Exposé III. Smooth Morphisms: Extension Properties

1. Formally Smooth Homomorphisms

In II, we limited ourselves to homomorphisms of finite type and, consequently, in local homomorphisms $A\to B$ of local rings, to the case where B is isomorphic to a localization of an A-algebra of finite type. This case is insufficient for various applications, notably in formal geometry or analytic geometry. For example, the formal power-series ring $B=A[[t_1,...,t_n]]$ has, from the point of view of formal geometry, the properties of a smooth algebra over A. In analytic geometry, the same is true of the local ring of a point (y,z) of a product $Y\times {\mathbb{C}}^n$, regarded as an algebra over the local ring of y; moreover, the completion of this algebra is isomorphic to the algebra of formal power series in n indeterminates over the completion of the base ring ${\mathcal{O}}_x$. This leads to the following definition.

Definition.

Let $u:A\to B$ be a local homomorphism of local rings, noetherian as recalled. Suppose that $\kappa (B)$ is finite over $\kappa (A)$. We say that u is a formally smooth homomorphism, or that the algebra B is formally smooth over A, if there exists a local finite $\bar{A}$-algebra $A^{\prime }$, free over $\bar{A}$, such that the local components of the semi-local ring $\bar{B}{\otimes }_{\bar{A}}{A}^{\prime }=B^{\prime }$ are $A^{\prime }$-isomorphic to algebras of formal power series over $A^{\prime }$.¹

Here $\bar{A}$ and $\bar{B}$ denote the completions of A and B. Since $B^{\prime }$ is finite and free over $\bar{B}$, it is indeed a semi-local ring, a direct sum of complete local rings, each of which is still a free module over $\bar{B}$, hence has the same dimension as $\bar{B}$, and therefore as B. It follows that the number of variables $t_i$ in the formal power-series rings considered in III.1.1 is equal to dim B̄ − dim Ā = dim B − dim A, and in particular is independent of the local component chosen. One sees at once that it is also the dimension of the ring $B\otimes k=B/\mathfrak{m}B$, where $k=A/\mathfrak{m}$ is the residue field of A; we shall call it the relative dimension of B with respect to A.

Remarks.

It is clear that Definition III.1.1 depends only on the homomorphism on completions $\bar{A}\to \bar{B}$ deduced from $A\to B$, which justifies the terminology to some extent. We repent here of Definitions I.3.2 b) and I.4.1 b), which risk being misleading, and prefer to say “formally unramified” and “formally étale” in the cases considered in those definitions, reserving the terminology “unramified” and “étale” for the case where B is a localization of an A-algebra of finite type.² The reader will immediately verify that “formally étale” is equivalent to “formally smooth and quasi-finite”. Finally, let us point out that there is a reasonable definition of “formally smooth” without any prior hypothesis on the residual extension $\kappa (B)/\kappa (A)$, supposed finite here, encompassing among others the local homomorphisms $A\to B$ such that B is flat over A and $B/\mathfrak{m}B$ is a separable extension of $A/\mathfrak{m}=k$, not necessarily of finite type. For example, a Cohen p-ring is formally smooth over the ring of p-adic integers. It is the lifting property for homomorphisms, compare III.2.1, that should become the definition in this general case. For the applications we have in view, the case treated in Definition III.1.1 will suffice; in what follows, in “formally smooth” we shall understand “with finite residual extension”.

Lemma.

If B is formally smooth over A, then B is flat over A.

Since flatness is invariant under completion, we may suppose A and B complete. Since flatness is invariant under a local flat, hence faithfully flat, extension of the base ring, Definition III.1.1 reduces us to the case where B is a formal power-series algebra over A. But then, as an A-module, B is isomorphic to a product of A-modules isomorphic to A; hence, since the base ring A is noetherian, B is A-flat as a product of flat A-modules.

Let us place ourselves under the conditions of III.1.1. Since the residual extensions of the local components of $B^{\prime }$ over $A^{\prime }$ are trivial, it follows that $L{\otimes }_k{k}^{\prime }$ is an artinian $k^{\prime }$-algebra whose local components have trivial residual extensions, where L, k, $k^{\prime }$ are the residue fields of A, B, and $A^{\prime }$. This necessary condition for the finite free extension $A^{\prime }$ to satisfy the condition stated in III.1.1 is also sufficient, as follows at once from III.1.4 (i) and III.1.5 below.

Proposition.

Let $A\to B$ be a local homomorphism of local rings with finite residual extension, and let $A^{\prime }$ be a finite local A-algebra over A, so that $B^{\prime }=B{\otimes }_A{A}^{\prime }$ is finite over B and hence is a semi-local ring, also noetherian.

1. If B is formally smooth over A, then the localizations of $B^{\prime }$ at its maximal ideals are formally smooth over $A^{\prime }$.
2. The converse is true if $A^{\prime }$ is free over A.

We are immediately reduced to the case where A and B are complete.

For (i), let $A^{\prime \prime }$ be a finite free local extension of A such that the local components of $B^{\prime \prime }=B{\otimes }_A{A}^{\prime \prime }$ are formal power-series algebras over $A^{\prime \prime }$. Extending scalars $A^{\prime \prime }\to A^{\prime \prime }{\otimes }_A{A}^{\prime }\to A^{\prime \prime \prime }$, where $A^{\prime \prime \prime }$ is a local component of $A^{\prime \prime }{\otimes }_A{A}^{\prime }$, one sees that the local components of $B^{\prime \prime }{\otimes }_{A^{\prime \prime }}{A}^{\prime \prime \prime }=B{\otimes }_A{A}^{\prime \prime \prime }$ are formal power-series algebras over $A^{\prime \prime \prime }$. But we also have

B ⊗_A A‴ = (B ⊗_A A′) ⊗_{A′} A‴ = B′ ⊗_{A′} A‴.

Moreover, since $A^{\prime \prime }$ is free over A, $A^{\prime \prime }{\otimes }_A{A}^{\prime }$ is free over $A^{\prime }$, and consequently so is $A^{\prime \prime \prime }$, which is a direct factor of it. This proves that $B^{\prime }$ is formally smooth over $A^{\prime }$.

For (ii), let $A^{\prime \prime }$ be a finite free local $A^{\prime }$-algebra such that the local components of $B^{\prime }{\otimes }_{A^{\prime }}{A}^{\prime \prime }=B{\otimes }_A{A}^{\prime \prime }$ are formal power-series algebras over $A^{\prime \prime }$. Since $A^{\prime }$ is free over A, so is $A^{\prime \prime }$; hence B is formally smooth over A.

Proposition.

Let $A\to B$ be a local homomorphism of local rings with trivial residual extension. In order that B be formally smooth over A, it is necessary and sufficient that $\bar{B}$ be isomorphic to a formal power-series algebra over $\bar{A}$.

Only the necessity has to be proved, and we may suppose A and B complete. Let $\mathfrak{m}$ and $\mathfrak{n}$ be the maximal ideals of A and B, respectively, and let $t_1,...,t_n$ be elements of $\mathfrak{n}$ defining a basis of the vector space

$$(\mathfrak{n}/{\mathfrak{n}}^2)/Im(\mathfrak{m}/{\mathfrak{m}}^2)=\mathfrak{n}/({\mathfrak{n}}^2+\mathfrak{m}B).$$

These elements therefore define a homomorphism of local A-algebras

$$B_1=A[[t_1,...,t_n]]\to B.$$

We prove that it is an isomorphism. It suffices to prove that for every power ${\mathfrak{m}}^q$ of $\mathfrak{m}$, one obtains an isomorphism after reducing modulo ${\mathfrak{m}}^q$, since $B_1$ and B are the projective limits of the corresponding rings reduced modulo ${\mathfrak{m}}^q$, with q variable. Since B and $B_1$ are flat A-modules, the graded rings associated with the $\mathfrak{m}$-adic filtration are obtained by tensoring over $k=A/\mathfrak{m}$ with $gr(A)$ the rings $B_1/\mathfrak{m}{B}_1$ and $B/\mathfrak{m}B$, respectively. We are thus reduced to showing that $B_1/\mathfrak{m}{B}_1\to B/\mathfrak{m}B$ is an isomorphism. Taking III.1.3 into account, we are thereby reduced to the case where A is a field k. On the other hand, if $A^{\prime }$ is a finite free local A-algebra such that $B{\otimes }_A{A}^{\prime }$ is a formal power-series algebra over $A^{\prime }$, note that this algebra is local since the residual extension of B over A is trivial. To prove that $B_1\to B$ is an isomorphism, it suffices to prove that $B_1{\otimes }_A{A}^{\prime }\to B{\otimes }_A{A}^{\prime }$ is one. This reduces us to the case where B is already a formal power-series algebra; this reduction should have been made first, before reducing to the case of a base field. But then B is a regular local ring with coefficient field k, and it is well known, and immediate by considering the graded rings associated with the ${\mathfrak{n}}_1$-adic and $\mathfrak{n}$-adic filtrations on $B_1$ and B, that $B_1\to B$ is an isomorphism. This completes the proof.

