# Serre’s theorem

*1958–59*

#### Translator’s note

*This page is a translation into English of the following:*

Gabriel, P. “Le théorème de Serre.” *Séminaire Claude Chevalley* **4** (1958–59), Talk no. 2. `http://www.numdam.org/item/SCC_1958-1959__4__A2_0`

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `f36a214`

# 1 Ringed topological spaces

From now on, and unless otherwise mentioned, the rings we consider will be assumed to be commutative, with a unit element, and *Noetherian*.
We define a *ringed topological space*

- a continuous map
\psi\colon V\to W ; - for every open subset
U ofW , a ring homomorphism\varphi_U\colon{\mathscr{B}}(U)\to{\mathscr{A}}(\varphi^{-1}(U)) that is compatible with the restriction maps.

The composition of two morphisms is defined in the evident way, and we speak of the category of ringed topological spaces.
In what follows, *the scheme of V*, and which is defined in the following way:

- the points of
S(V) are the closed irreducible subsets ofV ; - the closed subsets of
S(V) are the sets{\mathscr{F}} , whereF is a closed subset ofV , and{\mathscr{F}} denotes the set of closed irreducible subsets ofV that are contained inF .

It is then clear that the correspondence *scheme of (V,{\mathscr{A}})*, which is defined to be the ringed topological space given by

If

# 2 Sheaves of ideals of an algebraic set

If *support* is the set of points *dimension of {\mathscr{F}}* to be the dimension of its support.
When

The notion of an ideal generalises in the following way: we define a *sheaf of ideals* of

If

x\in W({\mathscr{I}}) if and only if{\mathscr{I}}_x , the fibre of{\mathscr{I}} atx , is a proper ideal of{\mathscr{O}}_x (i.e.{\mathscr{I}}_x\neq{\mathscr{O}}_x ), or if and only if the germs of the functions defined by{\mathscr{I}} atx are zero.W({\mathscr{I}}) is thus the support of{\mathscr{A}}|{\mathscr{I}} .

Conversely, if

The sections of

{\mathscr{I}}(W) on an open subsetU are the regular functions defined onU that are zero onU\cap W . If{\mathscr{I}}(U,x) denotes the inverse image in{\mathscr{A}}(U) of the maximal ideal of{\mathscr{O}}_x , then\Gamma(U,{\mathscr{I}}(W)) is the intersection of the{\mathscr{I}}(U,x) wherex runs overU\cap W .

If

—

- The map
W\mapsto{\mathscr{I}}(W) gives a bijective correspondence between the closed subsets ofV and the sheaves of ideals{\mathscr{I}} that satisfy the following condition: for everyx\in V ,{\mathscr{I}}_x is either equal to{\mathscr{O}}_x or equal to an intersection of prime ideals of{\mathscr{O}}_x . - The map
W\mapsto{\mathscr{I}}(W) associates the closed subsets whose connected components are all irreducible to the sheaves of prime ideals (for everyx ,{\mathscr{I}}_x is either equal to{\mathscr{O}}_x or equal to a prime ideal).

*Proof*. —

- It suffices to give a proof in the case where
(V,{\mathscr{A}}) is an affine algebraic set. So letA={\mathscr{A}}(V) , and\mathfrak{a}=\Gamma(V,{\mathscr{I}}(W)) . We have already seen that\mathfrak{a} is then the intersection of the prime ideals{\mathscr{I}}(V,x) , wherex runs overW . So\mathfrak{a} is an intersection of prime ideals. Since the correspondence between intersections of prime ideals ofA and closed subsets ofV is bijective, so too is the correspondence between ideals ofA and sheaves of ideals of(V,{\mathscr{A}}) ; it remains only to show, conversely, that the sheaf associated to an intersection\mathfrak{a} of prime ideals satisfies the condition of the proposition: this follows from the conservation properties of the prime decomposition under localisation. - The proof is analogous.

# 3 The ring of rational functions of an algebraic set

We recall that, if *rational function on V* to be a regular map

The sheaf

to every open subset

U ofV , we associate the ringK(U) of rational functions onU , with the restrictions being obvious.

It is clear that this defines a quasi-coherent sheaf on

# 4 Characterisation of affine algebraic sets

Let

We will now define a morphism

The map

It follows from the above that

The

It is clear that

The following are equivalent:

(V,{\mathscr{A}}) is an affine algebraic set.- If
0\to{\mathscr{F}}\to{\mathscr{G}}\to{\mathscr{H}}\to0 is an exact sequence of quasi-coherent sheaves, then the sections overV form an exact sequence. - If
0\to{\mathscr{F}}\to{\mathscr{G}}\to{\mathscr{H}}\to0 is an exact sequence of coherent sheaves, then the sections overV form an exact sequence. - There exist sections
f_i of{\mathscr{A}} overV such that:- the ideal generated by the
f_i is equal toA ; and - the open subsets
V_{f_i} ofV , where thef_i are non-zero, are affine open subsets, and they coverV .

- the ideal generated by the

*Proof*. It remains only to show that (c) implies (d), and that (d) implies (a).

For (c)

If

*Proof*. Indeed, the support of

The fibre

It thus follows that we have an epimorphism from

With this proven, let

If

*Proof*. Let

The quasi-compactness of

For (d)

Note first of all that, if

But the

From this we see that the map

Also, the

But

Indeed, by taking a suitable power of the formula

If

U is an affine open subset.- If
M is anA -module, then the canonical homomorphismM\otimes_A{\mathscr{U}}(U) \to {\mathscr{M}}(U) is bijective.

Furthermore, if one of these equivalent properties is satisfied, then

*Proof*. (a)

Since

(b)

But there exists an exact sequence

# Bibliography

*Bull. Soc. Math. France*.

**86**(1958), 97–136.

*Bull. Soc. Math. France*.

**86**(1958), 137–154.

*Séminaire Cartan–Chevalley: Géométrie algébrique*. 1955-56.

**8**.

*Séminaire Chevalley: Classification des groupes de Lie algébrique*. 1958.

**1**.