# I. Étale morphisms

To simplify the exposition, we assume that all preschemes in the following are locally Noetherian (at least, starting from §2).

## 1. Basics of differential calculus

Let *rings* on * n-th infinitesimal neighbourhood of X/Y*.
The polysyllogism is entirely trivial, even if rather long

^{4}; it seems wise to not discuss it until we use it for something helpful, with smooth morphisms.

## 2. Quasi-finite morphisms

Let

B/{\mathfrak{m}}B is of finite dimension overk=A/{\mathfrak{m}} .{\mathfrak{m}}B is an ideal of definition, andB/{\mathfrak{r}}(B)=k(B) is an extension ofk=k(A) .- The completion
\widehat{B} ofB is finite over the completion\widehat{A} ofA .

We then say that *quasi-finite* over *isolated in its fibre f^{-1}(x)*.
A morphism is said to be quasi-finite if it is quasi-finite at each point

^{5}.

If

We could also give the usual polysyllogism (i), (ii), (iii), (iv), (v) for quasi-finite morphisms, but that doesn’t seem necessary here.

## 3. Unramified morphisms

Let

{\mathcal{O}}_x/{\mathfrak{m}}_y{\mathcal{O}}_x is a finite separable extension ofk(y) .\Omega_{X/Y}^1 is zero atx .- The diagonal morphism
\Delta_{X/Y} is an open immersion on a neighbourhood ofx .

*Proof*. For the implication (i)

—

- Let
f satisfy one of the equivalent conditions of 3.1. We then say thatf is*unramified*atx , or thatX is unramified atx onY . - Let
A\to B be a local homomorphism. We say that it is*unramified*, or thatB is a local*unramified*algebra onA , ifB/{\mathfrak{r}}(A)B is a finite separable extension ofA/{\mathfrak{r}}(A) , i.e. if{\mathfrak{r}}(A)B={\mathfrak{r}}(B) andk(B) is a separable extension ofk(A) .^{6}

The fact that

The set of points where

Let

Indeed, this is the inverse image of the diagonal morphism *different ideal* of

—

- An immersion is ramified.
- The composition of two ramified morphisms is also ramified.
- Base extension of a ramified morphisms is also ramified.

We are rather indifferent about (ii) and (iii) (the second seems more interesting to me). We can, of course, also be more precise, by giving some one-off statements; this is more general only in appearance (except for in the case of definition (b)), and is boring. We obtain, as per usual, the corollaries:

—

- The cartesian product of two unramified morphisms is unramified.
- If
gf is unramified then so too isf . - If
f is unramified then so too isf_\text{red} .

Let

—

- In the case where we don’t suppose that the residue extension is trivial, we can reduce to the case where it is by taking a suitable finite flat extension of
A which destroys the aforementioned extension. - Consider the example where
A is the local ring of an ordinary double point of a curve, andB a point of its normalisation: thenA\subset B ,B is unramified overA with trivial residue extension, and\widehat{A}\to\widehat{B} is surjective but*not injective*. We are thus going to strengthen the notion of unramified-ness.

## 4. Étale morphisms. Étale coverings

We are going to suppose that everything concerning flat morphisms that we need to be true is indeed true;
these facts will be proved later, if there is time^{7}.

—

- Let
f\colon X\to Y be a morphism of finite type. We say thatf is*étale*atx iff is both flat and unramified atx . We say thatf is étale if it is étale at all points. We say thatX is étale atx overY , or that it is aY -prescheme which is étale atx etc. - Let
f\colon A\to B be a local homomorphism. We say thatf is étale, or thatB is étale overA , ifB is flat and unramified overA .^{8}

For

*Proof*. This is true individually for both “unramified” and “flat”.

Let

Suppose that the residue extension

We can combine flatness with 3.7.

Let

*Proof*. Again, this is true individually for both “unramified” and “flat”.

This proposition shows that we can forget about the “one-off” comments in the study of morphisms of finite type that are somewhere étale.

—

- An open immersion is étale.
- The composition of two étale morphisms is étale.
- The base extension of an étale morphism is étale.

*Proof*. Indeed, (i) is trivial, and for (ii) and (iii) it suffices to note that it is true for “unramified” and “flat”.

As a matter of fact, there are also corresponding comments to make about local homomorphisms (without the finiteness condition), which in any case should appear in the multiplodoque (starting with the case of unramified).

*[Trans.] Grothendieck’s “multiplodoque d’algèbre homologique” was the final version of his Tohoku paper — see (2.1) in ‘Life and work of Alexander Grothendieck’ by Ching-Li Chan and Frans Oort for more information.*

The cartesian product of two étale morphisms is étale.

Let

*Proof*. Indeed,

We say that a cover of

The first condition means that *and*, further, that, for all

Let *discriminant section*

*Proof*. We can reduce to the case where

We will have a less trivial statement to make later on, when we do not suppose a priori that

## 5. Fundamental property of étale morphisms

Let *étale* and *radicial* morphism.