Corollary.

If B is formally smooth over A, then there exists a finite local A-algebra $A^{\prime }$ such that the local components of

B̄ ⊗_{Ā} A′̄ = completion of (B ⊗_A A′)

are isomorphic to formal power-series algebras over $A\overline{{}^{\prime }}$.

Indeed, if L/k is the residual extension of B/A, consider an extension $k^{\prime }/k$ such that the residual extensions in the $k^{\prime }$-algebra $L{\otimes }_k{k}^{\prime }$ are trivial. Let $A^{\prime }$ be an algebra finite and free over A such that $A^{\prime }/\mathfrak{m}{A}^{\prime }=k^{\prime }$; one knows that such an algebra exists, for example by reducing step by step to the case where $k^{\prime }/k$ is monogenic, and then lifting to A the coefficients of the minimal polynomial of a generator of $k^{\prime }$ over k. It is local. Then $B{\otimes }_A{A}^{\prime }$ has, at its maximal ideals, trivial residual extensions over that $k^{\prime }$ of $A^{\prime }$, and the conclusion follows with the help of III.1.5.

Corollary.

Let $A\to B$ be a local homomorphism of local rings. In order that B be formally smooth over A, it is necessary and sufficient that B be flat over A and that $B/\mathfrak{m}B$ be formally smooth over $k=A/\mathfrak{m}$.

Making a suitable finite free local extension $A^{\prime }$ of A and using III.1.4 (ii), we are reduced to the case where the residual extension of B/A is trivial. We know moreover by III.1.4 (i) and III.1.3 that the stated conditions are necessary. For the sufficiency, it suffices to observe that the proof of III.1.5 proves, under the hypotheses made here, that B is a formal power-series algebra over A, supposing A and B complete, which is permissible.

Remark.

It would not be difficult to develop, for formally smooth homomorphisms, the analogue of all the properties of smooth morphisms studied in II. For the differential properties, however, this requires a modification of the usual definition of Kähler differentials, cf. I.1, with completed tensor products replacing ordinary tensor products. We shall content ourselves with evoking these abysses here, what precedes being sufficient for our purpose.

It remains to make the link between formal smoothness and the notion of smoothness developed in II, which we have not yet used at all.

Proposition.

Let $A\to B$ be a local homomorphism, with B a localization of an A-algebra of finite type. In order that B be smooth over A, it is necessary and sufficient that it be formally smooth over A.

Using III.1.7 and II.2.1, we are reduced to the case where A is a field.

Using III.1.4 (ii) and II.4.13, a suitable extension $k^{\prime }$ of k reduces us to the case where the residual extension for B/k is trivial. By III.1.5, respectively II.5.2, B is smooth over k, respectively formally smooth over k, if and only if B is a regular local ring, respectively its completion is a formal power-series algebra over k. But it is well known that these two conditions are equivalent when the residual extension is trivial.

2. The Lifting Property Characteristic of Formally Smooth Homomorphisms

Theorem.

Let $A\to B$ be a local homomorphism of local rings defining a finite residual extension. The following conditions are equivalent:

1. B is formally smooth over A.
2. For every local homomorphism $A\to C$, where C is a complete local ring, every ideal J of C contained in the radical $\mathfrak{r}(C)$, and every local A-homomorphism $B\to C/J$, there exists an A-homomorphism, necessarily local, $B\to C$ lifting it.
3. For every A-algebra C, not necessarily noetherian, every nilpotent ideal J of C, and every continuous A-homomorphism $B\to C/J$, i.e. one vanishing on a power of $\mathfrak{r}(B)$, there exists an A-homomorphism $B\to C$, necessarily continuous as well, lifting it.
4. The same statement as (ii) and (iii), but with C a local artinian ring finite over A.
5. As in (iv), but with J moreover square-zero.

Remark. For the rest of this exposé, we shall use only the implication (iv) ⇒ (i), or (iv bis) ⇒ (i). The direct implication (i) ⇒ (ii) will be proved by another method in the next number when B is a localization of an algebra of finite type over A. Recall that in the “good” theory of Cohen theorems,³ property (ii) or (iii) becomes the definition of formally smooth homomorphisms, while III.1.1 becomes a characteristic property valid only in the case of a finite residual extension. Care should be taken that neither of properties (ii) and (iii) is more general than the other. One could give an equivalent property covering both by introducing a linearly topologized ring C, separated and complete, a closed topologically nilpotent ideal of C, and a continuous homomorphism $A\to C$, thus making C a topological A-algebra; we leave this modification to the reader.

Proof of III.2.1. We shall prove (i) ⇒ (iii) ⇒ (ii), then (iv) ⇒ (i). Since (ii) ⇒ (iv) is trivial, and the equivalence of (iv) and (iv bis) is seen by an immediate induction on the integer n such that $J^n=0$, this will finish the proof.

(i) ⇒ (iii). An immediate induction reduces us to the case J² = 0. Since C is finite over A, some power ${\mathfrak{m}}^q$ of the maximal ideal of A annihilates C. Dividing by ${\mathfrak{m}}^q$, and noting that $B/{\mathfrak{m}}^{qB}$ is still formally smooth over $A/{\mathfrak{m}}^q$ by III.1.4 (i), we may suppose A artinian. Since B is flat over A by III.1.3, B is free over A because A is artinian. Thus there exists an A-module homomorphism

$$w:B\to C$$

lifting the given homomorphism $u:B\to C/J$. Put

f(x,y) = w(xy) − w(x)w(y),     x,y ∈ B.

Then f(x,y) ∈ J, and f is therefore an A-bilinear map from $B\times B$ to J. For there to exist a lift $v:B\to C$ of u that is an algebra homomorphism, it is necessary and sufficient that there exist an A-linear map $g:B\to J$ such that $v=w+g$ is an algebra homomorphism, which is written

g(1) = 1 − w(1),
g(xy) − u(x)g(y) − u(y)g(x) = −f(x,y),     x,y ∈ B.

This is a system of linear equations in ${\mathrm{Hom}}_A(B,J)$, with right-hand sides in J. Hence it has a solution if and only if the corresponding system in ${\mathrm{Hom}}_A(B,J){\otimes }_A{A}^{\prime }$, with right-hand sides in $J^{\prime }=J{\otimes }_A{A}^{\prime }$, has a solution, where $A^{\prime }$ denotes a faithfully flat algebra over A. Let $A^{\prime }$ be an algebra finite and free over A, local, such that $B^{\prime }=B{\otimes }_A{A}^{\prime }$ is a formal power-series algebra over $A^{\prime }$. In our proof we may suppose A and B complete, as is immediately checked. Since $A^{\prime }$ is free of finite type over A, we have

Hom_A(B,J) ⊗_A A′ = Hom_{A′}(B′,J′),

and one verifies that the system of equations obtained in ${\mathrm{Hom}}_{A^{\prime }}(B^{\prime },J^{\prime })$ is the one that determines the homomorphisms of $A^{\prime }$-algebras $B^{\prime }\to C^{\prime }=C{\otimes }_A{A}^{\prime }$ lifting the homomorphism $u^{\prime }:B^{\prime }\to C^{\prime }/J^{\prime }$ deduced from u by extension of scalars, by “correcting” by an $A^{\prime }$-module homomorphism $g^{\prime }:B^{\prime }\to J^{\prime }$ the $A^{\prime }$-module homomorphism $w^{\prime }:B^{\prime }\to C^{\prime }$ deduced from w by extension of scalars. Note that B generates $B^{\prime }$ as an $A^{\prime }$-module. We are thereby reduced to proving (iii) when B is a formal power-series algebra over A, $B=A[[t_1,...,t_n]]$. Lift arbitrarily the images in C/J of the $t_i$ to elements $z_i$ of C. Since the $z_i$ modulo J are nilpotent, $u:B\to C/J$ being continuous, the $z_i$ themselves are nilpotent, since J is nilpotent. Thus the $z_i$ define a continuous homomorphism of topological A-algebras from B to the discrete ring C, evidently lifting u, as required.

(iii) ⇒ (ii). Let $\mathfrak{n}$ be the maximal ideal of C, and for every integer q > 0 put

C_q = C/𝔫^q,    J_q = (J + 𝔫^q)/𝔫^q.

Thus $C_q/J_q$ identifies with a quotient algebra of C/J. On the other hand, the composite homomorphism $u_q:B\to C/J\to C_q/J_q$ is continuous from B to the discrete ring $C_q/J_q$, and $J_q$ is a nilpotent ideal in $C_q$. We then construct, step by step, A-homomorphisms

$$v_q:B\to C_q$$

such that (a) $v_q$ lifts $u_q$ and (b) $v_q$ lifts $v_{q-1}$. The possibility of the induction is checked easily: since

u_q: B → C/(J + 𝔫^q)     and     v_{q−1}: B → C/𝔫^{q−1}

define the same homomorphism

B → C/((J + 𝔫^q) + 𝔫^{q−1}) = C/(J + 𝔫^{q−1}) = C_{q−1}/J_{q−1},

namely $u_{q-1}$, they define a homomorphism

B → C/J′_q,    where J′_q = (J + 𝔫^q) ∩ 𝔫^{q−1} ⊃ 𝔫^q,

from which both arise by reduction. We are therefore reduced to lifting a homomorphism $B\to C/J_q^{\prime }$ from B into a quotient of $C_q$ by an ideal $J_q^{\prime }/{\mathfrak{n}}^q$ contained in $J_q$, hence nilpotent; this is possible by hypothesis (iii).

