# Sheaves of Modules II

This post is about the Ogus version of basically the results II.5.5, II.5.6, II.5.7 in Hartshorne.

#### Quick Results

- is an exact functor (pf: check exactness on stalks, and , and localization is exact (can show by unwrapping definition of and exactness).
- commutes with tensor products and direct sums (pf: because isomorphism can be checked on stalks and localization commutes with both operations)
- is (Ogus def) quasicoherent (pf: apply 1 to presentation of M).

Now time to get some more serious stuff. Fix a ring A, and set S = Spec A…

#### Two Lemmas

**Lemma1**: If X is quasicompact, is a collection of sheaves of modules and , then .

pf: injectivity: there is a factorization

.

The first map is the presheaf to sheaf map, the second map exists because maps into , and projecting to each factor, we get an element of . As is injective, so is the map we want.

surjectivity: Let . Let be the ith part of e, locally nonzero for finitely many i. X is quasicompact, so pick a finite cover on which e is finitely nonzero. For each , get a finite set on which is nonzero. The finite union of all these finite sets is finite, so for any i not in this set, must have . shows surjectivity.

**Lemma 2****:** If E is globally presented on S = Spec A, then E is quasicoherent.

pf: Apply global sections functor to

The previous lemma says the first two terms are , so have exact seq.

Apply functor, which is exact, so have

.

The 5-lemma gives , which is quasicoherent by quick results. QED.

THM 1: For S = Spec A, the functor between the category of A-modules and quasicoherent sheaves on S is an equivalence.

pf: Have that is is fully faithful, just need essentially surjective, i.e. if E quasicoherent on S, then for some . Enough to show isom. locally on distinguished affines. The idea is to fit and into short exact sequences where the other terms are finite products of or , using the previous two isomorphisms and the 5-lemma, conclude , hence QED.

RMK: the proof in Hartshorne is similar. For F quasicoherent on S = Spec A, and , there is always a natural map , and Hartshorne argues directly that it is injective and surjective (lemma II.5.3 and prop. II.5.4).

## About this entry

You’re currently reading “Sheaves of Modules II,” an entry on Math Meandering

- Published:
- May 29, 2009 / 12:18 am

- Category:
- Ogus Excerpts, Ogus/Hartshorne/Miranda

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