# On coherent algebraic and analytic sheaves

*4th and 11th of February, 1957*

#### Translator’s note

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

Grothendieck, A. “Sur les faisceaux algébriques et les faisceaux analytiques cohérents.” *Séminaire Henri Cartan* **9** (1956–1957), Talk no. 2. `numdam.org/item/SHC_1956-1957__9__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: `94f6dce`

The aim of this talk is to generalise certain theorems of Serre. It makes fundamental use of the techniques of Serre [1–3].

# 1 Generalities on coherent algebraic sheaves

Let *of finite type* if, on every small-enough open subset, it is isomorphic to a quotient of *coherent* if it is of finite type and if, for every homomorphism

Let

—

{\mathscr{O}}_X is a coherent sheaf of rings.- If
X is affine with coordinate ringA(X) , then, for every coherent{\mathscr{O}} -module{\mathscr{A}} onX , the stalks{\mathscr{A}}_x are generated by the canonical image of\Gamma(X,{\mathscr{A}}) . Furthermore,\Gamma(X,{\mathscr{A}}) is anA(X) -module of finite type, and everyA(X) -module of finite type comes from an essentially unique coherent{\mathscr{O}} -module. (Recall that\Gamma(X,{\mathscr{A}}) denotes the module of sections of{\mathscr{A}} overX ). - Under the conditions of b), we have that
\mathrm{H}^i(X,{\mathscr{A}})=0 fori>0 .

*Proof*. For the proofs, which are very elementary, see [1, chapitre 2, paragraphes 2,3,4], or a talk of Cartier in the 1957 *Séminaire Grothendieck*.

# 2 A dévissage theorem

Let *exact subcategory* if, for every exact sequence

Let *exact* subcategory

*Proof*. The proof is done by induction on

Let

*Proof*. By “compactness” reasons, we can restrict to the case where

Under the above conditions,

This implies that *if \dim\operatorname{supp}{\mathscr{A}}<n, then {\mathscr{A}}\in{\mathcal{K}}.*

Suppose first of all that

If

*Proof*. We can immediately restrict to the case where

Using the exact sequence

Let *globally* isomorphic to

On any irreducible algebraic set

*Proof*. This is an easy consequence of the fact that every open subset of

We will thus identify *if and only if* *if and only if* *if and only if* *if and only if*

Now, if *injective*;
let

We say that the subcategory *left exact* if, for every exact sequence *provided that* the

# 3 Complements on sheaf cohomology

Let

If *inverse image of {\mathscr{B}}*, as well as the canonical homomorphism

Now let *direct image* *Leray spectral sequence of the continuous map f*, i.e. there is a cohomological spectral sequence starting with

*the sheaf on*Y associated to the presheaf V\mapsto\mathrm{H}^q(f^{-1}(V),{\mathscr{A}}) .

From the Leray spectral sequence, we get homomorphisms
*if \mathrm{R}^qf_*({\mathscr{A}})=0 for q>0, then the homomorphisms in Equation (1) are isomorphisms*.
This follows immediately from the spectral sequence, or, even more simply, from the fact that

For the results of this section, see the 1957 *Séminaire Grothendieck*.

# 4 Supplementary results on algebraic sheaves on projective space

Let *dual* bundle is denoted

—

- Let
Y be an affine algebraic set, and{\mathscr{A}} a coherent algebraic sheaf on\mathbf{P}\times Y . Then, for everyn large enough,{\mathscr{A}}(n) is generated by the module of its sections, i.e.{\mathscr{A}}(n) is isomorphic to some quotient of{\mathscr{O}}_{\mathbf{P}\times Y}^k , for some integerk . - For
n large enough,\mathrm{H}^i(\mathbf{P},{\mathscr{O}}(n))=0 .

*Proof*. The proof is elementary;
for (a), see [1, théorème 1] (where the proof is given for the case where *Séminaire Grothendieck* for more on this point), and using the well-known cover of

Now suppose that

—

- Let
{\mathscr{A}}^h be a coherent{\mathscr{O}}^h -module on\mathbf{P}^h . Then, for alln large enough,{\mathscr{A}}^h(n) is isomorphic to a quotient of({\mathscr{O}}^h)^k , for some integerk . - For
n large enough,\mathrm{H}^i(\mathbf{P}^h,{\mathscr{O}}^h(n))=0 .

*Proof*. The proof is distinctly deeper: see [2, lemme 5, page 12, and lemma 8, page 24].
It works by induction on the dimension, and makes essential use of the fact that the cohomology

# 5 The finiteness theorem: statement

Let *Séminaire Grothendieck*), we can easily show that, for every *affine* open subset

A morphism *proper* if, for every irreducible component *Séminaire Cartan-Chevalley*).

A more geometric definition is the following: *closed*.
Let

Let

*Proof*. The proof will be given in §7.

We state here the following corollary, obtained by taking

Let

# 6 An algebraic-analytic comparison theorem: statement

Let *analytic sheaf associated to {\mathscr{A}}*.
It is shown in [2] that the covariant functor

*exact*. We have a functorial homomorphism

We will see that, if

This functorial homomorphism can be extended, in a unique way, to functorial homomorphisms (that commute with the coboundary operators):

These homomorphisms have all the functorial properties that we might desire, but whose precise statements will not be given here (even though they will, of course, be essential in the proofs.)

Suppose that the morphism of algebraic sets

*Proof*. The proof will be given in the following section.

Taking

If

Since

Under the conditions of Theorem 5, the

It is very plausible that, more generally, if *Séminaire Cartan*), or if *loc. cit.*).

Under the conditions of Theorem 5, suppose further that *affine* algebraic set, and let

*Proof*. We have already said that *Séminaire Cartan*).
It thus suffices, by Theorem 5, to prove that, if

# 7 Proof of Theorems 4 and 5

The proofs follow mainly from Theorem 3, the “dévissage” of Theorem 2 (which is necessary since there is no reason for

*(Chow’s lemma.) —*
Let

Recall (§5) that "proper implies, in this case, that the graph of *closed* subset of *proper* and *surjective*.

*Proof*. We cover

Theorem 4 and Theorem 5 say that every coherent algebraic sheaf *exact* subcategory (§2, Definition 1), by using the exact sequence of the *on Z*, with support equal to

*one*coherent

Let *proper* morphism of algebraic sets, with

First we will show how this lemma will imply the previous one.
Applying the lemma to

It thus remains only to prove Lemma 5.
Since the graph

We first prove Lemma 5 in the case where *vectorial-topological variant of the Künneth theorem* (using the fact that the space

To prove Lemma 5 in the general case, we proceed by induction on

The last paragraph of this proof can be simplified if we use the fact that

# 8 Algebraic and analytic sheaves on a compact algebraic variety

We are going to complete Corollary 1 of Theorem 5:

Let

The uniqueness of

With

*Proof*. This homomorphism comes from, by taking sections, the monomorphism of sheaves

From Corollary 1 and the exactness of the functor *projective* (Serre).

Let *Séminaire Grothendieck*), from the elementary local relations
*sheaf*

*Proof*. *(Proof of Theorem 6.) —*
We can now prove Theorem 6, by induction on *projective* variety, and *birational* morphism.
For every analytic sheaf *on Y*.
These quotients are in fact “algebraic,” by the induction hypothesis;
thus so too are their extensions

# Bibliography

*Annals of Mathematics*.

**61**(1955), 197–278.

*Annales de l’Institut Fourier*.

**6**(1955-56), 1–42.

*Journal de Math. P. Et Appl.*(1957).