Sheaves of Modules

So now that we have abstract varieties on hand, we’re going to do a bit more with sheaves, leading to some of the intimate connections between sheaf theory and geometry. Sadly, this often gives students a lot of trouble (I know I had a bit of trouble with it at first) because things are presented very algebraically and the geometry gets lost. So we’ll be making a point of the connections between the geometry and the algebra.

As usual, we start out with a definition. We take X to be a ringed space (say, a variety) with structure sheaf \mathscr{O}_X. Then we say that a sheaf \mathscr{F} is an \mathscr{O}_X-module (or sometimes a sheaf of \mathscr{O}_X-modules) if for each U, we have \mathscr{F}(U) is an \mathscr{O}_X(U)-module, and the restriction maps on \mathscr{F} are compatible with the module structures induces by the restriction maps in \mathscr{O}_X. A morphism \mathscr{F}\to\mathscr{G} is a homomorphism of \mathscr{O}_X-modules if over each open set it is a module homomorphism.

Now, pretty much anything we can do with R-modules we can do with \mathscr{O}_X-modules. We can take kernels, cokernels, images, quotients and the like and all we get are more \mathscr{O}_X-moduels. We can talk about exact sequences of \mathscr{O}_X-modules by just looking to see if they are exact as sequences of sheaves. In fact, we can even make a sheaf out of morphisms of \mathscr{O}_X-modules so that this sheaf is one too, by defining it as U\mapsto Hom_{\mathscr{O}_X|_U}(\mathscr{F}|_U,\mathscr{G}|_U), where restriction to U just means that we consider it as a sheaf on U and so only care about open sets contained in U.

We can even define the tensor product of two sheaves, though we do have to perform a sheafification, because it won’t generally be a sheaf itself.

However, now, we’re going to focus for a moment on what these can tell use geometrically. To start with, we will look at an affine variety V, which has coordinate ring k[V]. Now, any closed subset of V is defined by an ideal I\subset k[V]. In fact, this gives an ideal in every localization of k[V], which is to say, for each open subset of V, we have an ideal in the coordinate ring of that subset. Now, ideals are always modules, so in fact, this collection of ideals forms a sheaf of modules!

We can, in fact, generalize this. Let Y\subset X be a closed subset of a variety. Then, to each U\subset X open, we assign \mathscr{I}_Y(U)=\{f\in\mathscr{O}_X(U)|f(Y\cap U)=0\}, that is, on each open set, we assign the regular functions that are zero on Y\cap U. This gives a sheaf of modules (in fact, this example is so important that we call these sheaves of ideals) and it is determined by the closed subset. So if we took a subvariety, that is, an irreducible closed subset of a variety, we’d even get a sheaf of prime ideals.

Now, we should note that the correspondence doesn’t quite go both ways. A closed subset of a variety defines a unique sheaf of ideals. However, if we are given a sheaf of ideals \mathscr{I}, and then we take the set of points in each U where the elements of \mathscr{I}(U) vanish, we might not even get a closed subset. There are sheaves of ideals which bear little to no resemblance to the example of an ideal for an affine variety, so we will fix this with a condition called quasi-coherence. We say that a sheaf of modules is quasicoherent if we can cover X with open affine subsets such that when we restrict the sheaf to each of these sets we get particularly nice sheaves of modules, which we will now describe:

Given an affine variety V and a module M over its coordinate ring, we get a sheaf of modules called \tilde{M} on V, by looking at the localizations of the modules. We will call a sheaf of modules quasi-coherent if it is locally isomorphic to sheaves of this form. If we can even take the modules to be finitely generated, we will call the \mathscr{O}_X-module coherent.

So what makes the ideals we want special? We only care about quasi-coherent sheaves of ideals, that is, sheaves of ideals that locally look like they come from the localizations of actual ideals. In fact, we can do better, because the Hilbert Basis Theorem tells us that ideals are finitely generated for polynomial rings and their quotients. Thus, the sheaves of ideals that we want are coherent.

So now given a coherent sheaf of ideals, we do in fact get a closed subset. However, many coherent sheaves of ideals give the same one, just as many ideals give the same closed subset of an affine variety. We can fix this by taking the radical ideal over each open set. This still gives us a coherent sheaf of ideals, and one which is equal to its radical will be called reduced. But now we get the same correspondence we used to have: closed subsets of V are in one-to-one correspondence with reduced coherent sheaves of ideals. Subvarieties, in fact, correspond to reduced coherent sheaves of prime ideals, and any other operation on ideals can still be performed, taking all the notions from affine geometry with us into this new land of abstract algebraic geometry.

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About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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18 Responses to Sheaves of Modules

  1. Doesn’t the internal-hom need to be sheafified (and coproduct, and…)? I thought that colimit operations usually broke the sheaf property (giving mere presheaves) and needed to be sheafified to land back in the category of sheaves.

    Or is the case of sheaves of modules special enough that this extra step usually isn’t needed?