This done, the $v_q$ define a homomorphism from B into the projective limit C of the $C_q$. Since J is closed, J is the projective limit of the $J_q$; hence v lifts u, as required.

(iv) ⇒ (i). First one observes at once that if (iv) holds, then (iv) remains true for the local components of $B{\otimes }_A{A}^{\prime }$ over $A^{\prime }$, if $A^{\prime }$ is a finite local algebra over A. Taking $A^{\prime }$ free over A and such that the residual extensions of $B^{\prime }$ over $A^{\prime }$ are trivial, we are reduced, by III.1.4 (ii), to the case where the residual extension of B over A is trivial. We shall then prove the slightly more precise result:

Corollary.

Under the conditions of III.2.1, suppose moreover that the residual extension of B over A is trivial. Then the equivalent conditions of III.2.1 are also equivalent to the following two conditions, supposing in (v) that A and B are complete:

1. As in (iv), but with the local artinian ring C finite over A restricted to have trivial residual extension; and moreover, if one wants, with the ideal J square-zero.
2. There exists a local A-homomorphism, where $n=\dim \mathfrak{n}/({\mathfrak{n}}^2+\mathfrak{m}B)$,

u: B → B₁ = A[[t₁,...,t_n]]

inducing an isomorphism

𝔫/(𝔫² + 𝔪B) → 𝔫₁/(𝔫₁² + 𝔪B₁),

where $\mathfrak{n}$ and ${\mathfrak{n}}_1$ are the maximal ideals of B and $B_1$, respectively, and $\mathfrak{m}$ is that of A.

Proof. Since (iv bis) evidently implies (iv ter), setting aside the square-zero-ideal joke, it will suffice to prove (iv ter) ⇒ $(v)\Rightarrow (i)$.

For (iv ter) ⇒ (v), choose a basis $a_1,...,a_n$ of $\mathfrak{n}/({\mathfrak{n}}^2+\mathfrak{m}B)$. This therefore defines a local homomorphism of A-algebras

B → B₁/(𝔫₁² + 𝔪B₁) = k[t₁,...,t_n]/(t₁,...,t_n)²,

which can be lifted step by step, by (iv ter), to homomorphisms of A-algebras from B into $B_1/{\mathfrak{n}}_1^2$, $B_1/{\mathfrak{n}}_1^3$, and so on; passing to the projective limit gives the homomorphism $B\to B_1$ with the desired property.

For $(v)\Rightarrow (i)$, in the commutative diagram

𝔪/𝔪² → 𝔫/𝔫² → 𝔫/(𝔫² + 𝔪B) → 0
↓       ↓       ↓
𝔪/𝔪² → 𝔫₁/𝔫₁² → 𝔫₁/(𝔫₁² + 𝔪B₁) → 0

the two rows are exact, and the extreme vertical arrows are surjective; the middle arrow is therefore surjective, and it follows, since B is complete, that $B\to B_1$ is surjective. Let $x_i$, $1\le i\le n$, be elements of B lifting the $t_i$. They define a homomorphism of A-algebras $B_1\to B$, which is surjective for the same reason as u, and whose composite with u is the identity by construction. Thus $B_1\to B$ is also injective, and consequently is an isomorphism. We obtain:

Corollary.

Under the conditions of III.2.2 (v), u is necessarily an isomorphism.

This finishes the proof that B is formally smooth over A. At the same time we have recovered III.1.5, though there is little merit in that.

3. Local Infinitesimal Extension of Morphisms into a Smooth S-Scheme

Theorem.

Let $f:X\to Y$ be a morphism locally of finite type. The following conditions are equivalent:

1. f is smooth.
2. For every prescheme $Y^{\prime }$ over Y, every closed sub-prescheme $Y_0^{\prime }$ of $Y^{\prime }$ having the same underlying space as $Y^{\prime }$, every Y-morphism $g_0:Y_0^{\prime }\to X$, and every $z\in Y_0^{\prime }$, there exists an open neighborhood U of z in $Y^{\prime }$ and an extension g of $g_0{|}_{Y_0^{\prime }\cap U}$ to a Y-morphism $U\to X$.
3. For $Y^{\prime }$, $Y_0^{\prime }$, and z as in (ii), putting $X^{\prime }=X{\times }_Y{Y}^{\prime }$ and $X_0^{\prime }=X{\times }_Y{Y}_0^{\prime }$, every section of $X_0^{\prime }$ over $Y_0^{\prime }$ extends to a section of $X^{\prime }$ over an open neighborhood U of z.
4. For every Y-scheme $Y^{\prime }$ that is the spectrum of a local artinian ring finite over some ${\mathcal{O}}_y$, with $y\in Y$, every nonempty closed sub-prescheme $Y_0^{\prime }$ of $Y^{\prime }$, and every Y-morphism $g_0:Y_0^{\prime }\to X$, there exists a Y-morphism $g:Y^{\prime }\to X$ extending $g_0$.
5. For every $Y^{\prime }$ and $Y_0^{\prime }$ as in (iii), putting $X^{\prime }=X{\times }_Y{Y}^{\prime }$ and $X_0^{\prime }=X{\times }_Y{Y}_0^{\prime }$, every section of $X_0^{\prime }$ over $Y_0^{\prime }$ extends to a section of $X^{\prime }$ over $Y^{\prime }$.

Proof. The equivalence of (ii) and (ii bis), on the one hand, and of (iii) and (iii bis), on the other, is trivial, as is the implication (ii) ⇒ (iii). It remains to prove (i) ⇒ (ii) and (iii) ⇒ (i).

(i) ⇒ (ii). Let $x=g_0(z)$. Replacing X by a suitable open neighborhood of x, and $Y^{\prime }$ by the prescheme induced on the open inverse image of the latter under $g_0$, we may suppose that X is étale over $Y[t_1,...,t_n]$. Consider the composite Y-morphism $Y_0^{\prime }\to X\to Y[t_1,...,t_n]$; it is defined by n sections of the sheaf ${\mathcal{O}}_{Y_0^{\prime }}$, which can therefore be extended in a neighborhood of z to sections of ${\mathcal{O}}_{Y^{\prime }}$. Thus we may suppose that the morphism in question has been extended to a Y-morphism $Y^{\prime }\to Y[t_1,...,t_n]$. By I.5.6, there is then a unique Y-morphism $g:Y^{\prime }\to X$ lifting the preceding one and at the same time extending $g_0$.

(iii) ⇒ (i). Since the set of points where f is smooth is open, it suffices to prove that it contains every $x\in X$ that is closed in its fiber. Let y = f(x). Then ${\mathcal{O}}_x$ is an algebra over ${\mathcal{O}}_y$, a localization of an algebra of finite type, with finite residual extension. On the other hand, hypothesis (iii) implies that every homomorphism from ${\mathcal{O}}_x$ into an algebra A/J, where A is a local artinian algebra finite over ${\mathcal{O}}_y$ and J is an ideal contained in its radical, lifts to a homomorphism from ${\mathcal{O}}_x$ into the algebra A, taking into account that a morphism from $\mathrm{Spec}(B)$, with B local, into X is determined bijectively by a local homomorphism from some ${\mathcal{O}}_x$, $x\in X$, into B. By III.2.1 it follows that ${\mathcal{O}}_x$ is formally smooth over ${\mathcal{O}}_y$, hence smooth over ${\mathcal{O}}_y$ by III.1.9.

Corollary.

Let $f:X\to Y$ be as in III.3.1. The following conditions are equivalent:

1. f is étale.
2. Condition (ii) of III.3.1 holds with uniqueness of the extension g of $g_0$ to U.
3. Condition (iii) of III.3.1 holds with uniqueness of g.

It suffices to note, in the proof of (i) ⇒ (ii) above, that one can have uniqueness, when $Y_0^{\prime }$ is not identical to $Y^{\prime }$ in a neighborhood of z, only if $n=0$, a condition that is known to be sufficient.

Corollary.

Let X be a prescheme locally of finite type over a complete local ring A, let y be the closed point of $Y=\mathrm{Spec}(A)$, and let x be a point of $f^{-1}(y)$ rational over $\kappa (y)$. If X is smooth over A at x, then there exists a section s of X over Y “passing through x”, i.e. such that s(y) = x.

