# 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 , we have functor, sending an -module to the sheaf described as

Hartshorne defines Quasicoherent sheaves of modules to be things that are locally , but Ogus’ definition is a little more involved…

Now, given an A-module map , we get defined on distinguished affines by the diagram below

On the other hand, starting with a map of sheaves , taking global sections we get . Its natural to ask, ? The answer is yes: both and give a commutative square as the one above where the top map is just , 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 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 etc.). Also a sequences of sheaves is exact iff its exact on stalks (H.II.1.2).

In the following, is an index set and E is a sheaf of modules on a scheme X. Given get a global section of E, namely the image of the identity under the global sections map . Conversely, a global section defines a map sending . So there is a natural isomorphism . A similar argument shows

## (Quasi) Coherence

Ogus introduced the following definitions:

- Generated by Global Sections: for an index set I, there is an I-tuple such that the associated map is surj. on stalks.
- Globally Presented: there exists an exact sequence .
- Locally Generated by Global Sections: there is an open cover , s.t. each is globally generated.
- Quasicoherent: there is an open cover s.t. each is globally presented.
- Locally of finite type: there is open cover s.t. for each , there is finite I, and surjection .
- Locally finitely presented: for each there are finite sets .
- Coherent: locally of finite type and for I finite, and , is locally of finite type.

In contrast, Hartshorne defines F to be quasicoherent if there exists open (affine!) cover such that , for a -module. F is coherent if the modules are finitely generated.

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

## About this entry

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

- Published:
- May 28, 2009 / 5:29 pm

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

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