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


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