# 2 Formal geometry and algebraic geometry

Grothendieck, A.
“Géométrie formelle et géométrie algébrique”.
*Séminaire Bourbaki* **11** (1958–59), Talk no. 182.

The substance of §1 to §5 is contained in the published part of EGA III; that of §6 and §7 is contained in SGA III. For the study of the fundamental group, see SGA V, IX, X, and XI, as well as SGA 1962 (talks X, XII, and XIII) for the Lefschetz-type theorems and numerous open questions. Only the theory of moderately ramified coverings (cf. Theorem 14) has not yet been the subject of a dedicated talk. The corollary to Theorem 14, which completely determines Galois coverings of order coprime to the characteristic of an algebraic curve over an algebraically closed field, has been used in an essential manner on three separate occasions:

in the proof by Igusa of the Picard inequality for non-singular projective surfaces in arbitrary characteristic;

in the study (developed independently by Ogg and Šafarevič) of the group of homogeneous principal bundles over an abelian variety defined over a function field in one variable, in arbitrary characteristic; and

in the recent proof, by Artin, of certain key theorems concerning the “Weil cohomology” of algebraic varieties.

## 2.1 Schemes

We know that an affine algebraic space defined over a field

Let *Zariski topology*”, or the spectral topology.
Also, there is a sheaf of commutative rings *ringed space*, and is called the *prime spectrum* of

We define an *affine scheme* to be a ringed space that is isomorphic to some *prescheme* to be a locally-affine ringed space, i.e. such that every point has an open neighbourhood that is an affine scheme for the induced structure.
We define, in an evident way, *morphisms* of preschemes;
locally, they correspond to ring homomorphisms.

When we fix a prescheme * S-prescheme*;
if

*. So every prescheme can be regarded in a unique way as a*A -algebras

*change of base*of an

We say that *separated* over *scheme* to be a prescheme that is separated over *Noetherian*, i.e. finite unions of affine opens that are spectra of Noetherian rings.
We say that *of finite type* over

Inspired by well-known ideas, we can define the notion of a *projective morphism*, and, more generally, of a *proper morphism*.
Such a morphism is of finite type, and further sends closed subsets to closed subsets, and retains this property under an arbitrary change of base.

With *coherent sheaf of rings* in the sense of [8].
The coherent sheaves of modules on

## 2.2 Formal schemes

Let *closed subscheme of X*.
We can also, for any

*formal completion of*{\mathscr{O}}_X along X' . Endowed with this sheaf of rings,

*formal completion of*X along X' , and is thus a ringed space, but not a scheme in general. For every coherent sheaf

*formal sections of*{\mathscr{F}} along X , and can be identified with elements of

We define a *formal scheme* (implicitly assumed to be Noetherian) to be a topological space *coherent* sheaf of *local Noetherian* rings.

By the definitions, a formal completion *locally* of this type.
In fact, the data of a formal *affine* scheme (i.e. such that

The usual definitions (morphism, morphism of finite type, proper morphism, etc.) for ordinary schemes generalise without problem to formal schemes.

## 2.3 The three fundamental theorems

Let

*Finiteness theorem.* —

The

\operatorname{R}^q\overline{f}_*(\overline{{\mathscr{F}}}) are coherent sheaves on\overline{Y} .The natural homomorphisms

\operatorname{R}^q\overline{f}_*(\overline{{\mathscr{F}}}) \to \varprojlim_n\operatorname{R}^q (f_n)_*({\mathscr{F}}_n) are isomorphisms.

In this statement, we suppose that we already have some coherent subsheaf

Theorem 1 deals only with “formal cohomology”; the following theorem relates it with “algebraic cohomology”, and resembles a well-known theorem of Serre [23] on the comparison between algebraic cohomology and analytic cohomology.