  2. Charles says:

    Sheaf hom DOES manage to be a sheaf, which is a wonderful thing. It’s true of sheaves of abelian groups on a topological space, and this is actually a standard exercise (it’s not that hard). In general, direct limits on Noetherian spaces are always sheaves (Noetherian means ACC on open sets) and inverse limits are always sheaves as well. So in the category of varieties (or in nice categories of schemes) most of these operations play well with the sheaf axioms, and it’s not that hard to check this. (though inverse limit involves a nasty diagram)

    The big example where a further sheafification is actually necessary is tensor product.

    So the special situation isn’t sheaves of modules, but just sheaves of abelian groups on a Noetherian topological space.

    In fact significantly stronger statements can be formulated if we look at Grothendieck topologies, but I’m not ready to go there quite yet.

  3. Pingback: Locally Free Sheaves and Vector Bundles « Rigorous Trivialities

  4. kwan says:

    So how do I check that the image sheaf of a morphism of sheaves of modules is again a sheaf of modules? I can see that the image presheaf is a “presheaf” of modules, but how do I extend it to the sheafification? Since the scalar multiplication is not exactly a morphism of abelian groups, I can’t seem to apply the universal property of sheafification…

  5. I may be missing something, but how is scalar multiplication not a morphism of abelian groups? What sense do you mean?

    The image sheaf is just f_*\mathscr{F}(U)=\mathscr{F}(f^{-1}(U)), so any open cover on the bottom becomes an open cover above, and you can check the sheaf conditions on \mathscr{F}.

  6. kwan says:

    Oh, I didn’t mean the direct image sheaf given by a continuous map f between spaces. I meant the sheafification of “presheaf image,” that is, given a fixed space X, and a morphism between two presheaves on X, the presheaf image is a presheaf that gives the image of the homomorphisms. Sorry about the ambiguity.
    I said scalar multiplication was not homomorphism because it’s bilinear rather than linear.

  7. Ok, I’m again confused. What do you mean by scalar multiplication being bilinear? Let f\in R and M be an R-module, then mult_f:M\to M by x\mapsto fx is certainly a linear map.

  8. kwan says:

    Sorry for not being so precise — I just tried to be succinct. By scalar multiplication I meant the map R \times M \to M. I guess my question is, in a more general sense, that given a sheaf of rings O_X and a presheaf of O_X-modules, how does one check its sheafification is a module? I’m trying to check scalar multiplication commutes with restriction, using the universal property of the sheafification, and I’m having trouble.

  9. kwan says:

    I think I figured it out, but I am astounded by the amount of work involved to check such a simple property.

  10. Matt DeLand says:

    Hi,

    You should use the description of sheafification to show that whatever “structure” your presheaf had, the sheaf you make still has it. (I don’t know if this follows from the universal property). If \mathcal{F} is your presheaf, then the sheaf \mathcal{F}^s(U) is a sub “object” of \prod_{x \in U} \mathcal{F}_x consisting of those elements in the stalks that “fit together” correctly. Using this description you can check that sections can be added or multiplied by functions, or etc. This all true in more general “topologies” but you have to use a different construction of the sheafification.

    In general you do have to sheafify images, quotients, cokernels, etc.

  11. T says:

    In Hartshorne page 111 about sheaves of modules, property (e)states that $f^*(M^~)$ is isomorphic to $(M\otimes_A B)^~$. Can someone give me a detail proof of that? Thanks.

  12. Well, remember that f^*(\tilde{M})=f^{-1}(\tilde{M})\otimes_{f^{-1}(\mathscr{O}_Y)}\mathscr{O}_X. As we’re dealing with affine schemes, we lose nothing by passing to global sections and then doing \tilde{\hphantom{w}}, and the global sections of this will be M\otimes_B A.

  13. T says:

    Well, that’s right, but this just changes the question to why the global sections of $f^*(\widetilde{M})$ is isomorphic to $M\otimes_B A$?

  14. Well, M\otimes_B A will certainly be a submodule of the global sections, all you need to do is show that it’s all of them. Any global section of f^*(M) can be represented locally by elements of \Gamma(U,f^{-1}(\tilde{M})\otimes_{\mathscr{O}_Y}\mathscr{O}_X) (taken as a sheaf, which means just elements of f^{-1}(M)(U)\otimes_{\mathscr{O}_Y(U)}\mathscr{O}_X(U)) and then you need to show that collections of sections like this always patch together to give elements of M\otimes_B A, which just means keeping careful track of some localizations.

  15. Anonymous says:

    In the definition of sheaf module, one condition is that the restriction maps on F must be compatible with the module structures, what means this?

  16. I just wanted to say thank you for this excellent series of articles (and resulting discussion in some of the comment threads). I’ve definitely passed this on several other people who are trying to work through algebraic geometry. This post, in particular, was pretty excellent as sheaves have given me nothing but trouble in the past.

    • I’ve noticed that sheaves seem to be a common sticking point with people, which is terrible, considering that sheaves are, in a sense, such natural objects, just people seem to want to treat them very abstractly, without talking about the most concrete instances and intuitions behind them. Likely this is a bit of legacy from the super-abstract style of Grothendieck and EGA, but it does seem to be improving in recent years.

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