')

is bijective for every $n$ (resp. for $n⩽m$).

If $u$ is flat, then so is $v:X^{\prime }\to X$, which is deduced from it by base change, and one knows already in this case that $Gr(u)$ is bijective (16.2.2, (iii)). If the ${\mathcal{G}\mathcal{R}}_n(f)$ are flat for $n⩽m$, one sees first by induction on $n$ that the same holds for the $\mathcal{I}/{\mathcal{I}}^{n+1}$ for $n⩽m$, by virtue of the exact sequences

$$0\to {\mathcal{I}}^n/{\mathcal{I}}^{n+1}\to \mathcal{I}/{\mathcal{I}}^{n+1}\to \mathcal{I}/{\mathcal{I}}^n\to 0$$

$(0_I,\mathrm{6.1.2})$; furthermore one then has commutative diagrams

  0 → (𝓘^n/𝓘^{n+1}) ⊗ 𝒪_{Y'} → (𝓘/𝓘^{n+1}) ⊗ 𝒪_{Y'} → (𝓘/𝓘^n) ⊗ 𝒪_{Y'} → 0
           │                            │                       │
           ↓                            ↓                       ↓
  0 ──>  𝓘'^n/𝓘'^{n+1}    ─────>    𝓘'/𝓘'^{n+1}    ─────>   𝓘'/𝓘'^n   ────> 0

in which the rows are exact (the first by flatness $(0_I,\mathrm{6.1.2})$) and the last two vertical arrows are bijective by virtue of (16.2.3, (ii)); whence the conclusion.

Remarks (16.2.5).

(i) The reasoning of (16.2.2, (i)) still applies when in (16.2.1.1) one is dealing with morphisms of spaces ringed in local rings $(Er{r}_{III},(\mathrm{1.8.2}))$.

(ii) In (16.2.2, (ii)), the conclusion is no longer necessarily valid when one only supposes that $v$ and $f$ are morphisms of preschemes ($f$ satisfying the condition of (16.1.1)). For example (with the notation of the proof of (16.2.2, (ii))), it may happen that $\mathfrak{J}=0$ while the kernel ${\mathfrak{J}}^{\prime }$ of $A^{\prime }\to B^{\prime }=B{\otimes }_A{A}^{\prime }$ is not zero and $B^{\prime }\ne 0$, in which case one has $Y^{(n)}=Y$ for all $n$, but $Y^{\prime (n)}\ne Y^{\prime }$. One has an example of this by taking $A=\mathbb{Z}$, $B=\mathbb{Q}$, $A^{\prime }={\prod }_{h=1}^{\infty }(\mathbb{Z}/m^h\mathbb{Z})$ where $m>1$.

(16.2.6).

Consider the particular case of the diagram (16.2.1.1) where $X^{\prime }=X$, $v$ is the identity, $X$ is a prescheme, $Y$ a sub-prescheme of $X$, $Y^{\prime }$ a sub-prescheme of $Y$, and $f$, $u$, $f^{\prime }=f\circ u$ are the canonical injections; the di-homomorphism (16.2.1.3) gives, by tensoring with ${\mathcal{O}}_{Y^{\prime }}$ over $\rho \ast ({\mathcal{O}}_Y)$, a homomorphism of graded ${\mathcal{O}}_{Y^{\prime }}$-Algebras

$$(\mathrm{16.2.6.1})u\ast ({\mathcal{G}\mathcal{R}}_{\bullet }(f))\to {\mathcal{G}\mathcal{R}}_{\bullet }(f^{\prime }).$$