In particular, if X is smooth over A, then the natural map

Γ(X/Y) → Γ(X ⊗_A k / k)

from sections of X over Y to the set of points of the fiber $f^{-1}(y)=X{\otimes }_{A}k$ rational over k is surjective. This fact was especially well known and used when A is a discrete valuation ring and X is proper over A, in fact projective over A. In that case the sections of X over Y, i.e. the “points of X with values in A”, also identify with the rational sections, i.e. with the points of $X{\otimes }_{A}K=X_K$, which is a proper smooth scheme over K, with values in K, the field of fractions of A; in other words, with the points of X rational over K.

4. Local Infinitesimal Extension of Smooth S-Schemes

Theorem.

Let Y be a locally noetherian prescheme, let $Y_0$ be a closed sub-prescheme with the same underlying space, let $X_0$ be a smooth $Y_0$-prescheme, and let x be a point of $X_0$. Then there exist an open neighborhood $U_0$ of x, a prescheme X smooth over Y, and a $Y_0$-isomorphism

h: U₀ → X ×_Y Y₀.

Moreover, if (U′₀, $X^{\prime }$, h′) is another solution of this problem, then “it is isomorphic to the first in a neighborhood of x”.

We leave it to the reader to make precise what is meant by this. One may note that, for $U_0$ given, a solution of the stated problem amounts to giving on $U_0$ a sheaf of algebras $\mathcal{B}$ over $f_0^{-1}({\mathcal{O}}_Y)$, where $f_0$ is the continuous map underlying the structural morphism $U_0\to Y_0$, together with a homomorphism of rings $\mathcal{B}\to {\mathcal{O}}_{U_0}$ compatible with the homomorphism $f^{-1}({\mathcal{O}}_Y)\to f^{-1}({\mathcal{O}}_{Y_0})$, such that:

1. This homomorphism induces an isomorphism

ℬ ⊗_{f⁻¹(𝒪_Y)} f⁻¹(𝒪_{Y₀}) → 𝒪_{Y₀}.

1. $U_0$ equipped with $\mathcal{B}$ becomes a smooth Y-prescheme.

In this way the precise meaning of the assertion of local uniqueness becomes particularly evident.

Proof. We may already suppose that $X_0$ is étale over some $Y_0[t_1,...,t_n]=Y_0^{\prime }$. But the latter may be regarded as a closed sub-prescheme of $Y^{\prime }=Y[t_1,...,t_n]$ having the same underlying space. By I.8.3, there exists an X étale over $Y^{\prime }$ and a $Y_0^{\prime }$-isomorphism $X{\times }_{Y^{\prime }}{Y}_0^{\prime }\to X^{\prime }$. We have won existence. For uniqueness, use property III.3.1 (ii) of smooth morphisms, taking into account the following lemma.

Lemma.

Let Y be a prescheme, let $Y_0$ be a closed sub-prescheme defined by a locally nilpotent sheaf of ideals $\mathcal{J}$, let X and $X^{\prime }$ be Y-preschemes, and let $u:X\to X^{\prime }$ be a Y-morphism. Suppose X is flat over Y. In order that u be an isomorphism, it is necessary and sufficient that

u₀: X ×_Y Y₀ → X′ ×_Y Y₀

be an isomorphism.

The proof is easy, by passing to the affine case and looking at associated graded rings. One should note moreover that the analogous statement obtained by replacing “isomorphism” by “closed immersion” is also valid, and without the flatness hypothesis.

Remark.

It is essential to note that the local extension X obtained in III.4.1 is not canonical; in other words, the local isomorphism between two solutions is not unique, i.e. in general there exist nontrivial Y-automorphisms of X inducing the identity on the closed sub-prescheme $X_0=X{\times }_Y{Y}_0$. This is why, for the construction of global infinitesimal extensions of smooth preschemes, one must expect the existence of an obstruction of cohomological nature, which will be made precise below in III.6.

5. Global Infinitesimal Extension of Morphisms

Let T be a topological space, let $\mathcal{G}$ be a sheaf of groups on X, and let $\mathcal{P}$ be a sheaf of sets on T on which $\mathcal{G}$ acts, on the right to fix ideas. We say that $\mathcal{P}$ is formally principal homogeneous under $\mathcal{G}$ if the familiar homomorphism

𝒢 × 𝒫 → 𝒫 × 𝒫

of sheaves of sets, deduced from the operations of $\mathcal{G}$ on $\mathcal{P}$, is an isomorphism. This is equivalent to saying that for every $x\in T$, ${\mathcal{P}}_x$ is empty or a principal homogeneous space under the ordinary group ${\mathcal{G}}_x$; or also that for every open U of T, $\mathcal{P}(U)$ is empty or a principal homogeneous space under the ordinary group $\mathcal{G}(U)$. We say that $\mathcal{P}$ is a principal homogeneous sheaf under $\mathcal{G}$ if it is so formally and if, in addition, the ${\mathcal{P}}_x$ are nonempty; in other words, if all the ${\mathcal{P}}_x$ are principal homogeneous spaces, hence nonempty, under the ${\mathcal{G}}_x$.⁴ Recall that the set of classes, up to isomorphism, of principal homogeneous sheaves under $\mathcal{G}$ identifies with the cohomology set $H^1(T,\mathcal{G})$, which is also the usual cohomology group of T with coefficients in $\mathcal{G}$ when $\mathcal{G}$ is commutative. Thus, for every principal homogeneous $\mathcal{P}$, there is a characteristic class $c(\mathcal{P})\in H^1(T,\mathcal{G})$, whose triviality is necessary and sufficient for $\mathcal{P}$ to be trivial, i.e. isomorphic to $\mathcal{G}$, on which $\mathcal{G}$ acts by right translations, or equivalently for $\mathcal{P}$ to have a section.

Proposition.

Let S be a prescheme, let X and Y be preschemes over S, and let $Y_0$ be a closed sub-prescheme of Y defined by an ideal $\mathcal{J}$ on Y of square zero. Let $g_0$ be an S-morphism from $Y_0$ to X, and let $\mathcal{P}(g_0)$ be the sheaf on Y whose sections on an open U are the extensions $g:U\to X$ of $g_0{|}_{U\cap Y_0}$ to an S-morphism g. Then $\mathcal{P}(g_0)$ is, naturally, a formally principal homogeneous sheaf under the commutative sheaf of groups

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

Put $\mathcal{P}=\mathcal{P}(g_0)$. For every open U of Y we must define a map

$$\mathcal{P}(U)\times \mathcal{G}(U)\to \mathcal{P}(U)$$

so that: for fixed $g\in \mathcal{P}(U)$, the map s ↦ gs from $\mathcal{G}(U)$ to $\mathcal{P}(U)$ is bijective; $\mathcal{P}(U)$ becomes a set with group of operators $\mathcal{G}(U)$; and these maps are compatible with restriction operators for an open $V\subset U$. The verification of the last point is trivial, so for simplicity we may suppose $U=Y$. The verification of the second point, which is local if one wants, is left to the reader. We shall limit ourselves, for a given $g\in \mathcal{P}(Y)$, to defining a natural bijection from $\mathcal{G}(Y)$ onto $\mathcal{P}(Y)$. Thus suppose already given an S-morphism $g:Y\to X$, and seek a canonical bijection

$${\mathrm{Hom}}_{{\mathcal{O}}_{Y_0}}(g_0\ast ({\Omega }_{X/S}^1),\mathcal{J})\to \mathcal{P}(Y),$$

where $\mathcal{P}(Y)$ is the set of S-morphisms $g^{\prime }$ from Y to X inducing the same morphism $g_0:Y_0\to X$ as g. Giving such a $g^{\prime }$ is equivalent to giving an S-morphism $h:Y\to X{\times }_{S}X$ such that $p{r}_1\circ h=g$ and $h\circ i=(g_0,g_0)$, where $p{r}_1:X{\times }_{S}X\to X$ is the first projection, $i:Y_0\to Y$ is the canonical immersion, and $(g_0,g_0):Y_0\to X{\times }_{S}X$ is the morphism ${\Delta }_{X/S}{g}_0$ with components $g_0,g_0:$

Y₀ --h₀=(g₀,g₀)--> X ×_S X
|                         |
i                         pr₁
v                         v
Y  --------g----------->  X

Since $h_0$ factors through the diagonal immersion ${\Delta }_{X/S}$, and since Y is in the first-order infinitesimal neighborhood of $Y_0$, i.e. ${\mathcal{J}}^2=0$, the desired h necessarily factor, uniquely, through the first-order infinitesimal neighborhood of the diagonal. This neighborhood identifies, as an X-prescheme via $p{r}_1$, with the spectrum $X^{\prime }$ of the sheaf of algebras ${\mathcal{O}}_X+{\Omega }_{X/S}^1$, where the second term is regarded as a square-zero ideal; the diagonal morphism $X\to X^{\prime }$ corresponds to the canonical augmentation of this sheaf of algebras. Put $Y^{\prime }=X^{\prime }{\times }_{X}Y$ and $Y_0^{\prime }=Y^{\prime }{\times }_Y{Y}_0=X^{\prime }{\times }_X{Y}_0$. The desired h are then in bijective correspondence with sections u of $Y^{\prime }$ over Y extending a given section $u_0$ of $Y_0^{\prime }$ over $Y_0$. We may moreover identify $Y^{\prime }$ with the spectrum of the sheaf of algebras on Y

𝒜 = g*(𝒪_X + Ω¹_{X/S}) = 𝒪_Y + g*(Ω¹_{X/S}),

and $Y_0^{\prime }$ with the sheaf of algebras

𝒜₀ = 𝒜 ⊗_{𝒪_Y} 𝒪_{Y₀} = 𝒪_{Y₀} + g₀*(Ω¹_{X/S}).