*Proof*. Recall what “radicial” means: injective, with radicial residual extensions (and we also recall that it means that the morphism remains injective for any base extension).
The necessity is trivial, and the sufficiency remains to be shown.
We are going to give two different proofs: the first is shorter, the second is more elementary.

A flat morphism is open, and so we can suppose (by replacing

Y withf(X) ) thatf is an onto*homeomorphism*. For any base extension, it remains true thatf is flat, radicial, surjective, thus a homeomorphism, and a fortiori closed. Thusf is*proper*. Thusf is*finite*(reference: Chevalley’s theorem), defined by a coherent sheaf{\mathscr{B}} of algebras. Now{\mathscr{B}} is locally free, and further, by hypothesis, of rank1 everywhere, and soX=Y .We can suppose that

Y andX are*affine*. We can further easily reduce to proving the following: ifY=\operatorname{Spec}(A) , withA local, and iff^{-1}(y) is non-empty (wherey is the closed point ofY ), thenX=Y (indeed, this would imply that everyy\in f(X) has an open neighbourhoodU such thatX|U=U ). We will then have thatX=\operatorname{Spec}(B) , and wish to prove thatA=B . But, for this, we can reduce to proving the analogous claim where we replaceA by\widehat{A} , andB byB\otimes_A\widehat{A} (taking into account the fact that\widehat{A} is faithfully flat overA ). We can thus suppose thatA is*complete*. Letx be the point overy . By 2.2,{\mathcal{O}}_x is finite overA , and is thus (being flat and radicial overA ) identical toA . SoX=Y\coprod X' (disjoint sum). But sinceX is radicial overY ,X' is empty.

Let *closed immersion* and *étale*.
If

*Proof*. Indeed,

We thus deduce:

Let

*Proof*. Only the first claim demands a proof;
by 5.2, it suffices to note that a section is a closed immersion (since

Let

*Proof*. This follows from 5.3 by reducing to the case where

Here is a particularly important variant of 5.3:

Let *bijective*.

*Proof*. We can again reduce to the case where

This result contains a claim of *uniqueness* and of *existence* of *morphisms*.
It can also be expressed (if *fully faithful*, i.e. establishes an *equivalence* between the first and a *full subcategory* of the second.
We will see below that it is in fact an equivalence between the first and the second (which will be a theorem of *existence for étale S-schemes*).

The following form (which looks more general) of 5.5 is often useful:

**(Extension of coverings).**
Consider a commutative diagram of morphisms

*Proof*. By replacing

We also note the following immediate consequence of 5.1 (which we did not give as a corollary, in order to not interrupt the line of ideas developed following 5.1:

Let

*Proof*. It suffices to prove sufficiency;
since it is true for the property of being a surjection, we can reduce to the case of an open immersion.
By 5.1, we have to show that *radicial* (which is trivial) and *étale* (which follows from 5.9 below).

Let

*Proof*. The two algebras over

With the notation of 5.8, suppose that

## 6. Application to étale extensions of complete local rings

This section is a particular case of the results on formal preschemes, which should appear in the multiplodoque. Nevertheless, here we get away without much difficulty, i.e. without the explicit local determination of the étale morphisms in §7 (using the Main Theorem). This is perhaps sufficient reason to keep this current section (even in the multiplodoque) where it is.

Let *equivalence* between the category of *finite and étale over A* and the category of algebras that are

*finite rank and separable over*k .

Firstly, the functor in question is fully faithful, as follows from the more general fact:

Let

*Proof*. We can reduce to the case where

It remains to prove that, for every finite and separable

## 7. Local construction of unramified and étale morphisms

Let

In other words, taking

The different ideal of

*Proof*. Let

We thus find:

Under the conditions of 7.1, and supposing that

*Proof*. Since

Under the conditions of 7.3, in order for

*Proof*. The second claim follows from the first along with Nakayama (in

A monic polynomial *separable polynomial* (if

Let *finite* over the *local* ring

*Proof*. We only have to prove necessity.
Suppose that

Let *unramified* over

(Of course, these conditions are more than sufficient …)

Before proving 7.6, we first state some nice corollaries:

For *étale* over

*Proof*. We can take

Let

*Proof*. This is a simple translation of 7.7.

We will now show how the jargon of 7.6 follows from the main theorem:
there exists, by 7.7, an epimorphism

*Proof* (of Theorem 7.6). This repeats a proof from the *Séminaire Chevalley*.
By the “Main Theorem”, we have that

Suppose, for the moment, that we have both 7.9 and 7.10.
Let

Let

Let

N.B. 7.10 should have appeared as a corollary to 7.1, and before 7.5 (which it implies).