On the other hand, one identifies ${\mathcal{O}}_Y$ with $\psi \ast ({\mathcal{O}}_X)/{\mathcal{I}}_f$ and ${\mathcal{O}}_{Y^{\prime }}$ with $\rho \ast ({\mathcal{O}}_Y)/{\mathcal{I}}_u$; since $\rho \ast$ is an exact functor, one has $\rho \ast ({\mathcal{O}}_Y)=\rho \ast (\psi \ast ({\mathcal{O}}_X))/\rho \ast ({\mathcal{I}}_f)={\psi }^{\prime }\ast ({\mathcal{O}}_X)/\rho \ast ({\mathcal{I}}_f)$, and since ${\mathcal{O}}_{Y^{\prime }}$ is on the other hand identified with ${\psi }^{\prime }\ast ({\mathcal{O}}_X)/{\mathcal{I}}_{f^{\prime }}$, one sees that ${\mathcal{I}}_u={\mathcal{I}}_{f^{\prime }}/\rho \ast ({\mathcal{I}}_f)$. One deduces from this, for every integer $n$, a canonical homomorphism ${\mathcal{I}}_{f^{\prime }}^n/{\mathcal{I}}_{f^{\prime }}^{n+1}\to {\mathcal{I}}_u^n/{\mathcal{I}}_u^{n+1}$, whence a canonical homomorphism of graded ${\mathcal{O}}_{Y^{\prime }}$-Algebras

$$(\mathrm{16.2.6.2}){\mathcal{G}\mathcal{R}}_{\bullet }(f^{\prime })\to {\mathcal{G}\mathcal{R}}_{\bullet }(u).$$

Proposition (16.2.7).

Let $X$ be a prescheme, $Y$ a sub-prescheme of $X$, $Y^{\prime }$ a sub-prescheme of $Y$, $j:Y^{\prime }\to Y$ the canonical injection. Then one has an exact sequence of conormal sheaves (${\mathcal{O}}_{Y^{\prime }}$-Modules)

$$(\mathrm{16.2.7.1})j\ast ({\mathcal{N}}_{Y/X})\to {\mathcal{N}}_{Y^{\prime }/X}\to {\mathcal{N}}_{Y^{\prime }/Y}\to 0$$

where the arrows are the degree-1 components of the canonical homomorphisms (16.2.6.1) and (16.2.6.2).

The question being local, one can restrict to the case where $X=\mathrm{Spec}(A)$, $Y=\mathrm{Spec}(A/\mathfrak{J})$, $Y^{\prime }=\mathrm{Spec}(A/\mathfrak{K})$, with $\mathfrak{J}$ and $\mathfrak{K}$ ideals of $A$ such that $\mathfrak{J}\subset \mathfrak{K}$; everything reduces to showing

that the sequence of canonical homomorphisms $\mathfrak{J}/\mathfrak{KJ}\to \mathfrak{K}/{\mathfrak{K}}^2\to (\mathfrak{K}/\mathfrak{J})/(\mathfrak{K}/\mathfrak{J}{)}^2\to 0$ is exact, which is immediate, since the image of $\mathfrak{J}/\mathfrak{KJ}$ in $\mathfrak{K}/{\mathfrak{K}}^2$ is $(\mathfrak{J}+{\mathfrak{K}}^2)/{\mathfrak{K}}^2$ and $(\mathfrak{K}/\mathfrak{J})/(\mathfrak{K}/\mathfrak{J}{)}^2$ is identified with $\mathfrak{K}/(\mathfrak{J}+{\mathfrak{K}}^2)$.

It is easy to give examples where the sequence (16.2.7.1) extended on the left by a 0 is no longer exact; with the preceding notation, it suffices to take $A=k[T]$, $\mathfrak{J}=A{T}^2$, $\mathfrak{K}=AT$, because then one has $(\mathfrak{J}+{\mathfrak{K}}^2)/{\mathfrak{K}}^2=0$ and $\mathfrak{J}/\mathfrak{KJ}\ne 0$. See, however, (16.9.13) and (19.1.5) for useful cases where the extended sequence remains exact.

16.3. Fundamental differential invariants of a morphism of preschemes

Definition (16.3.1).

