Sheaves of Modules III

This post focuses on Ogus’ proof of 

Thm2: if \ 0 \to E' \to E \to E'' \to 0 is exact on affine scheme S, E’ qcoh, then \ 0 \to E'(S) \to E(S) \to E'(S) \to 0 exact.


Thm3: In the above sequence, if any two are qcoh, then so it the third.

But before that, recall from Thm1 from last time or H.II.5.4 show that on an affine scheme S = Spec A, quasicoherent sheaves are just \ \sim of the global sections, the proof banks on the fact that, for E quasicoherent, and

\ E(S_a) \cong E(S)_a, \ a \in E(S).

Having established this result, a more general result can be proved ( quasi separated = intersection of any two quasi compact sets is quasi compact.):

THM. X a scheme, quasi compact and quasi separated, E qcoh on X. Then for \ a \in E(X),

\ E(X)_a \cong E(X_a)

pf: multiplication by a is an isomorphism on \ E(X_a) so (as in the affine case) there is a natural map \ E(X)_a \to E(X_a).  

Injectivity: say \ E(X)_a \ni e \mapsto 0 \in E(X_a).  Choose n so \ a^ne =: e' \in E(X),  Cover X with a finite # of affines \ \{X_i\}.  Using the result for affines: \ E(X_i)_a \cong E((X_i)_a) conclude that for some power, \ a^me \equiv 0 (since it restricts to 0 on the affines X_i). As multiplication by a is an isomorphism, e = 0.  

surjectivity: fix \ e \in E(X_a), again, e has a restriction to \ E((X_i)_a), again use that the result holds for affines to get a finite  \ n_i \mbox{ and } e_i \in E(X_i) such that

\ e_i \mapsto a^{n_i}e|_{(X_i)_a} \in E((X_i)_a)

If necessary multiply the \ e_i by some power of a so that all the \ n_i are equal.  Then (here is where we use quasi separatedness), on \ X_{ij} = X_i \cap X_j have

\ e_i - e_j \mapsto 0 \in E((X_{ij})_a)

Now \ X_{ij} is quasi compact, can use injectivity argument above to conclude that \ a^m(e_i - e_j) \equiv 0 for all i,j and some m, hence \ a^me_i glue to give a global section, and this global section maps to \ a^le \in E(X_a) which is enough to show surjectivity.QED

The goal now is to prove Thm2, Thm3.

Note for Thm3, given that \ \sim is fully faithful and exact, the only nontrivial statement is that if E’,E” are qcoh then so is E.  Hartshorne proves this briefly using the 5-lemma.  Ogus develops many (relatively simple) lemmas that makes Thm2 quick to prove, in contrast Hartshorne proves Thm2 directly in a somewhat lengthy argument (Its analogues to ex. II.2.16b). 

Lemma1: If \ E',E, E'' all qcoh, then \ E(S) \to E''(S) surjective.

pf: Set Q to be the cokernel of the map. Using \ E \cong \widetilde{E(S)} etc. and exactness of \ \sim, it follows \ \widetilde{Q} is the cokernel of \ E \to E'', i.e. 0, so Q is 0.


Lemma2: If \ E',E'' qcoh, \ a \in \mathcal{O}_S(S) and \ 0\to E'(S)_a \to E(S)_a \to E''(S)_a \to 0 exact, then E is qcoh.

pf: use the 5-lemma and \ E'(S)_a \cong E'(S_a), \ E''(S)_a \cong E''(S_a) to conclude \ E(S)_a \cong E(S_a).


Lemma3: If \ E' qcoh and \ E'' = \mathcal{O}_S, then \ E is qcoh. Hence \ E(S) \to E''(S) is surjective (Lemma1).

pf: locally \ E \to E'' is surjective, so can find affine basis \ \{S_a\} s.t. \ 0 \to E'(S_a) \to E(S_a) \to A_a \to 0 exact. For any \ a' \in A_a:

\ 0 \to E(S_a)_{a'} \to E(S_a)_{a'} \to A_{aa'} \to 0,

since localization is exact; the 5-lemma gives

\ 0 \to E(S_{aa'}) \to E(S_{aa'}) \to A_{aa'} \to 0

is exact.  Lemma2 says \ E|_{S_a} is qcoh, use Lemma1 and 5-lemma to get \ E(S)_a \cong E(S_a), hence E qcoh.


Proof of Thm2: let \ e'' \in E''(S).  Recall from sheaves of modules I \ e'' determines a unique map \ \mathcal{O}_S \to E''.  We have two maps to \ E'' so we can take the fiber product, call it \ \overline{E} and get:

\ \begin{array}{ccccccccc} 0 & \to & F' & \to & \overline{F} & \to & \mathcal{O}_S & \to & 0 \\ \mbox{} & \mbox{} & \downarrow & \mbox{} & \downarrow & \mbox{} & \downarrow & \mbox{} & \mbox{} \\ 0 & \to & F' & \to & F & \to & F'' & \to & 0 \end{array}

The map  \ E' \to \overline{E} is induced by the given \ E' \to E and the 0-map \ E' \to \mathcal{O}_S.  The top row is exact essentially by properties of fiber product and exactness of the bottom row (think!).  By construction \ e'' lifts to the identity in \ \mathcal{O}_S(S) and Lemma3 says we can further lift \ 1 \in A to \ \overline{E}(S), and applying \ \overline{E}(S) \to E(S) we get a lift of \ e'' in \ E(S).


Proof of Thm3: assuming E’,E” are qcoh. The restriction to \ S_a preserves qcoh, so by Thm2, \ 0 \to E'(S_a) \to E(S_a) \to E''(S_a) \to 0 is exact, the 5-lemma gives

\ 0 \to E'(S)_a \to E(S)_a \to E''(S)_a \to 0

is exact, so E is qcoh by Lemma2.


Lastly, some comments about pullback and push forwards.  Let \ f \colon X \to Y by a map of schemes.  Let G qcoh on Y.  Then \ f^*G is qcoh on X (pf: can first reduce to X affines, then map check on distriguished affines of X mapping into affines of Y, and so can reduce to X,Y affine and its true in this case).  For E qcoh on X, for \ f_* E to be qcoh on Y, need f to be quasi separated and quasi compact (usually for a morphism to have P means if U has P then its inverse image has P).  The proof for push forwards is essentially showing 

\ f_*E(U)_a \cong f_*E(U_a)

which is done by unravelling definitions (in particular that \ f^{-1}(U) is quasi compact and quasi separated ) and using the THM that started this post.


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