# 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.

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 $\ \sim$ 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.

QED

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 $\ E(S)_a \cong E(S_a)$.

QED

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.

QED.

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 $\ E(S)$.

QED

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.

QED

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.