Let $f:X\to S$ be a morphism of preschemes, ${\Delta }_f:X\to X{\times }_{S}X$ the corresponding diagonal morphism, which is an immersion (I, 5.3.9 and Err_III, 10). One denotes by ${\mathcal{P}}_f^n$ or ${\mathcal{P}}_{X/S}^n$, and calls the sheaf of principal parts of order $n$ of the $S$-prescheme $X$, the ${\mathcal{O}}_X$-augmented sheaf of rings $n$-th normal invariant of ${\Delta }_f$ (16.1.2). One sets ${\mathcal{P}}_f^{\infty }={\mathcal{P}}_{X/S}^{\infty }=\lim {\leftarrow }_n{\mathcal{P}}_{X/S}^n$, ${\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_f)={\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_{X/S})={\mathcal{G}\mathcal{R}}_n({\Delta }_f)$ (16.1.2); the ${\mathcal{O}}_X$-Module ${\mathcal{G}\mathcal{R}}_1({\Delta }_f)$, augmentation ideal of ${\mathcal{P}}_{X/S}^1$, is also denoted ${\Omega }_f^1$ or ${\Omega }_{X/S}^1$, and is called the ${\mathcal{O}}_X$-Module of 1-differentials of $f$, or of $X$ with respect to $S$, or of the $S$-prescheme $X$.

It follows from this definition that ${\mathcal{P}}_{X/S}^0$ is canonically identified with ${\mathcal{O}}_X$ (16.1.2).

One has (16.1.2.2) a canonical surjective homomorphism of graded ${\mathcal{O}}_X$-Algebras

$$(\mathrm{16.3.1.1}){\mathbf{S}}_{{\mathcal{O}}_X}^{\bullet }({\Omega }_{X/S}^1)\to {\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X/S}).$$

It also follows from Definition (16.3.1) that for every open $U$ of $X$, one has ${\mathcal{P}}_{f|U}^n={\mathcal{P}}_f^n|U$, ${\mathcal{P}}_{f|U}^{\infty }={\mathcal{P}}_f^{\infty }|U$, ${\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_{f|U})={\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_f)|U$, ${\Omega }_{f|U}^1={\Omega }_f^1|U$ (in other words, the notions introduced are local on $X$).

(16.3.2).

Denote by $p_1$, $p_2$ the two canonical projections of the product $X{\times }_{S}X$; since ${\Delta }_f$ is an $X$-section of $X{\times }_{S}X$ for both morphisms $p_1$ and $p_2$, each of these morphisms defines, for every $n$, a homomorphism of sheaves of rings ${\mathcal{O}}_X\to {\mathcal{P}}_{X/S}^n$, right inverse to the augmentation ${\mathcal{P}}_{X/S}^n\to {\mathcal{O}}_X$ (16.1.7); one can also say that one thus defines on ${\mathcal{P}}_{X/S}^n$ two quasi-coherent augmented ${\mathcal{O}}_X$-Algebra structures; the corresponding ${\mathcal{O}}_X$-Module structures on ${\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_{X/S})$ are the same. One similarly has, by passage to the limit, two ${\mathcal{O}}_X$-Algebra structures on ${\mathcal{P}}_{X/S}^{\infty }$.

(16.3.3).

The morphism $s=(p_2,p_1{)}_S:X{\times }_{S}X\to X{\times }_{S}X$ is an involutive automorphism of $X{\times }_{S}X$, called the canonical symmetry, such that

  (16.3.3.1)    p_1 ∘ s = p_2,   p_2 ∘ s = p_1,   s ∘ Δ_f = Δ_f.

If one sets $s=(\rho ,\lambda )$, $p_i=({\pi }_i,{\mu }_i)$ ($i=1,2$), ${\Delta }_f=(\delta ,\nu )$, then ${\lambda }^{♯}$ is an isomorphism of $\rho \ast ({\pi }_1\ast ({\mathcal{O}}_X))$ onto ${\pi }_2\ast ({\mathcal{O}}_X)$, and $\delta \ast ({\lambda }^{♯})$ leaves invariant $\delta \ast ({\mathcal{O}}_{X{\times }_{S}X})$ and the kernel $\mathcal{I}$ of the homomorphism ${\nu }^{♯}:\delta \ast ({\mathcal{O}}_{X{\times }_{S}X})\to {\mathcal{O}}_X$. Therefore:

Proposition (16.3.4).

