# Sheaves of Modules I

After doing presheaves, sheaves, sheafifacation, locally ringed spaces, and schemes, the next big topic was shaves of modules over a scheme.

For $\ S = \mbox{Spec} A$, we have $\ \sim$ functor, sending an  $\ A$-module  $\ M$ to the sheaf  $\ \widetilde{M}$ described as

•  $\ \widetilde{M}(D_f) = M_f = A_f \otimes_A M$
•  $\ (\widetilde{M})_p = M_p := A_p \otimes_A M$

Hartshorne defines Quasicoherent sheaves of modules to be things that are locally $\ \widetilde{M}$, but Ogus’ definition is a little more involved…

Now, given an A-module map $\ g \colon M \to N$, we get $\ \tilde g \colon \widetilde{M} \to \widetilde{N}$ defined on distinguished affines by the diagram below

$\ \begin{array}{ccc} M & \to & N \\ \downarrow & \mbox{} & \downarrow \\ M_f & \to & N_{f} \end{array}$

On the other hand, starting with a map of sheaves  $\ G \colon \widetilde{M} \to \widetilde{N}$, taking global sections we get  $\ \Gamma(G) \colon M \to N$.  Its natural to ask,  $\ G = \widetilde{\Gamma(G)}$?  The answer is yes: both  $\ G$ and  $\ \widetilde{\Gamma(G)}$ give a commutative square as the one above where the top map is just  $\ \Gamma(G)$, and by the universal property of localization, there is unique bottom map making everything commute QED.

With S as before,  Mod_A is the category of A-modules, and Mod_S is the category of sheaves of modules on S.  The  $\ \sim$ functor from Mod_A to Mod_S  is fully faithful but not an equivalence.  It turns out, it is an equivalence to a subcategory of Mod_S, namely the category of QUASICOHERENT sheaves of modules.

But first some fun facts: For maps of sheaves of modules, kernels, cokernels, and images commute with talking stalk (i.e. have equality  $\ \ker \theta_p = (\ker \theta)_p$ etc.).  Also a sequences of sheaves is exact iff its exact on stalks (H.II.1.2).

In the following, $\ I$ is an index set and E is a sheaf of modules on a scheme X.  Given  $\ a \in \mbox{\underline{Hom}}(\mathcal{O}_X, E)$ get a global section of E, namely the image of the identity under the global sections map  $\ \Gamma(X,\mathcal{O}_X) \to E$.  Conversely, a global section  $\ e \in E(X)$ defines a map  $\ \mathcal{O}_X(U) \to E(U)$ sending  $\ 1 \mapsto e|_U$.  So there is a natural isomorphism  $\ \mbox{\underline{Hom}}(\mathcal{O}_X, E) \cong E(X)$.  A similar argument shows  $\ \mbox{\underline{Hom}}(\mathcal{O}^{(I)}_X, E) \cong E(X)^I$

## (Quasi) Coherence

Ogus introduced the following definitions:

• Generated by Global Sections: for an index set I, there is an I-tuple $\ e. \in E(X)$ such that the associated map  $\ \mathcal{O}^{(I)}_X \to E$ is surj. on stalks.
• Globally Presented: there exists an exact sequence  $\ \mathcal{O}_X^{(J)} \to \mathcal{O}_X^{(I)} \to E$.
• Locally Generated by Global Sections: there is an open cover  $\ \{U_i\}$, s.t. each  $\ E|_{U_i}$ is globally generated.
• Quasicoherent: there is an open cover s.t. each $\ E|_{U_i}$ is globally presented.
• Locally of finite type: there is open cover s.t. for each $\ U_i$, there is finite I, and surjection $\ \mathcal{O}_{U_i}^{(I)} \to E|_{U_i}$.
• Locally finitely presented: for each $\ U_i$ there are finite sets $\ \mathcal{O}_{U_i}^{(J)} \to \mathcal{O}_{U_i}^{(I)} \to E$.
• Coherent: locally of finite type and for I finite, and $\ \mathcal{O}_X \xrightarrow{\theta} E$,  $\ \ker(\theta)$ is locally of finite type.

In contrast, Hartshorne defines F to be quasicoherent if there exists open (affine!) cover such that  $\ F|_{U_i} \cong \widetilde{M}_i$, for  $\ M_i$ a  $\ \mathcal{O}_X(U_i)$-module.  F is coherent if the modules are finitely generated.

The equivalence of these two definitions is the content of exercise II.5.4