Then $u_0$ is the section defined by the canonical augmentation of ${\mathcal{A}}_0$ into ${\mathcal{O}}_{Y_0}$. Thus $\mathcal{P}(Y)$ identifies with the set of algebra homomorphisms $\mathcal{A}\to {\mathcal{O}}_Y$ inducing the canonical augmentation ${\mathcal{A}}_0\to {\mathcal{O}}_{Y_0}$. But the algebra homomorphisms $\mathcal{A}\to {\mathcal{O}}_Y$ correspond bijectively to module homomorphisms $\mathcal{M}\to {\mathcal{O}}_Y$, putting for simplicity $\mathcal{M}=g\ast ({\Omega }_{X/S}^1)$, and we are interested in those inducing the zero homomorphism ${\mathcal{M}}_0\to {\mathcal{O}}_{Y_0}$, where ${\mathcal{M}}_0=\mathcal{M}{\otimes }_{{\mathcal{O}}_Y}{\mathcal{O}}_{Y_0}$; that is, those sending $\mathcal{M}$ into the augmentation ideal $\mathcal{J}$. We therefore find the set

$${\mathrm{Hom}}_{{\mathcal{O}}_Y}(\mathcal{M},\mathcal{J})={\mathrm{Hom}}_{{\mathcal{O}}_{Y_0}}({\mathcal{M}}_0,\mathcal{J}),$$

since $\mathcal{J}$ is annihilated by $\mathcal{J}$. This is the desired canonical bijection $III.5.1.\ast$.

Taking into account the implication (i) ⇒ (iii) in III.3.1, one obtains:

Corollary.

If X is smooth over S, at least at the points of $g_0(Y_0)$, then $\mathcal{P}$ is even a principal homogeneous sheaf under the commutative sheaf of groups $\mathcal{G}$, which in this case may also be written

𝒢 = g₀*(𝔤_{X/S}) ⊗_{𝒪_{Y₀}} 𝒥,

where ${\mathfrak{g}}_{X/S}$ is the sheaf on X dual to ${\Omega }_{X/S}^1$, i.e. the tangent sheaf, or sheaf of derivations, of X relative to S. This last formula comes from the fact that ${\Omega }_{X/S}^1$ is then free of finite type.

In particular, this principal homogeneous sheaf determines a cohomology class in $H^1(Y_0,\mathcal{G})$, whose vanishing is necessary and sufficient for the existence of an S-morphism g extending $g_0$. And if such an extension exists, the set of all possible extensions is a homogeneous space under the group $H^0(Y_0,\mathcal{G})$.

In applying the methods of formal geometry, the situation is most often the following. We are given two S-preschemes X and Y, and a coherent ideal $\mathcal{I}$ on S. Let $S_n$ denote the closed sub-prescheme of S defined by ${\mathcal{I}}^{n+1}$, and put

X_n = X ×_S S_n,    Y_n = Y ×_S S_n.

Suppose we have an $S_n$-morphism

$$g_n:Y_n\to X_n$$

or, what amounts to the same thing, an S-morphism $Y_n\to X$, or again an $S_{n+1}$-morphism $Y_n\to X_{n+1}$, since such a morphism necessarily induces $Y_n\to X_n$. We seek to extend it to an $S_{n+1}$-morphism

$$g_{n+1}:Y_{n+1}\to X_{n+1}.$$

If this can be continued indefinitely, one obtains a morphism $\hat{Y}\to \hat{X}$ for the formal preschemes obtained by completing Y and X for the ideals $\mathcal{I}{\mathcal{O}}_Y$ and $\mathcal{I}{\mathcal{O}}_X$. We may apply III.5.1 with $(S,X,Y,Y_0,g_0)$ replaced by (S_{n+1}, $X_{n+1}$, $Y_{n+1}$, $Y_n$, g_n). The sheaf $\mathcal{G}$ here becomes the sheaf of module homomorphisms from $g_n\ast ({\Omega }_{X_{n+1}/S_{n+1}}^1)$ into

$$\mathcal{J}={\mathcal{I}}^{n+1}{\mathcal{O}}_Y/{\mathcal{I}}^{n+2}{\mathcal{O}}_Y.$$

Since $\mathcal{J}$ is annihilated by $\mathcal{I}{\mathcal{O}}_Y$, we may then replace $g_n\ast ({\Omega }_{X_{n+1}/S_{n+1}}^1)$ by the sheaf it induces on $Y_0$, namely $h_0\ast ({\Omega }_{X/S}^1)$, where $h_0$ is the composite $Y_0\to Y_n\to X_{n+1}$, or again the composite $Y_0\to X_0\to X_{n+1}$, where $g_0:Y_0\to X_0$ is induced by $g_n$. Since the inverse image of ${\Omega }_{X_{n+1}/S_{n+1}}^1$ on $X_0=X_{n+1}{\times }_{S_{n+1}}{S}_0$ is ${\Omega }_{X_0/S_0}^1$, one sees that one also has

𝒢 = Hom_{𝒪_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), 𝓘^{n+1}𝒪_Y / 𝓘^{n+2}𝒪_Y).

Thus we obtain:

Corollary.

Let S, X, Y, $\mathcal{I}$, and $g_n$ be as above, and let $\mathcal{P}(g_n)$ be the sheaf on Y whose sections on an open U are the extensions $g_{n+1}$ of $g_n$ to an $S_{n+1}$-morphism $Y_{n+1}\to X_{n+1}$. Then $\mathcal{P}(g_n)$ is a formally principal homogeneous sheaf under the sheaf of groups

$$\mathcal{G}={\mathrm{Hom}}_{{\mathcal{O}}_{Y_0}}(g_0\ast ({\Omega }_{X_0/S_0}^1),g{r}_{\mathcal{I}{\mathcal{O}}_Y}^{n+1}({\mathcal{O}}_Y)).$$

In particular:

Corollary.

If moreover X is smooth over S, at least at the points of $g_0(Y_0)$, then $\mathcal{P}(g_n)$ is even a principal homogeneous sheaf. In particular, it defines an obstruction class in $H^1(Y_0,\mathcal{G})$, whose vanishing is necessary and sufficient for the existence of a global extension $g_{n+1}$ of $g_n$. And if such an extension exists, the set of all global extensions is a principal homogeneous space under $H^0(Y_0,\mathcal{G})$. Finally, in the case considered, the sheaf $\mathcal{G}$ may also be written

𝒢 = g₀*(𝔤_{X₀/S₀}) ⊗_{𝒪_{Y₀}} gr^{n+1}_{𝓘𝒪_Y}(𝒪_Y).

Proceeding step by step, one sees therefore that if all the $H^1(Y_0,{\mathcal{G}}_n)$ vanish, where

$${\mathcal{G}}_n=g_0\ast ({\mathfrak{g}}_{X_0/S_0})\otimes g{r}_{\mathcal{I}{\mathcal{O}}_Y}^n({\mathcal{O}}_Y),$$

then, starting with an arbitrary $g_k$, one can extend it successively to $g_{k+1}$, and so on. In particular, if $\mathcal{I}$ is nilpotent, one will be able to find an extension g of $g_k$ to Y. The vanishing condition for the H¹ is satisfied in particular if $Y_0$ is affine. Thus one obtains:

Corollary.

In the statement of Theorem III.3.1, one obtains a necessary and sufficient condition equivalent to the others by supposing that the $Y^{\prime }$ occurring in (ii), or (ii bis), is affine, and by requiring the existence of a global extension g of $g_0$ to all of $Y^{\prime }$.

Note that the proof of III.3.1 could not have given this result directly.

An important case is that where Y is flat over S. Then one has

gr^n(𝒪_Y) = gr^n(𝒪_S) ⊗_{𝒪_{S₀}} 𝒪_{Y₀},

and when, moreover, the $g{r}^n({\mathcal{O}}_S)$ are locally free on S, one finds

𝒢_n = Hom_{𝒪_{Y₀}}(g₀*(Ω¹_{X₀/S₀}), 𝒪_{Y₀}) ⊗_{𝒪_{S₀}} gr^n(𝒪_S),

or again, if ${\Omega }_{X_0/S_0}^1$ is itself locally free, for example if X is smooth over S,

𝒢_n = g₀*(𝔤_{X₀/S₀}) ⊗_{𝒪_{S₀}} gr^n(𝒪_S).