The homomorphism $\sigma =\delta \ast ({\lambda }^{♯})$ deduced from $s$ (and also called the canonical symmetry) is an involutive automorphism of the projective system $({\mathcal{P}}_{X/S}^n)$ of ${\mathcal{O}}_X$-augmented

sheaves of rings, and consequently also of their projective limit ${\mathcal{P}}_{X/S}^{\infty }$. This automorphism permutes the two ${\mathcal{O}}_X$-Algebra structures on the ${\mathcal{P}}_{X/S}^n$ and on ${\mathcal{P}}_{X/S}^{\infty }$.

(16.3.5).

In what follows, the two ${\mathcal{O}}_X$-Algebra structures defined on the ${\mathcal{P}}_{X/S}^n$ and on ${\mathcal{P}}_{X/S}^{\infty }$ will play very different roles: we shall agree from now on, unless expressly stated otherwise, that when ${\mathcal{P}}_{X/S}^n$ or ${\mathcal{P}}_{X/S}^{\infty }$ is considered as an ${\mathcal{O}}_X$-Algebra, it is the algebra structure defined by $p_1$ that is meant.

For every open $U$ of $X$ and every section $t\in \Gamma (U,{\mathcal{O}}_X)$, one will denote simply by $t\cdot 1$ or even $t$ the image of $t$ under the structural homomorphism $\Gamma (U,{\mathcal{O}}_X)\to \Gamma (U,{\mathcal{P}}_{X/S}^n)$ (resp. $\Gamma (U,{\mathcal{O}}_X)\to \Gamma (U,{\mathcal{P}}_{X/S}^{\infty })$) (that is to say, the homomorphism corresponding to $p_1$).

Definition (16.3.6).

One denotes by $d_f^n$, or $d_{X/S}^n$ (resp. $d_f^{\infty }$, or $d_{X/S}^{\infty }$), or simply $d^n$ (resp. $d^{\infty }$), the homomorphism of sheaves of rings ${\mathcal{O}}_X\to {\mathcal{P}}_f^n={\mathcal{P}}_{X/S}^n$ (resp. ${\mathcal{O}}_X\to {\mathcal{P}}_f^{\infty }={\mathcal{P}}_{X/S}^{\infty }$) deduced from $p_2$. For every open $U$ of $X$ and every $t\in \Gamma (U,{\mathcal{O}}_X)$, $d^{n}t$ (resp. $d^{\infty }t$) is called the principal part of order $n$ (resp. principal part of infinite order) of $t$. One sets $dt=d^{1}t-t\cdot 1$, and dt is called the differential of $t$ (an element of $\Gamma (U,{\Omega }_{X/S}^1)$, also denoted $d_{X/S}(t)$).

It follows at once from this definition that one has

  (16.3.6.1)    d(t_1 t_2) = t_1 dt_2 + t_2 dt_1

for any $t_1$, $t_2$ in $\Gamma (U,{\mathcal{O}}_X)$, that is to say, $d$ is a derivation of the ring $\Gamma (U,{\mathcal{O}}_X)$ into the $\Gamma (U,{\mathcal{O}}_X)$-module $\Gamma (U,{\Omega }_{X/S}^1)$.

In all the notation introduced in (16.3.1) and (16.3.6), one will sometimes replace $S$ by $A$ when $S=\mathrm{Spec}(A)$.

(16.3.7).