*First comparison theorem.* —
The

There are canonical isomorphisms

This corollary is, for

Let

Finally, applying this corollary to

Let

Of course, the natural map

Now suppose that

The latter identity shows that the category of coherent sheaves on *subcategory* (with morphisms being the induced morphisms) of the category of coherent sheaves on

For a sheaf of modules on

The closed subschemes of

Indeed, they correspond to coherent subsheaves of

Let

In other words, proper algebraic schemes over *there exist proper formal schemes over \overline{Y} that are not “algebraisable”*, i.e. isomorphic to some

Let

the local rings of

{\mathscr{O}}_{{\mathfrak{X}}} are*flat*A -modules, or, equivalently, that, if we endow{\mathscr{O}}_{{\mathfrak{X}}} andA with the filtration given by powers of the maximal ideal ofA , then the associated graded algebras satisfy\operatorname{gr}({\mathscr{O}}_{{\mathfrak{X}}}) \simeq \operatorname{gr}^0({\mathscr{O}}_{{\mathfrak{X}}})\otimes_k\operatorname{gr}(A); H^2({\mathfrak{X}}_0,{\mathscr{O}}_{{\mathfrak{X}}_0})=0 , where we consider{\mathfrak{X}}_0={\mathfrak{X}}\otimes_Ak as an algebraic scheme overk ;{\mathfrak{X}}_0 is projective.

Then, under these conditions,

Conditions (ii) and (iii) will be satisfied if, in particular, *simple curve* over

For the proof of Theorem 1, Theorem 2, and Theorem 3, we refer the reader to [3].

## 2.4 Applications to Zariski’s connection theorem and “main theorem”

Let *the Stein factorisation* of

*Zariski’s “connection theorem”* —
Let

From this, we immediately deduce the usual variants of the connection theorem. We state here only the following:

For a point

To show that *isomorphism*

For

If this is so, then

Let

Let ^{5}, then

Since a morphism of finite type is locally affine, and *a fortiori* locally quasi-projective, we immediately deduce from Theorem 6 the usual local variants of the main theorem.

## 2.5 Application to the cohomological study of proper and flat morphisms

Let

Let

In particular, if

Let

If

In particular:

Let

Thus:

Suppose that

It thus follows that:

Let

Corollary 3 above proves that

Proposition 2 allows us to determine the

Let

Let *sheaves* of groups

## 2.6 Application to existence and uniqueness theorems for sheaves and schemes over a complete \mathscr{J} -adic ring

Theorem 7 gave a uniqueness result for locally free coherent sheaves, by using Theorem 1 and Theorem 2.
Using Theorem 3, we now obtain *existence* theorems for sheaves, for morphisms of schemes, or for schemes.
In the following, *complete*.
The general method still consists of making *formal* construction, which consists essentially of doing *algebraic geometry over an Artinian ring*, and deducing conclusions from this that are “algebraic” in nature, by using the three fundamental theorems.

Let

We construct, step by step, locally free sheaves

Let

Let *simple* (by which we mean *absolutely* simple) over *theory of modules*”, or of “structure variations” of

{\mathscr{O}}_X is a sheaf ofA -algebras;{\mathscr{O}}_X is endowed with an augmentation homomorphism{\mathscr{O}}_X\to{\mathscr{O}}_{X_0} (that is compatible with theA -algebra structures);

and with the above data being subject to the following conditions: the augmentation induces an isomorphism

Let

Note that the isomorphic in question is not canonical, since *equal characteristics*), where we can take *obstruction of constructing X_n lifting X_{n-1}*, which lives in

Let

Also, if we can find *one* *that are compatible with the augmentation*.
The filtration of

Let

(Note that, in general, there is no privileged choice of origin in the latter homogeneous principal space, since there is no privileged way of lifting

Let *there exists at most one flat A-scheme X* (up to isomorphism) such that

Corollary 1 and Corollary 3 immediately imply the claims, which seem more general, obtained by supposing only that *complete local Noetherian ring with residue field k*, provided that we introduce

Let *simple* scheme of finite type over

- if
H^1(X_0,{\mathfrak{G}}_{X_0/k})=0 thenF(A) has at most one element; - if
H^2(X_0,{\mathfrak{G}}_{X_0/k})=0 thenF(A) has at least one element.

Suppose that

We can use Corollary 2 of Theorem 3. For example:

If

(We can also deduce this result from Corollary 1 of Proposition 3).

Let

We can combine Theorem 9 (ii) with Theorem 4. In particular:

Let

—

- Corollary 3 and Corollary 4 are above all interesting if
k is of characteristicp\neq0 , takingA to be a discrete valuation ring*of characteristic*, with residue field0 k ; for example, the “smallest possibleA ”, i.e. that for whichp generates the maximal ideal. (In fact, by theorems of Cohen, it suffices to have Corollary 3 and Corollary 4 for such a ringA ). We note that, concerning this point, according to the specialists, we do not know if there exist schemes over a fieldk that are not reductions\mod p of a flat scheme defined over such a ringA . At the least, the results of this section give a way of systematically investigating this question. We must start by seeing if the first obstruction that we have, inH^2(X_0,{\mathfrak{G}}_{X_0/k}) , is necessarily zero.