If, for example, S is affine with affine ring A, and $\mathcal{I}$ is defined by an ideal I of A, one finds

H^i(Y₀,𝒢_n) = H^i(Y₀,𝒢₀) ⊗_A gr_I^n(A)

for every i; indeed, the question is local on $S_0$, and one is reduced to the case where one tensors by a free module. In this case, the vanishing of $H^1(Y_0,{\mathcal{G}}_0)$ implies that all obstructions to the successive extensions of $g_n$ vanish. Thus one obtains:

Corollary.

Let $(S,X,Y,\mathcal{I},g_n)$ be as above. Suppose moreover that X is smooth over S and Y is flat over S, and finally that S is affine and the

$$g{r}^n({\mathcal{O}}_S)={\mathcal{I}}^n/{\mathcal{I}}^{n+1}$$

are locally free. Then the obstruction to constructing $g_{n+1}$ lies in

$$H^1(Y_0,{\mathcal{G}}_0){\otimes }_{A}g{r}_I^{n+1}(A),$$

where A is the ring of S and I the ideal of A defining $\mathcal{I}$, with

$${\mathcal{G}}_0=g_0\ast ({\mathfrak{g}}_{X_0/S_0}).$$

If $H^1(Y_0,{\mathcal{G}}_0)=0$, then $g_n$ can be extended to an Ŝ-morphism ĝ: $\hat{Y}\to \hat{X}$.

Of course, this result would remain valid exactly as stated if, instead of starting with ordinary S-preschemes X and Y, one started with formal $\hat{\mathcal{I}}$-adic Ŝ-preschemes $\mathfrak{X}$ and $\mathfrak{Y}$. It allows one to prove, for example, that certain formal schemes proper over a complete local ring are in fact algebraic. Indeed, proceeding as in Lemma III.4.2, one finds:

Corollary.

Under the conditions of III.5.6, if $g_0$ is an isomorphism, then so is ĝ.

The same result holds for closed immersions.

Thus one obtains:

Proposition.

Let A be a complete local ring with maximal ideal $\mathfrak{m}$ and residue field k. Let $\mathfrak{X}$ and $\mathfrak{Y}$ be two $\mathfrak{m}$-adic formal preschemes over A, flat over A, meaning that for every n, $X_n$ and $Y_n$ are flat over $A_n=A/{\mathfrak{m}}^{n+1}$. Suppose $X_0=\mathfrak{X}{\otimes }_{A}k$ is smooth over k and $H^1(X_0,{\mathfrak{g}}_{X_0/k})=0$. Then every k-isomorphism from $Y_0$ onto $X_0$ extends to an A-isomorphism from $\mathfrak{Y}$ onto $\mathfrak{X}$; this extension is unique if moreover $H^0(X_0,{\mathfrak{g}}_{X_0/k})=0$.

This gives in particular a result on the uniqueness of a smooth formal prescheme over A reducing to a given prescheme $X_0$, provided $H^1(X_0,{\mathfrak{g}}_{X_0/k})=0$. Moreover, if $\mathfrak{X}$ and $\mathfrak{Y}$ come from ordinary proper schemes over A, say X and Y, then by the existence theorem for sheaves in formal geometry, cf. the Bourbaki seminar exposé no. 182,⁵ there is a bijective correspondence between the A-isomorphisms $Y\to X$ and the A-isomorphisms of the formal completions. Hence:

Corollary.

The preceding statement III.5.8 remains valid when $\mathfrak{X}$ and $\mathfrak{Y}$ are replaced by ordinary A-schemes X and Y, proper over A.

Finally, when $\mathfrak{X}$ is a formal scheme proper over A, and $\mathfrak{Y}$ is of the form Ŷ where Y is an ordinary proper scheme over A, Proposition III.5.8 gives sufficient conditions for finding an isomorphism of $\mathfrak{X}$ with Ŷ, and hence for the formal scheme $\mathfrak{X}$ to be in fact “algebraic”, i.e. isomorphic to an $\hat{X}$, with X an ordinary proper scheme over A, which will then be canonically determined. This happens notably if $X_0={\mathbb{P}}_k^r$, or more generally if $X_0$ is a Severi-Brauer scheme, i.e. becomes isomorphic to the standard projective space over the algebraic closure of k: every formal scheme proper and flat over A, with fiber ${\mathbb{P}}_k^r$, is algebraizable, and more precisely is isomorphic to the $\mathfrak{m}$-adic formal completion of ${\mathbb{P}}_A^r$. In particular, thanks to the “existence theorem”, every ordinary proper scheme over A with fiber ${\mathbb{P}}_k^r$ is isomorphic to ${\mathbb{P}}_A^r$, where A is a complete local ring. Using descent theory, one can prove that if A is not complete, X becomes isomorphic to ${\mathbb{P}}^r$ after making a finite étale extension $A\to A^{\prime }$ of the base; in this form, the result remains valid for a fiber that is a Severi-Brauer scheme.

6. Global Infinitesimal Extension of Smooth S-Schemes

Under the conditions of Theorem III.4.1, we propose to seek whether there exists a prescheme X smooth over Y such that $X{\times }_Y{Y}_0$ is $Y_0$-isomorphic to $X_0$, knowing that such a scheme “exists locally on $X_0$”. Taking up again the step-by-step construction method, we are led to replace Y by the letter S, to suppose given a closed sub-prescheme $S_0$ of S defined by a sheaf of ideals $\mathcal{I}$, which it is no longer necessary to suppose locally nilpotent, to introduce the closed sub-preschemes $S_n$ of S defined by the ${\mathcal{I}}^{n+1}$, and to suppose given a sub-prescheme $X_n$ smooth over $S_n$. We propose to find an $S_{n+1}$-prescheme $X_{n+1}$ “reducing to $X_n$”, i.e. equipped with an isomorphism

X_n → X_{n+1} ×_{S_{n+1}} S_n

that is smooth over $S_{n+1}$, or equivalently by II.2.1, flat over $S_{n+1}$. As we noted in III.4, such data amount to giving a sheaf of algebras $\mathcal{B}$ over $f^{-1}({\mathcal{O}}_{S_{n+1}})$, where f is the continuous map underlying the structural morphism $X_n\to S_n$, equipped with an augmentation $\mathcal{B}\to {\mathcal{O}}_{X_n}$ compatible with the augmentation $f^{-1}({\mathcal{O}}_{S_{n+1}})\to f^{-1}({\mathcal{O}}_{S_n})$, and satisfying two conditions (a) and (b) that we shall not rewrite, merely noting that they are local in nature on the topological space underlying $X_n$. By III.4.1, a solution exists locally. It is moreover unique up to nonunique isomorphism, at least locally. Let us begin by making this point precise.

Proposition.

Let $X_{n+1}$ over $S_{n+1}$ reduce to $X_n$ over $S_n$. Then the sheaf, on the topological space underlying $X_n$, or equivalently $X_0$, of $S_{n+1}$-automorphisms of $X_{n+1}$ inducing the identity on $X_n$ is canonically isomorphic to

𝒢 = 𝔤_{X₀/S₀} ⊗_{𝒪_{S₀}} gr^{n+1}_𝓘(𝒪_S)

as a sheaf of groups.

Indeed, by III.5.4 and III.4.2 this sheaf is a principal homogeneous sheaf under $\mathcal{G}$. Since it has a distinguished section, the identity automorphism of $X_{n+1}$, it identifies as a sheaf of sets with $\mathcal{G}$. One must verify that this identification is compatible with the group structures. This is easy, and is moreover a special case of a more general result on the compatibility of the principal-bundle structures in III.5.1 and III.5.3 with composition of morphisms, a result that we do not state here but that ought to appear in the hyperplodocus.

In particular, the sheaf on $X_0$ of germs of automorphisms of $X_{n+1}$, with the structures just made explicit, is commutative. It follows that if $X_{n+1}^{\prime }$ is another solution of the problem, isomorphic to $X_{n+1}$ over the open U of $X_0$, then the isomorphism from $\mathrm{Aut}(X_{n+1})|U$ to $\mathrm{Aut}(X_{n+1}^{\prime })|U$ deduced by transport of structure from an isomorphism $X_{n+1}|U\to X_{n+1}^{\prime }|U$ does not depend on the choice of the latter. It is in fact nothing but the identity isomorphism of $\mathcal{G}$, when both automorphism sheaves are identified with $\mathcal{G}$ by III.6.1.

From III.6.1 one deduces:

Corollary.

Let $X_{n+1}$ and $X_{n+1}^{\prime }$ be smooth over $S_{n+1}$ and “reduce to $X_n$”. Then the sheaf, on the space underlying $X_0$, of $S_{n+1}$-isomorphisms from $X_{n+1}$ to $X_{n+1}^{\prime }$ inducing the identity on $X_n$ is naturally a principal homogeneous sheaf under $\mathcal{G}$.