Suppose in particular that $S=\mathrm{Spec}(A)$ and $X=\mathrm{Spec}(B)$ are affine schemes, $B$ being therefore an $A$-algebra. Then ${\Delta }_f$ corresponds to the canonical surjective homomorphism $\pi :B{\otimes }_{A}B\to B$ such that $\pi (b\otimes b^{\prime })=b{b}^{\prime }$, with kernel $\mathfrak{J}={\mathfrak{J}}_{B/A}$ (0, 20.4.1); ${\mathcal{P}}_f^n$ is the structure sheaf of the prescheme $\mathrm{Spec}(P_{B/A}^n)$, where

  P_{B/A}^n = (B ⊗_A B) / 𝔍^{n+1};

${\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_f)$ is the quasi-coherent ${\mathcal{O}}_X$-Module corresponding to the graded $B$-module

  gr_𝔍^•(B ⊗_A B) = ⨁_{n ⩾ 0} (𝔍^n / 𝔍^{n+1});

in particular ${\Omega }_f^1={\Omega }_{X/S}^1$ is the quasi-coherent ${\mathcal{O}}_X$-Module corresponding to the $B$-module of 1-differentials of $B$ with respect to $A$, namely ${\Omega }_{B/A}^1$ (0, 20.4.3). The projection morphisms $p_1:X{\times }_{S}X\to X$, $p_2:X{\times }_{S}X\to X$ correspond to the two ring homomorphisms $j_1:B\to B{\otimes }_{A}B$, $j_2:B\to B{\otimes }_{A}B$ such that $j_1(b)=b\otimes 1$, $j_2(b)=1\otimes b$, so that (by the convention of (16.3.5)) $P_{B/A}^n$ is always considered as a $B$-algebra via the composite homomorphism $B\to B{\otimes }_{A}B\to P_{B/A}^n$; the ring homomorphism $B\to B{\otimes }_{A}B\to P_{B/A}^n$ deduced from $j_2$ is denoted $d_{B/A}^n$ and corresponds to $d_{X/S}^n$ acting on $\Gamma (X,{\mathcal{O}}_X)$; for every $t\in B$, dt is equal to $d_{B/A}t$, defined in (0, 20.4.6) (cf. $Er{r}_{IV},11$).

If ${\pi }_n:B{\otimes }_{A}B\to P_{B/A}^n$ is the canonical homomorphism, one therefore has, by virtue of the preceding definitions,

  (16.3.7.1)    π_n(b ⊗ b') = b · π_n(1 ⊗ b') = b · d_{B/A}^n(b')   for b ∈ B, b' ∈ B.

Proposition (16.3.8).

The image of the homomorphism $d_{X/S}^n:{\mathcal{O}}_X\to {\mathcal{P}}_{X/S}^n$ generates the ${\mathcal{O}}_X$-Module ${\mathcal{P}}_{X/S}^n$.

One reduces at once to the case where $X=\mathrm{Spec}(B)$ and $S=\mathrm{Spec}(A)$ are affine, and the proposition follows from (16.3.7.1) since ${\pi }_n$ is surjective. Note that in general $d_{X/S}^n$ is not surjective (even already for $n=1$).

Proposition (16.3.9).

Suppose that $f:X\to S$ is a morphism locally of finite type. Then the ${\mathcal{P}}_{X/S}^n$ and the ${\mathcal{G}\mathcal{R}}_n({\mathcal{P}}_{X/S})$ are quasi-coherent ${\mathcal{O}}_X$-Modules of finite type.

This follows from (16.1.6) and from the fact that ${\Delta }_f$ is locally of finite presentation (1.4.3.1).

16.4. Functorial properties of differential invariants

(16.4.1).

Consider a commutative diagram of morphisms of preschemes

  (16.4.1.1)
                  X   <──u──   X'
                  │            │
                  f            f'
                  ↓            ↓
                  S   <──w──   S'

One deduces a commutative diagram

                   X        <──u──    X'
                   │                  │
                   Δ_f                Δ_{f'}
                   ↓                  ↓
                X ×_S X   <──v──   X' ×_{S'} X'

where $v$ is the composite morphism (I, 5.3.5 and 5.3.15)

  (16.4.1.2)    X' ×_{S'} X' ──(p'_1, p'_2)_S──> X' ×_S X' ──u ×_S u──> X ×_S X.