We note that Theorem 3, and the corresponding technique, only works for a

*complete*(local, for simplicity) base ring.In order to go from known results concerning the completion of a local ring to the corresponding results for the local ring itself, we would need a fourth “fundamental theorem”, whose precise statement still needs to be found.

We will compare the results from this section (mainly the above Corollary 1 and Corollary 2), as well as those from the following, with the results of Kodaira–Spencer on the variation of complex structures. Using the conjectural theorem to which we have just alluded, we should be able to conclude, under the conditions of Corollary 1, but where

A is no longer assumed to be complete, that there exists a ringA' that containsA , and that is finite and unramified overA , such thatX\otimes_AA' andX'\otimes_AA' areA' -isomorphic (whereX andX' are given, and are proper flatA -schemes such thatX\otimes_Ak=X'\otimes_Ak=X_0 ). This is what we can prove, at least, whenX_0={\mathfrak{P}}_k^r , by using Corollary 1 of Proposition 2. In any case, ifH^1(X_0,{\mathfrak{G}}_{X_0/k})=0 , then we can prove that the fibres ofX andX' over any pointy ofY=\operatorname{Spec}(A) are isomorphic, or at least when we pass to the algebraic closure of the residue field\mathfrak{K}(y) . (We have a local result, seemingly stronger, when we don’t suppose thatA is necessarily local). As for “structure variations” of the projective space, we again point out the following question, suggested by a corresponding problem of Kodaira–Spencer. LetX be a proper flat scheme, over a local integral ringA with field of fractionsK and residue fieldk , and suppose thatX\otimes_AK is isomorphic to{\mathfrak{P}}_K^r . Is it then true thatX\otimes_Ak=X_0 is isomorphic to{\mathfrak{P}}_k^R (or at least, over the algebraic closure ofk )? In this question, we can assume thatA is a complete discrete valuation ring. There is an analogous question whenX_0 is an abelian variety.

*[Comp.]*
*(Concerning Remark 1 above).*
We note that J.-P. Serre has constructed a non-singular projective variety, of dimension ^{6}
Mumford would have found an analogous result, with a non-singular projective *surface*.

*[Comp.]*
*(Concerning Remarks 2 and 3 above.)*
I am now less optimistic concerning the results conjectured here.
However, the question concerning structure variations for projective space, mentioned at the end of Remark 3 above, has been positively resolved by Hironaka, and the analogous question for abelian varieties has been resolved by Koizumi.

## 2.7 Application to the “theory of modules”

Since the speaker has only recently encountered this theory himself, we will be obliged to limit ourselves to just cursory remarks.
For simplicity, we work over a *field* *equal characteristic*, even though Theorem 8 allows us to also discuss the more general case, without any fundamental changes, so it seems.
We have not yet gotten past the “formal” stage, but the speaker still hopes to be able to construct true schemes of modules in certain cases from this, and, in particular, construct, for every integer

*[Comp.]*
We note that Mumford has recently constructed schemes of modules for the curves of genus *Mumford–Tate seminar*, Harvard University, 1961–62).
Theorem 10 also proves that the “level

We continue to use the setting and notation of Theorem 9, and now suppose that *Suppose* that we can find a complete local Noetherian *formal scheme of modules for X_0*.
(Note that it does not necessarily exist).
Let

*a priori*have nilpotent elements, and it seems likely that there should exist cases where

*regular*rings) is

*a priori*inadequate in the general case.

It remains to give sufficient conditions for there to exist a formal scheme of modules for *sufficient* (even if not at all necessary) in order to guarantee the existence of a formal scheme of modules.
We restrict ourselves to stating here a theorem that deals with a particularly simple case (whose analogue in the theory of analytic spaces is well known, cf. Kodaira–Spencer), which can easily be proven using the results from the previous section:

Let

As we have already pointed out, it is not true in general that the formal scheme

Note also that methods such as those described in this section can be applied in the construction and study of Picard varieties, as well as in many other constructions. We will return to this soon.