This expresses indeed that $X_{n+1}$ and $X_{n+1}^{\prime }$ are locally isomorphic, and that the sheaf of germs of automorphisms of the first is $\mathcal{G}$.

Now note that by III.4.1 one can always find a covering $(U_i)$ of $X_n$ by opens, which may be supposed affine, and for each i a smooth scheme $X^i$ over $S_{n+1}$ reducing to $U_i$. Suppose for simplicity that $X_n$ is separated, so the $U_{ij}=U_i\cap U_j$ are still affine opens of $X_n$. Since H¹ of such an open with values in the quasi-coherent sheaf $\mathcal{G}$ is zero, Corollary III.6.2 implies that $X^i|U_{ij}$ is isomorphic to $X^j|U_{ij}$; let

$$f_{ji}:X^i|U_{ij}\to X^j|U_{ij}$$

be such an isomorphism. It is determined up to a section of $\mathcal{G}$ on $U_{ij}$. For every triple of indices put

f_{ji}^{(k)} = f_{ji}|U_{ijk},    where U_{ijk} = U_i ∩ U_j ∩ U_k.

If one had

$$f_{kj}^{(i)}{f}_{ji}^{(k)}=f_{ki}^{(j)},$$

it would follow that the $X^i$ glue by the $f_{ji}$, and hence define a solution $X=X_{n+1}$ of the desired problem. Such a solution exists more generally if one can modify the $f_{ji}$ into $f_{ji}^{\prime }:$

f′_{ji} = f_{ji} g_{ji},    g_{ji} ∈ Γ(U_{ij},𝒢),

so that the $f_{ji}^{\prime }$ satisfy the preceding transitivity condition. This sufficient condition for the existence of a solution is also necessary, as one sees by recalling that such a solution X must, on each $U_i$, be isomorphic to $X^i$; this allows one to choose isomorphisms

$$f_i:X|U_i\to X^i$$

and to define

f′_{ji} = (f_j|U_{ij})(f_i|U_{ij})^{-1}: X^i|U_{ij} → X^j|U_{ij},

satisfying the gluing condition.

Now put

f_{ijk} = (f_{ki}^{(j)})^{-1} f_{kj}^{(i)} f_{ji}^{(k)}.

This is an automorphism of $X^i|U_{ijk}$, which we identify with a section of $\mathcal{G}$ by III.6.1. One checks, by a small formal calculation left to the reader, that it is a 2-cocycle f of the open covering $\mathcal{U}=(U_i)$, with coefficients in $\mathcal{G}$. The same calculation shows that, under III.6.2, the gluing condition III.6.1 for the $f_{ij}^{\prime }$ is equivalent to the formula

f = dg,

where $g=(g_{ij})$ is regarded as a 1-cochain of $\mathcal{U}$ with coefficients in $\mathcal{G}$. Thus the necessary and sufficient condition for the existence of a solution of the problem is that the cohomology class in $H^2(\mathcal{U},\mathcal{G})$ defined by the cocycle III.6.3 be zero. Moreover, since $\mathcal{U}=(U_i)$ is an affine covering of $X_0$, which is a scheme, $H^2(\mathcal{U},\mathcal{G})$ identifies with $H^2(X_0,\mathcal{G})$. It is immediate that the cohomology class thus obtained in $H^2(X_0,\mathcal{G})$ does not depend on the affine covering considered. We shall call it the obstruction class to extending $X_n$ to a scheme $X_{n+1}$ smooth over $S_{n+1}$.

Suppose this obstruction is zero. Then the argument sketched above shows that every solution $X=X_{n+1}$ is isomorphic to a solution obtained by gluing from isomorphisms $f_{ji}^{\prime }$, which may be supposed of the form III.6.2, the gluing condition being just III.6.3. The set of admissible g is therefore a principal homogeneous space under the group $Z^1(\mathcal{U},\mathcal{G})$ of 1-cocycles of $\mathcal{U}$ with coefficients in $\mathcal{G}$. Moreover, one sees at once that two cochains g and $g^{\prime }$, with dg = $d{g}^{\prime }=f$, define isomorphic solutions if and only if the cocycle $g-g^{\prime }$ is of the form dh, with $h=(h_i)\in C^0(\mathcal{U},\mathcal{G})$. Thus one obtains:

Theorem.

Let $(S,\mathcal{I},X_n)$ be as above, with $X_n$ assumed separated.⁶ Then one can define canonically an obstruction class in $H^2(X_0,\mathcal{G})$, where $\mathcal{G}$ is defined in III.6.1, whose vanishing is necessary and sufficient for the existence of a scheme $X_{n+1}$, smooth over $S_{n+1}$, reducing to $X_n$. If this obstruction is zero, then the set of isomorphism classes, with isomorphisms inducing the identity on $X_n$, of $S_{n+1}$-preschemes $X_{n+1}$ reducing to $X_n$ is naturally a principal homogeneous space under $H^1(X_0,\mathcal{G})$.

Remarks.

Starting from III.6.1, the arguments made here are purely formal, and are advantageously transcribed in the setting of local categories, or even of general fibered categories. The obstruction class to the existence of a “global” object of a category, where one can find an object “locally”, any two objects are always “locally isomorphic”, and the automorphism group of any object is commutative, obtained in this general context, contains as a special case the “second boundary homomorphism” in an exact sequence of sheaves of not necessarily commutative groups, studied for example by Grothendieck in Kansas or Tôhoku. The silly cocycle calculation made here should therefore be regarded as a makeshift, due to the absence of a satisfactory reference text.

6.5

Note that in III.6.3 there is in general no distinguished element in the principal homogeneous space under $H^1(X_0,\mathcal{G})$ under consideration. This is reflected in particular by the fact that, after localizing on S, one obtains a principal homogeneous sheaf on $S_0$ with structural group $R^1{f}_{\ast }(\mathcal{G})$, which is not necessarily trivial, i.e. which defines a cohomology class in $H^1(S_0,R^1{f}_{\ast }(\mathcal{G}))$ that is not necessarily zero. This is when one supposes that the class d ∈ $H^2(X_0,\mathcal{G})$ is not zero, but is zero “locally over S”, i.e. defines a zero section of $R^2{f}_{\ast }(\mathcal{G})$, equivalently a zero element in $H^0(S_0,R^2{f}_{\ast }(\mathcal{G}))$.

6.6

For the moment we know almost nothing about the general algebraic mechanism of the cohomology classes introduced in this number and their relations with the preceding number, and we have nothing precise to say about them in the simplest particular cases, such as the case of abelian schemes over artinian rings.⁷ One hopes that people will be found to work the question out thoroughly; it seems particularly interesting. It is intimately linked, in particular, to the “module theory” of algebraic structures.

Corollary.

Suppose $H^2(X_0,\mathcal{G})=0$. Then an $X_{n+1}$ exists, and it is unique up to isomorphism if moreover $H^1(X_0,\mathcal{G})=0$.

In particular, proceeding step by step, and observing that an affine scheme is acyclic for a quasi-coherent sheaf, one concludes:

Corollary.

Under the conditions of Theorem III.4.1, if $X_0$ is affine, then there exists an X smooth over Y reducing to $X_0$, and this X is unique up to nonunique isomorphism.

Note that the direct proof of Theorem III.4.1 could not have given this result.

Corollary.

Under the conditions of III.6.3, suppose S is affine with ring A, $\mathcal{I}$ is defined by an ideal I of A, and finally the

$$g{r}_{\mathcal{I}}^n({\mathcal{O}}_S)={\mathcal{I}}^n/{\mathcal{I}}^{n+1}$$

are locally free. Then $H^i(X_0,\mathcal{G})$ identifies with

$$H^i(X_0,{\mathcal{G}}_0){\otimes }_{A}g{r}_I^{n+1}(A),$$

where

$${\mathcal{G}}_0={\mathfrak{g}}_{X_0/S_0}.$$

Thus the obstruction class to extending $X_n$ lies in $H^2(X_0,{\mathcal{G}}_0){\otimes }_{A}g{r}_I^{n+1}(A)$, and, if it is zero, the set of isomorphism classes of solutions is a principal homogeneous space under $H^1(X_0,{\mathcal{G}}_0){\otimes }_{A}g{r}_I^{n+1}(A)$.

In particular:

Corollary.

Under the conditions of III.6.9, suppose

$$H^2(X_0,{\mathfrak{g}}_{X_0/S_0})=0.$$

Then there exists an $\hat{\mathcal{I}}$-adic formal scheme $\mathfrak{X}$ over the $\mathcal{I}$-adic formal completion Ŝ of S, “smooth over S”, i.e. such that the ${\mathfrak{X}}_p$ are smooth over the $S_p$, and reducing to $X_n$, i.e. equipped with an isomorphism

X_n → 𝔛 ×_S S_n.

If moreover $H^1(X_0,{\mathfrak{g}}_{X_0/S_0})=0$, then such a $\mathfrak{X}$ is unique up to isomorphism.

