Given an -module , we may build a sheaf of -modules associated to on the affine scheme (with structure sheaf ). Often, this sheaf of modules is constructed by first defining it on a basis, and then using the fact that a sheaf on a basis uniquely defines a sheaf on the whole space.
In this post, I want to present a more “hands on” way of constructing that sheaf. This is part of my quest to “reconcile” with elementary constructions (i.e. those using elements instead of universal properties).
As mentioned in the introduction, we will build a sheaf , said to be the sheaf associated to on the affine scheme . This construction plays an important role in algebraic geometry because it’s a “local model” for quasicoherent sheaves.
Before we begin, we need the following technical definition. Let be some open set in , and let be an element of . We say that is a system of compatible germs if, for every , there exists an element having and , together with an element such that, for every , we have . Here is the further localization of via the canonical .
Now, we use this notion to define our object of interest. Let be an open set of . To this open set we associate the data , which is defined to be the subset of all systems of compatible germs in .
Notice that is an abelian group by adding two systems component-wise, using the abelian group structure on each . Moreover, we can give the structure of an -module in the following way. For a scalar and a system , we define using the fact that , and that each is an -module. We have a couple of things to check before we can claim this actually works:
First of all, we need to verify the resulting system is of compatible germs. This is not too hard. Let , and and be given for as in the definition of a system of compatible germs. There’s a canonical -module structure on , which allows us to talk about . Here is the image of via the restriction map . It doesn’t take much work to see that localizing further using preserves the action on the module, in the sense that for all (hint: the localization map corresponds to the obvious map ; write the appropriate commutative diagram). This shows everything works, and we have a system of compatible germs.
Then, we need to check that this respects the axioms for a module action. This is so because “taking the germ”, i.e. sending to , is a ring homomorphism for each .
If is an inclusion of open sets, we may define the obvious restriction , which simply sends a system to the smaller system . The only subtlety here is that it is not obvious the smaller system is of compatible germs. However, that’s not too bad to show (hint: the distinguished opens form a base for the topology on , and implies the existence of a canonical localization map ). Notice that this restriction map respects the module structure, in the sense that acting by followed by restriction gives the same result as restriction followed by acting by .
From the definition of the restriction maps as litteral restrictions, it’s obvious that we have a presheaf on . We want to show it is a sheaf. It’s clear that the identity axiom holds: if some section restricts locally everywhere to the zero system, that means each component of the system is zero, so the system as a whole is a bunch of zeroes.
We check the gluability axiom. Let be an open set, and let be a family of open sets that cover . Suppose we also have a family of systems with each . Suppose further that these systems agree on overlaps, in the sense that for each , we have To glue all of these together, the obvious choice is to set for each , where is some choice function such that each lies in the chosen open set . Because the systems agree on overlaps, this definition is independent of the actual choice function. It is also quite clear that this defines a system of compatible germs over , and that moreover its restriction to each gives back the corresponding system .
So we have a sheaf! In fact, a sheaf of -modules, since as we’ve remarked above the restriction maps respect the sheaf of ring’s action. Notice that we haven’t really used the fact each section of this sheaf is a system of compatible germs. We have only demonstrated that this property carries over our constructions. But this property will be useful in characterizing this sheaf via its stalks, which is the object of the next section.
The stalks of
For each , the stalk is an -module in the following way. Recall that each element in is an equivalence class of pairs, with an open neighborhood around and , and where two pairs and are equivalent if and only if there exists an open neighborhood of with such that .
Given two pairs and in , we may first find a small enough open around such that , and define, using the -module structure on : This is well-defined, as one can check quickly. Similarily, given and , we again find a small enough open and define Everything works well to give to the structure of a -module, as claimed.
We already know that the stalk at of the structure sheaf is isomorphic to the localization . In fact, a similar thing can be said of the sheaf : it is isomorphic to , as we now show. Let be an element of . The element is a system of compatible germs over , which we may write as . Let be the element . This makes a well-defined function This function is in fact an -linear map (or, which is the same thing, an -linear map).
The map is injective: suppose . Because is a system of compatible germs, we can find some and some such that , and for each , we have . We may write as a fraction where and is some integer. Then, because the image of in is zero, there must exist some such that in . It is clear that is an open neighborhood of which is contained in . Moreover, for any , the localization factorizes through . Now, the image of in is which is equal to zero by equation (1) above. Hence for every . This shows that , so is injective.
The map is also surjective. Pick some element . Now we consider the fraction as an element of . It’s clear that the system is of compatible germs, and that . Hence is surjective, as we wanted.
Therefore is an isomorphism!