## 2.8 Application to the fundamental group

The techniques described allows us to tackle the system study of the fundamental group, using the example of topological theory. The first two theorems stated in this section are generalisations of results in a recent work by Lang–Serre.

Let *unramified covering of X* if

X' is finite overX , i.e. it is defined by a coherent sheaf of algebras{\mathscr{A}}={\mathscr{A}}(X') onX ;{\mathscr{A}} is a locally free sheaf onX ;for all

x\in X , the quotient{\mathscr{A}}_x/{\mathfrak{m}}_x{\mathscr{A}}_x = {\mathscr{A}}_x\otimes_{{\mathscr{O}}_X}\pi(x) is a separable algebra over\mathfrak{K}(x) .

This notion of unramified covering (due to Serre and the speaker) posses all the elementary properties for which we can reasonably hope, and which we will not list.
We restrict ourselves to saying that it gives rise to a *Galois theory* modelled on classical Galois theory (and containing it; the proofs being overall simpler than the proofs generally seen for the latter) and the Galois theory of topological coverings.
More precisely, we define a *geometric point* of a scheme *locality* of the geometric point *fundamental group of the connected scheme X at the geometric point a*, and we denote it by

*covariant functor in the pointed scheme*(X,a) . Every statement concerning the classification of inseparable coverings can then be translated into the language of group theory, following the well-known dictionary (except that we must take into account the fact that here we have

*topological groups*).

Our goal is to develop an analogue of the homotopy exact sequence of fibre bundles, relative to a proper morphism

Let

Let

It follows, in particular, from these two lemmas that, for *every* algebraic extension

Let

X'_0\otimes_k\Omega is completely decomposed overX_0\otimes_k\Omega ;- the natural morphism
X'\to X\otimes_AA' is an isomorphism.

Under these conditions, *unramified* extension of

i bis.

When condition (ii) is satisfied, we say that the unramified covering *geometrically trivial*.

Let *surjective*.

What we need to show is effectively the following: if an unramified covering

Let

The fact that the first homomorphism is injective has already been shown with Lemma 1 and Lemma 2;
the exactness in the middle follows from Lemma 3;
finally, the surjectivity of the last homomorphism (which is the only thing to rely on the fact that

Let *separable scheme* over

The proof is easy, thanks to Theorem 2.

This proposition, combined with Lemma 1, practically reduces the homotopical study of proper and flat morphisms (with separable fibres) to the case where

A flat morphism of finite type whose fibres are separable (resp. simple) schemes is said to be *separable* (resp. *simple*).
We show that, if *Bertini’s theorem*).

Let

and let

induced by the natural morphisms

X' is unramified overX at the points ofF'_1 (and thusF'_1 is an unramified cover ofF ), andF'_1 is a geometrically trivial covering ofF (cf. Lemma 3).

Under these conditions, there exists an open neighbourhood

Of course, conditions (ii) and (iii) are also necessary for the conclusion of the theorem. The proof of the theorem is easy, thanks to Lemma 3 and Theorem 2.

Suppose that (i) is still satisfied.
For an unramified covering over

By Theorem 11, the set of points of

*The kernel of the homomorphism \pi_1(X)\to\pi_1(Y) (which is surjective, by Lemma 4) is the closed invariant subgroup generated by the images in \pi_1(X) of \pi_1(f^{-1}(y)), where f^{-1}(y) denotes the scheme f^{-1}(y)\otimes_{k(y)}\overline{\mathfrak{K}(y)} (where \overline{\mathfrak{K}(y)} denotes an algebraic closure of \mathfrak{K}(y)).*

We note that, since we cannot *choose* the same base point for all the fibres, the homomorphisms

Under the general conditions of Theorem 11, suppose further that

(We apply the last part of Lemma 3 to the generic fibre of *separable* morphism.
Then

Suppose, in addition to (i), that

Note that it was not necessary to suppose that *geometrically* trivial over *a posteriori*, even though *a priori* this condition is a lot stronger).
Corollary 3 is equivalent to the following statement:

Let

From this, we easily deduce the two following statements of Serre–Lang, with all normality hypotheses removed:

Let

(The surjectivity in the above claim is almost trivial). We thus deduce, with Serre–Lang:

Let

—

- Using Proposition 4, we see that the hypothesis that
f_*({\mathscr{O}}_X)={\mathscr{O}}_Y in Corollary 4 is not essential. In the general case, instead of putting the trivial groupe after\pi_1(Y) , one must continue by\pi_0(\overline{F})\to\pi_0(X)\to\pi_0(Y)\to e , as in algebraic topology.

