# 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

1. $\ M \mapsto \widetilde{M}$ is an exact functor (pf: check exactness on stalks, and $\ \widetilde{M}_p = M_p$, and localization is exact (can show by unwrapping definition of $\ S^{-1}M$ and exactness).
2. $\ M \mapsto \widetilde{M}$ commutes with tensor products and direct sums (pf: because isomorphism can be checked on stalks and localization commutes with both operations)
3. $\ \widetilde{M}$ 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, $\ E_i$ is a collection of sheaves of modules and $\ E:= \oplus_i E_i$, then $\ \Gamma(X, E) = \oplus_i \Gamma(X,E_i) \to \Gamma(X, E)$.

pf: injectivity: there is a factorization

$\ \oplus_i \Gamma(X,E_i) \to \Gamma(X, E) \to \prod_i \Gamma(X, E_i)$.

The first map is the presheaf to sheaf map, the second map exists because $\ s \in \Gamma(X,E)$ maps $\ p \in X$ into $\ \oplus_i (E_i)_p$, and projecting to each factor, we get an element of $\ \prod_i E_i(X)$.  As $\ \oplus_i E_i(X) \to \prod_i E_i(X)$ is injective, so is the map we want.

surjectivity: Let $\ e \in E(X)$.  Let $\ e_i$ be the ith part of e, locally $\ e_i$ nonzero for finitely many i.  X is quasicompact, so pick a finite cover $\ \{U_j\}$ on which e is finitely nonzero.  For each $\ j$, get a finite set $\ j_1, \dotsc , j_k$ on which $\ e$ is nonzero.  The finite union of all these finite sets is finite, so for any i not in this set, must have $\ e_i \equiv 0$.  shows surjectivity.

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

pf: Apply global sections functor to

$\ \mathcal{O}^{(J)}_S \xrightarrow{\theta} \mathcal{O}_S^{(I)} \to E \to 0$

The previous lemma says the first two terms are $\ A^{(J)} \xrightarrow{\Gamma(\theta)} A^{(I)}$, so have exact seq.

$\ A^{(J)} \to A^{(I)} \to \mbox{coker} \theta \to 0$

Apply $\ \sim$ functor, which is exact, so have

$\ \mathcal{O}^{(J)}_S \to \mathcal{O}_S^{(I)} \to \widetilde{\mbox{coker} \theta} \to 0$.

The 5-lemma gives $\ E \cong \widetilde{\mbox{coker} \theta}$, which is quasicoherent by quick results. QED.

THM 1: For S = Spec A, the $\ \sim$ 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 $\ E \cong \widetilde{M}$ for some $\ M$. Enough to show isom. locally on distinguished affines.  The idea is to fit $\ E(S)_a$ and $\ E(S_a)$ into short exact sequences where the other terms are finite products of $\ E(S_{a_ia}) \cong E(S_{a_i})_a$ or $\ E(S_{a_ia_ja}) \cong E(S_{a_ia_j})_a$, using the previous two isomorphisms and the 5-lemma, conclude $\ E(S)_a \cong E(S_a)$, hence $\ E \cong \widetilde{E(S)}$ QED.

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