# Sheaves of Modules III

This post focuses on Ogus’ proof of

Thm2: if is exact on affine scheme S, E’ qcoh, then exact.

and

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 of the global sections, the proof banks on the fact that, for E quasicoherent, and

, .

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 ,

pf: multiplication by a is an isomorphism on so (as in the affine case) there is a natural map .

Injectivity: say . Choose n so , Cover X with a finite # of affines . Using the result for affines: conclude that for some power, (since it restricts to 0 on the affines X_i). As multiplication by a is an isomorphism, e = 0.

surjectivity: fix , again, e has a restriction to , again use that the result holds for affines to get a finite such that

If necessary multiply the by some power of a so that all the are equal. Then (here is where we use quasi separatedness), on have

Now is quasi compact, can use injectivity argument above to conclude that for all i,j and some m, hence glue to give a global section, and this global section maps to which is enough to show surjectivity.QED

The goal now is to prove Thm2, Thm3.

Note for Thm3, given that 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 all qcoh, then surjective.

pf: Set Q to be the cokernel of the map. Using etc. and exactness of , it follows is the cokernel of , i.e. 0, so Q is 0.

QED

Lemma2: If qcoh, and exact, then E is qcoh.

pf: use the 5-lemma and , to conclude .

QED

Lemma3: If qcoh and , then is qcoh. Hence is surjective (Lemma1).

pf: locally is surjective, so can find affine basis s.t. exact. For any :

,

since localization is exact; the 5-lemma gives

is exact. Lemma2 says is qcoh, use Lemma1 and 5-lemma to get , hence E qcoh.

QED.

Proof of Thm2: let . Recall from sheaves of modules I determines a unique map . We have two maps to so we can take the fiber product, call it and get:

The map is induced by the given and the 0-map . The top row is exact essentially by properties of fiber product and exactness of the bottom row (think!). By construction lifts to the identity in and Lemma3 says we can further lift to , and applying we get a lift of in .

QED

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

is exact, so E is qcoh by Lemma2.

QED

Lastly, some comments about pullback and push forwards. Let by a map of schemes. Let G qcoh on Y. Then 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 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

which is done by unravelling definitions (in particular that is quasi compact and quasi separated ) and using the THM that started this post.

## About this entry

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

- Published:
- May 30, 2009 / 10:47 pm

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

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