# Quasi-coherent sheaves

*1958–59*

#### Translator’s note

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

Gabriel, P. “Faisceaux quasi-cohérents.” *Séminaire Claude Chevalley* **4** (1958–59), Talk no. 1. `http://www.numdam.org/item/SCC_1958-1959__4__A1_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`

We assume prior knowledge of the definitions and elementary properties of sheaves of modules on a topological space, i.e. [[2], chapitre I, §1; chapitre II, §§1–2].
We define a presheaf

- for every open subset
U in\mathscr{B} , an object{\mathscr{P}}(U) of{\mathcal{C}} , that we may also denote by\Gamma(U,{\mathscr{P}}) ; - for every pair
(U,V) of open subsetsU in\mathscr{B} such thatU\subset V , a morphism\rho_{UV}\colon{\mathscr{P}}(V)\to{\mathscr{P}}(U) . The morphism\rho_{UV} will be called the restriction ofV toU . We further suppose that, ifU\subset V\subset W are open subsets in\mathscr{B} , then\rho_{UV}\circ\rho_{VW}=\rho_{UW} .

The construction of the sheaf

# 1 Preliminaries on localisation

Let *complete* if it satisfies the following condition:
if

If now

The set

The addition in

We have a bilinear map

We can furthermore easily prove the following claims:
the correspondence

If

If

# 2 The prime spectrum of a commutative ring

Let

Then

The set

The sets *prime spectrum of A*.

If the ideal *special*.
Every special open subset is *quasi-compact*:
indeed, if

In particular, *Noetherian*, then every increasing sequence of ideals stabilises, and the same is true for every increasing sequence of open subsets.
The prime spectrum is thus a Zariski topological space.

# 3 Quasi-coherent sheaves on V(A)

Now let *presheaf \mathfrak{M}^* on the base of special open subsets of V(A)*.
For this, we will associate, to each

With this in mind, we will define the presheaf

We will denote by *algebraic sheaf* on *quasi-coherent sheaf* on

For this, we note that the prime ideals of

More generally, if

For every prime ideal

—

- If
\mathfrak{p} is a point ofV(A) , then the localised ring\mathfrak{A}_\mathfrak{p} (the fibre of the sheaf\mathfrak{A} over\mathfrak{p} ) is canonically isomorphic toA_\mathfrak{p} ; Identifying\mathfrak{A}_\mathfrak{p} withA_\mathfrak{p} , the localised module\mathfrak{M}_\mathfrak{p} of the sheaf\mathfrak{M} at\mathfrak{p} is canonically isomorphic toM_\mathfrak{p} , and thus toM\otimes_A A_\mathfrak{p} . - The canonical map from
M_f=\mathfrak{M}^*(U_f) to\mathfrak{M}(U_f) is an isomorphism.

*Proof*. The first claim of (a) follows from the fact that

We now show that the map *injective*:
Since

Similarly, the map *surjective*:
it suffices to prove this in the case where

For this, let

Also, the

If

*Proof*. Let

But if

If

*Proof*. We will exhibit an inverse map

It follows from this corollary, along with the exactness of the functor

If

{\mathscr{F}} is quasi-coherent.- For every special open subset
U_f , the natural map\Gamma(V,{\mathscr{F}})\otimes_A A_f \to \Gamma(U_f,{\mathscr{F}}) is bijective. - For every point
\mathfrak{p} ofV , the natural map\Gamma(V,{\mathscr{F}})\otimes_A A_\mathfrak{p} \to {\mathscr{F}}_\mathfrak{p} is bijective. {\mathscr{F}} is locally isomorphic to a quasi-coherent sheaf, i.e. every point\mathfrak{p} ofV has a special neighbourhoodU_f such that{\mathscr{F}}|U_f is a coherent sheaf.

*Proof*. We have already seen that (a)

We will first show that (b)

These maps induce a morphism of sheaves:

This morphism will be bijective if the induced maps

It remains only to show that (d)

We note first of all that the map

So take some algebraic sheaf

Since

The two latter terms of the exact sequence are the sections over

# 4 Coherent sheaves on V(A)

From now on we will suppose that * A is a Noetherian ring*.
Then it is well known that the category of Noetherian

*coherent sheaf on*V(A) to be any sheaf

*of finite type*if every point of

If

{\mathscr{F}} is a coherent algebraic sheaf onV(A) .{\mathscr{F}} is quasi-coherent and of finite type.{\mathscr{F}} is of finite type, and, for every open subsetU and every morphism\varphi\colon{\mathscr{G}}\to{\mathscr{F}}|U , where{\mathscr{G}} is a sheaf of finite type onU , the kernel\operatorname{Ker}\varphi is of finite type onU .

*Proof*. We already know that (a)

Now for (c)

Since the kernel of this surjection is also of finite type, we in fact have an exact sequence (provided that the special open subset

The sheaf

Finally, (a)

We then have the following diagram:

The sheaves

This latter exact sequence allows us to complete the diagram and show that, on

Claim (c) gives a characterisation of coherent sheaves, which does not require the base of special open subsets of

# 5 The maximal spectrum of a Jacobson ring

We are going to apply the above to algebraic geometry.
For this, we denote by *maximal spectrum of A*, which is defined to be the topological subspace of

*Jacobson ring*, i.e. that every prime ideal is the intersection of maximal ideals (see [1] and [3]). The following proposition then holds:

The following are equivalent:

A is a Jacobson ring.- The correspondence
U\mapsto U\cap\Omega(A) between open subsets ofV(A) and of\Omega(A) is bijective. - The correspondence
W\mapsto W\cap\Omega(A) between closed subsets ofV(A) and of\Omega(A) is bijective.

*Proof*. The equivalence of (b) and (c) is trivial.
On the other hand, the closed subsets of

Under this last hypothesis, the spaces *special open subset of \Omega* to be the restriction of any special open subset of

*quasi-coherent*(resp.

*coherent*)

*algebraic sheaf on*\Omega to be the restriction of any quasi-coherent (resp. coherent) algebraic sheaf on

In particular, every algebra of finite type over a field is a Jacobson ring.
If *sheaf of germs of regular functions on V*.

Similarly, a sheaf of modules over this sheaf of rings is said to be *algebraic* (resp. *quasi-coherent algebraic*, resp. *coherent algebraic*) if it is the restriction of a sheaf on

# 6 Quasi-coherent sheaves on an algebraic variety

More generally, if *quasi-coherent* (resp. *coherent*) *algebraic sheaf on X* to be any sheaf

In particular, the sheaf of rings *sheaf of germs of regular functions*.
Every sheaf of modules over *algebraic*.

# Bibliography

*Séminaire Cartan-Chevalley*.

**8**(1955-56), Talk no. 3.

*Topologie algébrique et théorie des faisceaux*. Hermann, 1958.

*Act. Scient. Et Ind.*

**1252**.

*Math. Z.*

**54**(1951), 354–387.

*Ann. Math.*

**61**(1955), 197–279.