So 7.6 now follows from the combination of 7.9 and 7.10; it remains only to prove 7.9.

*Proof* (of Lemma 7.9). Let

We must be able to state 7.6 for a ring *monogenous*

## 8. Infinitesimal lifting of étale schemes. Applications to formal schemes

Let

*Proof*. Let

We know that there exists a

The analogous claim holds for étale *coverings*, if we suppose the residue field

The functor described in 5.5 is an *equivalence* of *categories*.

*Proof*. By 5.5, it remains only to show that every étale

If

Let

*Proof*. Of course, we define an étale covering of a *formal* prescheme *locally free*, and such that the residue fibres *separable* algebras over

It was not necessary to restrict ourselves to the case of *coverings* in 8.4, but this is the only case that we will use for the moment.

## 9. Invariance properties

Let *quasi-finite* and *flat* over * A and B have the same Krull dimension, and the same depth* (Serre’s “cohomological codimension”, in the more modern language).
It also follows, for example, that

*. Also, for any prime ideal*A is Cohen–Macaulay if and only if B is

*every*prime ideal

*faithfully*flat over

*; also,*{\mathfrak{q}} and {\mathfrak{p}} have the same rank

*.*A has no embedded prime ideals if and only if B has none

We will thus content ourselves with more specific propositions dealing with the case of étale morphisms.

Let

*Proof*. Let

Let

Let

This is equivalent to the following:

Let

*Proof*. The necessity is trivial, since *étale* over

Let

*Proof*. Indeed, *finite* and étale over

Let

- If
f is étale, thenA is normal if and only ifB is normal. - If
A is normal, thenf is étale if and only iff is injective and unramified (and thenB is normal, by (i)).

We will give two different proofs of (i): the first using certain properties of quasi-finite flat morphisms (stated at the start of this section) and without using 7.6 (and thus the Main Theorem); the second proof does the opposite. For (ii), it seems like we do indeed need the Main Theorem, no matter what.

*Proof*. *(First proof).*
We use the following necessary and sufficient condition for a local Noetherian ring

—

- For every rank-
1 prime ideal{\mathfrak{p}} ofA ,A_{\mathfrak{p}} is normal (or, equivalently, regular); - For every rank-
\geqslant 2 prime ideal{\mathfrak{p}} ofA , the depth ofA_{\mathfrak{p}} is\geqslant 2 .^{10}

We assume this criterion here, but it should also appear in the section on flatness.
Its main advantage is that it does not suppose a priori that

By the statements at the start of this section, the rank-

*Proof*. *(Second proof).*
Suppose that

Now suppose that *since A is normal, the F_i are in A[t]* (supposing that they are monic).
Note that

*injective*; for this, we have used the fact that

Now recall the well-known lemma, taken from Serre’s lectures last year:

Let

The determinant of the matrix

Let

Suppose that

We can apply the above corollary to the situation that we have obtained in the proof: since

*Proof*. *(of ii).*
We proceed as in the above proof to show that we can choose

(From an editorial point of view, we should perform the two proofs above, and place the formal calculations of the lemma and of its corollaries in a separate section).

Let

Let

*Proof*. Let *injective* (taking into account the fact that

Let

(This is the “less trivial” statement which was alluded to in the remark in §4.)

We do not claim that a connected étale covering of an irreducible scheme is itself irreducible if we do not assume the base to be normal. This question will be studied in §11.

## 10. Étale coverings of a normal scheme

Let

*Proof*. By 9.10,

Under the conditions of 10.1,

*Proof*. We know that this normalisation is finite over

An algebra *unramified over X* (or simply

*unramified over*K if

*and*the normalisation

For every

*Proof*. The inverse functor is the normalisation functor.

Suppose that *fundamental bilinear form*

The syllogism 4.6 immediately implies the syllogism of being unramified in the classical case:

Let

K is unramified overY .If

L is an extension ofK that is unramified overY , and ifY' is a normal prescheme, of fieldL , that dominatesY (e.g. the normalisation ofY inL ), andM an extension ofL that is unramified overY' , thenM/K is unramified overX (this is the*transitivity*property).Let

Y' be a normal integral prescheme that dominatesY , of fieldK'/K ; ifL is an extension ofK that is unramified overY , thenL\otimes_K K' is an extension ofK' that is unramified overY' (this is the*translation*property)

Furthermore:

Under the conditions of (iii), if

Usually, people (those who are disgusted by the consideration of non-integral rings, even if they are direct sums of fields) state the translation property in the following (weaker) form:

Under the conditions of (iii), let *sum extension* of

This latter fact is actually false without the unramified hypothesis, even in the case of extensions given by direct sums of number fields…

To finish this section, we are going to give the intuitive interpretation of the notion of étale coverings: there should be the “maximal number” of points over the point