Indeed, one constructs $X_{n+1}$, $X_{n+2}$, and so on step by step, whence $\mathfrak{X}$ by passing to the inductive limit of the $X_i$. The uniqueness assertion already appears in the preceding number.

7. Application to the Construction of Formal Schemes and Ordinary Smooth Schemes over a Complete Local Ring A

The results of the preceding number sometimes make it possible to prove the existence of an $\mathfrak{m}$-adic formal scheme over such a ring, reducing to a given smooth scheme $X_0$ over k. Distinguish two cases.

1. A is “of equal characteristics”. This is the case in particular if k has characteristic 0. Then one knows that there exists a coefficient subfield of A, i.e. a subfield $k^{\prime }$ such that $A\to k$ induces an isomorphism $k^{\prime }\to k$. Then there even exists an ordinary smooth scheme over A reducing to $X_0$, namely $X=X_0{\otimes }_{k}A$, with A regarded as an algebra over k by the homomorphism $k\to k^{\prime }\to A$ defined by $k^{\prime }$. It should be noted, however, that this construction is not “natural”. It is easy to convince oneself, already in the case where A = k[t]/(t²), the algebra of dual numbers, that another lifting homomorphism $k\to A$, in this case defined by an absolute derivation of k into itself, defines an $X^{\prime }$ over A that in general is not isomorphic to X, if $H^1(X_0,{\mathfrak{g}}_{X_0/k})\ne 0$. It would moreover be interesting to study, for k of characteristic 0, or imperfect of characteristic p > 0, which X smooth over A are obtained in this way, and under what condition two homomorphisms $k\to A$ define isomorphic A-schemes. Nevertheless, the existence of $k^{\prime }$ is enough to imply that the first obstruction to lifting $X_0$, which lies in $H^2(X_0,{\mathfrak{g}}_{X_0/k}){\otimes }_k\mathfrak{m}/{\mathfrak{m}}^2$, is necessarily zero. Of course, once $X_0$ has then been lifted to $X_1$ smooth over $A/{\mathfrak{m}}^2$, the new obstruction to constructing $X_2$ will in general not be zero: it will depend on a variable element in a certain principal homogeneous space under $H^1(X_0,{\mathcal{G}}_0)\otimes \mathfrak{m}/{\mathfrak{m}}^2$ and lies in $H^2(X_0,{\mathcal{G}}_0)\otimes {\mathfrak{m}}^2/{\mathfrak{m}}^3$. The situation ought to be studied in detail.⁸

2. A is of unequal characteristics. In this case we know nothing, except if by luck $H^2(X_0,{\mathfrak{g}}_{X_0/k})=0$, in which case one can construct an $\mathfrak{m}$-adic formal smooth scheme over A reducing to k. Even if $A=\mathbb{Z}/p^2\mathbb{Z}$ and $X_0$ is an “abelian” scheme of dimension 2, one does not know whether it can be lifted to an $X=X_1$ smooth over A;⁹ on the other hand, we have no example of an $X_0$ that has been proved not to come from an ordinary scheme X smooth over A. I have the impression that such examples should exist, with $X_0$ a projective surface.¹⁰ Let us simply point out that by Cohen's theorem, there exists a Cohen p-ring B with residue field k and a homomorphism $B\to A$ inducing the identity isomorphism on residue fields. Consequently, the “strongest” lifting result would be obtained by taking A to be a Cohen p-ring: if there is an ordinary or formal solution over such a ring, there is one over every complete local ring with residue field k. In particular, since for a Cohen p-ring $\mathfrak{m}/{\mathfrak{m}}^2$ identifies canonically with k, one sees that for every smooth scheme $X_0$ over a field k of characteristic p > 0, there exists a cohomology class in $H^2(X_0,{\mathfrak{g}}_{X_0/k})$, the first obstruction to lifting $X_0$ to a smooth scheme over a Cohen p-ring. We do not know whether it can be nonzero.¹¹

Even if one succeeds step by step in constructing the $X_n$ reducing to $X_0$, this generally gives only a formal scheme $\mathfrak{X}$ smooth over A, reducing to $X_0$. When $X_0$ is proper over A, there remains the question whether $\mathfrak{X}$ is in fact algebraizable, in order to obtain an ordinary proper scheme over A, smooth over A, reducing to $X_0$. The only known criterion, noted in the Bourbaki seminar and appearing in the Éléments, Chapter III, 4.7.1, is the following: if $\mathfrak{X}$ is proper over A, and if $\mathcal{L}$ is an invertible sheaf on $\mathfrak{X}$ such that the induced sheaf ${\mathcal{L}}_0$ on $X_0$ is ample, i.e. some tensor power ${\mathcal{L}}_0^{\otimes n}$, n > 0, comes from a projective immersion of $X_0$, then there exists a scheme X projective over A, and an ample invertible sheaf on X, such that $(\mathfrak{X},\mathcal{L})$ is obtained from it by $\mathfrak{m}$-adic completion. This leads us, given a locally free sheaf ${\mathcal{E}}_0$ on $X_0$, which we shall choose invertible and ample for our purpose, to extend it to a locally free sheaf $\mathcal{E}$ on $\mathfrak{X}$. For this, one is reduced to constructing step by step locally free sheaves ${\mathcal{E}}_n$ on the $X_n$. The discussion is entirely analogous to that of III.6, cf. Remark III.6.4; the essential role is played by the sheaf of automorphisms of an ${\mathcal{E}}_{n+1}$ inducing the identity on ${\mathcal{E}}_n$. One shows at once that this sheaf identifies with

𝒢 = Hom_{𝒪_{X₀}}(𝓔₀, 𝓔₀ ⊗ gr^{n+1}_𝓘(𝒪_X))
  = Hom_{𝒪_{X₀}}(𝓔₀,𝓔₀) ⊗ gr^{n+1}_𝓘(𝒪_X),

which is again a sheaf of commutative groups. One obtains:

Proposition.

Let S be a prescheme equipped with a quasi-coherent sheaf of ideals $\mathcal{I}$, let X be a prescheme over S, let $S_n$ be the sub-prescheme of S defined by ${\mathcal{I}}^{n+1}$, and let $X_n=X{\times }_S{S}_n$ for every integer n. Let ${\mathcal{E}}_n$ be a locally free sheaf on $X_n$, and seek to extend it to a locally free sheaf ${\mathcal{E}}_{n+1}$ on $X_{n+1}$. Then ${\mathcal{E}}_n$ defines a canonical obstruction class in $H^2(X_0,\mathcal{G})$, where $\mathcal{G}$ is the quasi-coherent sheaf given by the formula above. The vanishing of this class is necessary and sufficient for the existence of an ${\mathcal{E}}_{n+1}$ extending ${\mathcal{E}}_n$. If this class is zero, then the set of isomorphism classes, with isomorphisms inducing the identity on ${\mathcal{E}}_n$, of solutions ${\mathcal{E}}_{n+1}$ is a principal homogeneous space under $H^1(X_0,\mathcal{G})$.

This proposition gives rise to the usual corollaries. Let us only point out that if X is flat over S, then one may write

𝒢 = Hom_{𝒪_{X₀}}(𝓔₀,𝓔₀) ⊗_{𝒪_{S₀}} gr^{n+1}_𝓘(𝒪_S),

whence, if S is affine with ring A and the ${\mathcal{I}}^n/{\mathcal{I}}^{n+1}$ are locally free, the sufficient condition

H²(X₀,𝒢₀) = 0,    with    𝒢₀ = Hom_{𝒪_{X₀}}(𝓔₀,𝓔₀),

for the existence of an ${\mathcal{E}}_{n+1}$, and hence, step by step, for the existence of successive extensions ${\mathcal{E}}_m$, $m=n$, $n+1$, etc.

Returning to the initial situation, we therefore find:

Proposition.

Let A be a complete local ring, and let $\mathfrak{X}$ be a formal scheme proper and flat over A, such that $X_0$ is projective and $H^2(X_0,{\mathcal{O}}_{X_0})=0$. Then there exists a scheme X projective over A whose $\mathfrak{m}$-adic formal completion is isomorphic to $\mathfrak{X}$.

Combining this with III.6.10, one finds:

Theorem.

Let A be a complete local ring with residue field k, and let $X_0$ be a projective smooth scheme over k such that

$$H^2(X_0,{\mathfrak{g}}_{X_0/k})=H^2(X_0,{\mathcal{O}}_{X_0})=0.$$

Then there exists a smooth and projective scheme X over A reducing to $X_0$.

More generally, if one is given an $X_n$ smooth over $A_n=A/{\mathfrak{m}}^{n+1}$ reducing to $X_0$, then there exists an X smooth and proper over A and an isomorphism $X{\otimes }_A{A}_n=X_n$.

Corollary.

Every smooth proper curve over k is obtained by reduction from a smooth proper curve over A.

This result will be the essential tool, together with the existence theorem for sheaves in formal geometry, for studying the fundamental group of $X_0$ by transcendental means.