One therefore deduces from $u$ and $v$, as was explained in (16.2.1), homomorphisms of augmented sheaves of rings

$$(\mathrm{16.4.1.3}){\nu }_n:\rho \ast ({\mathcal{P}}_{X/S}^n)\to {\mathcal{P}}_{X^{\prime }/S^{\prime }}^n$$

(where one has set $u=(\rho ,\lambda )$); these homomorphisms form a projective system, and consequently give at the limit a homomorphism of augmented sheaves of rings

$$(\mathrm{16.4.1.4}){\nu }_{\infty }:\rho \ast ({\mathcal{P}}_{X/S}^{\infty })\to {\mathcal{P}}_{X^{\prime }/S^{\prime }}^{\infty };$$

on the other hand, by passage to the quotients, the homomorphisms ${\nu }_n$ give a di-homomorphism of graded Algebras (relative to ${\lambda }^{♯}$):

$$(\mathrm{16.4.1.5})gr(u):\rho \ast ({\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X/S}))\to {\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X^{\prime }/S^{\prime }}).$$

(16.4.2).

If one has a commutative diagram

        X   <──u──   X'  <──u'──  X''
        │            │            │
        f            f'           f''
        ↓            ↓            ↓
        S   <──w──   S'  <──w'──  S''

one deduces a commutative diagram

              X       <──u──     X'     <──u'──     X''
              │                  │                  │
              Δ_f                Δ_{f'}             Δ_{f''}
              ↓                  ↓                  ↓
           X ×_S X   <──v──   X' ×_{S'} X'  <─v'─  X'' ×_{S''} X''

where $v^{\prime }$ is defined from $u^{\prime }$, $w^{\prime }$, $f^{\prime }$, f'' as $v$ was from $u$, $w$, $f$, $f^{\prime }$. One verifies at once that if $u^{\prime \prime }=u\circ u^{\prime }$, $w^{\prime \prime }=w\circ w^{\prime }$, then the composite morphism $v\circ v^{\prime }$ is equal to the morphism v'' defined from u'', w'', $f$, f'' as $v$ was from $u$, $w$, $f$, $f^{\prime }$. If one sets $u^{\prime }=({\rho }^{\prime },{\lambda }^{\prime })$, $u^{\prime \prime }=({\rho }^{\prime \prime },{\lambda }^{\prime \prime })$, it then follows from (16.2.1) that the homomorphism ${\nu }_n^{\prime \prime }:{\rho }^{\prime \prime }\ast ({\mathcal{P}}_{X/S}^n)\to {\mathcal{P}}_{X^{\prime \prime }/S^{\prime \prime }}^n$ is equal to the composite

  ρ'*(ρ*(𝒫_{X/S}^n)) ──ρ'*(ν_n)──> ρ'*(𝒫_{X'/S'}^n) ──ν'_n──> 𝒫_{X''/S''}^n,

and one has analogous transitivity properties for the homomorphisms (16.4.1.4) and (16.4.1.5), which allows one to say that the ${\mathcal{P}}_{X/S}^n$, ${\mathcal{P}}_{X/S}^{\infty }$ and ${\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X/S})$ depend functorially on $f$.

(16.4.3).

One verifies at once (for example by reducing to the affine case using (16.3.7)) that with the notation of (16.4.1), the diagram

  (16.4.3.1)
              ρ*(𝒪_X)  ──λ^#──>  𝒪_{X'}
                │                  │
                ↓                  ↓
            ρ*(𝒫_{X/S}^n) ──ν_n──> 𝒫_{X'/S'}^n

where the vertical arrows are those defining the algebra structures chosen in (16.3.5) (that is to say, those coming from the first projections), is commutative; the same holds for the diagram

  (16.4.3.2)
              ρ*(𝒪_X)         ──λ^#──>      𝒪_{X'}
                │                              │
                ρ*(d_{X/S}^n)                  d_{X'/S'}^n
                ↓                              ↓
            ρ*(𝒫_{X/S}^n)   ──ν_n──>      𝒫_{X'/S'}^n

the vertical arrows here defining the algebra structures coming from the second projections; moreover, if $\sigma$ and ${\sigma }^{\prime }$ are the canonical symmetries corresponding to $f$ and $f^{\prime }$ (16.3.4), one has