A morphism of finite type *universally open* if, for every base extension *universally* open (Chevalley’s theorem).
It thus follows, for example, that, if *quasi-finite* morphism, with

Let *finite* over ^{11}

If ^{12}, then the irreducible components of

Let *étale* morphism.
With the notation of 10.7, the function *finite* over *étale covering* of

For a separated étale morphism *étale covering* of

In 10.7 and its corollary, there was no normality hypothesis on

Let *quasi-finite* morphism.
Suppose that *irreducible*, that every component of *étale covering* of

*Proof*. The “only if” is trivial;
we will prove the “if”.
Let *finite* over

Let

*Proof*. It remains only to show that, if

In the general case (where

## 11. Various addenda

We have already said that a connected étale covering of an integral scheme is not necessarily integral. Here are two examples of this fact.

Let

C be an algebraic curve with an ordinary double pointx , and letC' be its normalisation, witha andb the two points ofC' overx . LetC'_1 andC'_2 be copies ofC' , witha_i (resp.b_i ) the point ofC'_i corresponding toa (resp.b ). In the curveC'_1\coprod C'_2 , we identifya_1 withb_2 , anda_2 withb_1 (we leave making this process of identification precise to the reader; it will be explained in Chapter VI of the multiplodoque, but, in the case of curves over an algebraically closed field, is already covered in Serre’s book on algebraic curves). We obtain a curveC'' which is*connected*and*reducible*, and which is a degree-2 étale covering ofC . The reader can verify that, generally, the “Galois” connected étale coveringsC'' ofC whose inverse imagesC''\times_C C' are*trivial*coverings ofC' (i.e. isomorphic to the sum of a certain number of copies ofC' ) are “cyclic” of degreen , and, conversely, for every integern>0 , we can construct a cyclic connected étale covering of degreen . In the language of the fundamental group (which will be developed later), this implies that the quotient of\pi_1(C) by the closed invariant subgroup generated by the image of\pi_1(C')\to\pi_1(C) (the homomorphism induced by the projection) is isomorphic to the compactification of\mathbb{Z} . More precisely, we should show that the fundamental group ofC is isomorphic to the (topological) free product of the fundamental group ofC with the compactification of\mathbb{Z} . We note that is was questions of this sort that gave birth to the “theory of descent” for schemes.Let

A be a complete integral local ring; we know that its normalisationA' is finite overA (by Nagata), and is thus a complete semi-local ring, and thus local, since it is integral. Suppose that the residue extensionL/k that it defines is non-radicial (in the contrary case, we say thatA is*geometrically unibranch*; cf. below). This will be the case, for example, for the ring\mathbb{R}[[s,t]]/(s^2+t^2)\mathbb{R}[[s,t]] , where\mathbb{R} is the field of real numbers. Then letk' be a finite Galois extension ofk such thatL\otimes_k k' decomposes; letB be a finite and étale algebra overA corresponding to the residue extensionk' (recall thatB is essentially unique). Then the residue algebra ofB'=A'\otimes_A B overB isL\otimes_k k' , which is not local, and soB' is not a local ring, and thusB has zero divisors (since it is complete). NowB' is contained in the total ring of fractions ofB (since it is free overA' , thus torsion free overA' , thus torsion free overA , thus contained inB'\otimes_A K=B'_{(K)}=A'_{(K)}\otimes_K B_{(K)}=B_{(K)} , sinceA'_{(k)}=K ), and soB is not integral. In the case of the ring\mathbb{R}[s,t]/(s^2+t^2)\mathbb{R}[s,t] , takingk'/k=\mathbb{C}/\mathbb{R} , we see thatB is the local ring of two secant lines at their point of intersection.We note also that, if there exists a connected étale covering

X ofY that is integral but not irreducible, then every irreducible component ofX gives an example of an unramified coveringX' ofY that dominatesY but is not étale overY . In the case of example (a), we thus see thatC' is unramified overC , without being étale at the two pointsa andb (note that, directly, by inspection of the completions of the local rings atx anda , from the “formal” point of view,C' at the pointa can be identified with a closed subscheme ofC at the pointx , i.e. one of the two “branches” ofC passing throughx ).

In both (a) and (b), we see that the fact that the conclusions of 9.5 (i) and (ii) fail to hold is directly linked with the fact that a point of *distinct* points of the normalisation (in (b), the fact that the residue extension is non-radicial should be interpreted geometrically in this way).
More precisely, we say that an integral local ring *geometrically unibranch* if its normalisation has only a single maximal ideal, with the corresponding residue extension being radicial;
a point *Examples:*
a normal point, an ordinary cusp point of a curve, etc.
It seems that, if *if all the points of Y are geometrically unibranch, then every unramified connected Y-prescheme that dominates Y is étale and irreducible*.
The proof follows that of 9.5, using the following generalisation of Theorem 8.3, which will be proved later by means of the technique of descent:

^{13}