Remark: the isomorphism is the canonical one identifying with as a colimit, since we’ve implicitely used morphisms in the construction of . In other words, this is what we get when we apply the universal property of as a colimit; here we defined it in an elementary fashion.
Behavior over distinguished open sets
Just as we were expecting each stalk to “be” the localization at a point, just by the way we defined things, we now expect to be the smaller localization . Moreover, for any inclusion of distinguished open sets , we expect that the isomorphisms make the following diagram commute:
However, this is better seen via an alternative construction.
Another construction
Another construction is given as an exercise in FoAG (Ravi Vakil). For any distinguished open set , let be the localization of at the multiplicative submonoid of the functions that do not vanish outside of , i.e. those such that . This definition obviously depends only on the set and not on the function itself; this point is the main technical advantage in defining this way instead of directly stating that it is . We still have an isomorphism between and , however:
Lemma. For any section , there is a canonical isomorphism of -modules Here, the isomorphism being “canonical” means that is the localization of at , in the sense that both and verify the same universal property.
Proof. Recall that for any multiplicative submonoid , we have a canonical isomorphism of -modules between and , under which corresponds to . (As usual, “canonical” means “coming from an universal property”, the “universal property” in this case being the tensor product’s).
Let now be the multiplicative submonoid of the functions having . Because , the section is in particular an invertible element of . Hence, the universal property of localization gives a canonical ring homomorphism which sends to (the inverse of in may be written as since ). On the other hand, any section verifies , so the element is invertible in . Again by the universal property, there exists a canonical ring homomorphism in the other direction , and it must be the inverse of by usual abstract nonsense. Therefore is an isomorphism of rings (in fact, an isomorphism of -algebras).
We give the structure of an -module using in the obvious way. We obtain a chain of isomorphisms of -modules under which corresponds to , and in the other direction, the element with for some integer corresponds to .
Given an inclusion of distinguished open sets , we want to define a restriction morphism from to . Since we have , by definition the section is invertible in (with inverse ). Of course, strictly speaking, we should say more formally that the map defined by is an automorphism of the -module , instead of saying that “ is invertible in ”. In reality, is invertible in , making into an -algebra. Even better, the universal property of localization for modules gives us an -linear map which we call the restriction morphism. Since it’s obtained via a universal property, everything is functorial. Moreover, recall that in a canonical way, i.e. they verify the same universal property, and so each -module is an -module; also, the ring homomorphism alluded to is precisely the restriction map for the sheaf of rings , and the fact it makes into an -module means precisely that for any and any , we have This makes into a presheaf of -modules on the distinguished base.
The restriction is the localization , in the sense that the following diagram commutes:
Lemma. The presheaf is a sheaf on the distinguished base.
Proof. We start with the base identity axiom. Suppose is some covering of a distinguished open ; by quasicompacity, we may take to be a finite set . Suppose is some section of over such that for each . To verify the identity axiom, it suffices to show in , given that for each .
As a section over , restricts to an element in , and this element is zero by hypothesis. Therefore, using the fact is a finite set, we may choose a large enough integer such that, for each , it holds in that .
Under the identification of with , each corresponds to where is the image of in . By laziness, we immediately stop writing “”. Because the sets cover , the elements generate the whole ring (hint: ). In particular, we may write as a linear combination: where each is an element of . Because , we find that This shows in , so the identity axiom is verified.
Now, we show the base gluability axiom. Fix an arbitrary covering of , which may be infinite. Suppose we have a collection of sections with such that these sections all “agree on overlaps”, that is, for every .
We break the proof in two parts: when is finite, and when it is not. First, suppose is the finite set . To simplify notation, set (notice that ). Each section restricts to the element in . The overlap condition then says that for each , there is some integer such that holds in . Using the finiteness of , we may in fact pick a single integer such that We further simplify the situation by setting and . We have and moreover in . The previously displayed equation becomes Since covers , we may express as a linear combination where each . We finally define the section that will be the “gluing” of the sections we started with. It’s the section over defined as Notice that for every , Therefore, becomes when restricted to , which shows that is the gluing of our original sections .
We still have to show the gluability axiom holds when is infinite. In that case, use quasicompactness of to choose a finite subset such that covers , and do the same construction as before to obtain a section over . Now pick some new index which is not in , and redo the same construction with to obtain yet again a section over . But and both restrict to the same sections over the covering afforded by . By the identity axiom, we must have , so that in particular restricted to is . This means we can take , obtained by gluing only a finite number of chosen sections, to play the role of the gluing of the whole collection of sections. This shows the gluing axiom holds.
As always, when we have a sheaf on a base, there’s a unique canonical way of extending it to a sheaf on the whole space. In fact, this extended sheaf, if defined using the concept of system of compatible germs, is exactly how we defined in the first section of this article.