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). 

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