  ν_n ∘ ρ*(σ) = σ' ∘ ν_n

which lets one pass from one of the preceding diagrams to the other. One therefore deduces from (16.4.3.1) a canonical homomorphism of augmented ${\mathcal{O}}_{X^{\prime }}$-Algebras

  (16.4.3.3)    P^n(u) : u*(𝒫_{X/S}^n) = 𝒫_{X/S}^n ⊗_{𝒪_X} 𝒪_{X'} → 𝒫_{X'/S'}^n

and it follows from (16.4.3.2) that the diagram

  (16.4.3.4)
              𝒪_{X'}             ──id──>          𝒪_{X'}
                │                                  │
                u*(d_{X/S}^n)                      d_{X'/S'}^n
                ↓                                  ↓
            u*(𝒫_{X/S}^n)     ──P^n(u)──>      𝒫_{X'/S'}^n

is commutative. One deduces from it a homomorphism of graded ${\mathcal{O}}_{X^{\prime }}$-Algebras

$$(\mathrm{16.4.3.5})G{r}_{\bullet }(u):u\ast ({\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X/S}))\to {\mathcal{G}\mathcal{R}}_{\bullet }({\mathcal{P}}_{X^{\prime }/S^{\prime }})$$

and in particular a homomorphism of ${\mathcal{O}}_{X^{\prime }}$-Modules

  (16.4.3.6)    Gr_1(u) : Ω_{X/S}^1 ⊗_{𝒪_X} 𝒪_{X'} → Ω_{X'/S'}^1

giving rise to a commutative diagram

  (16.4.3.7)
              𝒪_{X'}                ──id──>               𝒪_{X'}
                │                                            │
                d_{X/S} ⊗ 1                                  d_{X'/S'}
                ↓                                            ↓
            Ω_{X/S}^1 ⊗_{𝒪_X} 𝒪_{X'}    ────────────>    Ω_{X'/S'}^1

(16.4.4).

When $S=\mathrm{Spec}(A)$, $S^{\prime }=\mathrm{Spec}(A^{\prime })$, $X=\mathrm{Spec}(B)$, $X^{\prime }=\mathrm{Spec}(B^{\prime })$ are affine, so that one has a commutative diagram of ring homomorphisms

              B  ──>  B'
              ↑       ↑
              A  ──>  A'

the image of ${\mathfrak{J}}_{B/A}$ in $B^{\prime }{\otimes }_{A^{\prime }}{B}^{\prime }$ is contained in ${\mathfrak{J}}_{B^{\prime }/A^{\prime }}$, and the homomorphism ${\nu }_n$ corresponds to the ring homomorphism $P_{B/A}^n\to P_{B^{\prime }/A^{\prime }}^n$ deduced from the homomorphism $B{\otimes }_{A}B\to B^{\prime }{\otimes }_{A^{\prime }}{B}^{\prime }$ by passage to the quotients. The homomorphism (16.4.3.6) corresponds to the homomorphism defined in (0, 20.5.4.1), and the commutative diagram (16.4.3.7) to the diagram (0, 20.5.4.2).

Proposition (16.4.5).

Suppose that $X^{\prime }=X{\times }_S{S}^{\prime }$, with $f^{\prime }$ and $u$ the canonical projections.

Then the canonical homomorphisms $P^n(u)$ (16.4.3.3) and $G{r}_1(u)$ (16.4.3.6) are bijective.

One indeed has $X^{\prime }{\times }_{S^{\prime }}{X}^{\prime }=(X{\times }_{S}X){\times }_S{S}^{\prime }$, and it suffices to apply (16.2.3, (ii)) replacing $g$ by the first projection $p_1:X{\times }_{S}X\to X$ and $f$ by the diagonal ${\Delta }_f$.