- In general, we cannot say anything at the moment about the kernel of
\pi_1(\overline{F})\to\pi_1(X) , although it should involve a\pi_2(Y) . It seems, however, that we should be able to prove that\pi_1(\overline{F})\to\pi_1(X) is*injective*ifY is the spectrum of a local ringA , by appealing to Theorem 12 below (which shows that this is the case ifA is*complete*).

Theorem 11 used only Theorem 1 and Theorem 2; we will now use Theorem 3, along with the following elementary lemma:

Let

This lemma, which is of a purely local nature, plays a role analogous to that of Theorem 8 here, in the theory of modules. Combining it with the existence theorem (Theorem 3), we obtain:

Let

In other words:

Pick a geometric point in *isomorphism*.

Applying Lemma 5 to

Let *surjective*.

We might hope that this homomorphism is always *bijective*.
Unfortunately, this is not the case in general if

Let

We thus deduce, by a well-known technique using hyperplane sections:

Let

We wish to describe the kernel of the homomorphism *for X'_1\otimes_{K'}\overline{K}=\overline{X'_1} to come from an unramified covering of X_0, it is necessary and sufficient for there to exists a finite extension K'' of K' in \overline{K} such that X''_1=X'_1\otimes_{K'}K'' is of the form X''\otimes_{V''}K'', where V'' is the normal closure of V in K'', and where X'' is an unramified covering of X\otimes_V V''.*
Suppose, for example, that

*(i.e. the normalisation of*L'' is an unramified extension of the field of functions of X\otimes_VV''

*simple*, then it follows from the “

*purity theorem*” of Nagata–Zariski that it even suffices to show that

Under the above conditions, and with the above notation, for the unramified covering

Now note that *of order n, coprime to the characteristic p of \mathfrak{K}(y_0)* (which is also the characteristic of the residue field of

*Abhyankar’s lemma*”) that, if we adjoin an

Let *isomorphism*.

In other words:

The classification of unramified Galois coverings, of Galois group of order coprime to the characteristic

In particular, if

We finally point out that the techniques utilised also give the following result, which is more general than Theorem 13:

Let *surjective* homomorphism (defined up to inner automorphism)

From this we obtain corresponding variants of the corollaries of Theorem 13, and of Corollary 4 of Theorem 12. Similarly, using Corollary 3 of Theorem 9, we obtain, transcendentally:

Let *of order coprime to the characteristic* that is generated by elements

If *three point problem*”, at least for Galois coverings of order coprime to the characteristic.
(Here, Theorem 9 is actually useless, and it seems that we can deduce the above corollary from the particular case in question in the three point problem).

—

A more complete study, probably involving generalised Galois coverings of

X ,X_0 , andX_1 (of eventually infinitesimal Galois group), should allow one to recover the kernel in Corollary 2 of Theorem 12. However, a study of coverings admitting ramifications that are not “tame” seems much more difficult.Lemma 6, combined with a result of Grauert concerning the formal completion of a non-singular projective scheme along a hyperplane section (or with the theorem, as yet unproven, mentioned in Remark 2 after Theorem 11), would also allow us to prove, in “abstract” algebraic geometry, the classical

*Lefschetz theorem*on the fundamental group.

### References

[3] J. Dieudonné, A. Grothendieck. “Eléments de géométrie algébrique.” *Publications Mathématiques de L’Institut Des Hautes Etudes Scientifiques*. (n.d.).

[8] A. Grothendieck. “The cohomology theory of abstract algebraic varieties,” in: *International Congress of Mathematicians, 1958, Edinburgh*, n.d.

[23] J.-P. Serre. “Géométrie algébrique et géométrie analytique,” in: *Ann. Institut Fourier Grenoble*, 1956: pp. 1–42.

*[Comp.]*This hypothesis can be replaced by the weaker hypothesis “iff is separated”, by means of the following result (see SGA VIII, 6.2): every morphismf\colon X\to Y which is quasi-finite and separated is also projective.↩︎Serre, J.-P. “Exemples des variétés projectives en caractéristique

p non relevables en caractéristique zéro”.*Proc. Nat. Acad. Sc. U.S.A.***47**(1961), 108–109.↩︎