Sheaves are great generalizations of function spaces on a topological space. Yet not all sheaves have nice properties, especially when it comes to computing their topological invariants, the homological groups and cohomological groups. So we should try to find some nicely-behaved sheaves. It was perhaps due to J-P.Serre in his article ‘Faiseaux Algèbriques Cohérents’ that we found at last some class of sheaves which indeed have nice properties, it is the quasi-coherent sheaves and coherent sheaves that we are going to talk about in this post. In this post, we will give the definitions and basic properties of these sheaves. The references for this post are Hartshorne’s Algebraic Geometry and Qing Liu’s Algebraic Geometry and Arithmetic Curves as well as some posts from the Internet.
Given a topological space and a sheaf of (locally) ringed space, just as in the case of given a ring, where we would like to consider all the modules over this ring, here we also would like to consider the category of -modules. In other words, an -module is a sheaf over such that for any open subset of , is a module over . We define morphisms between -modules in an obvious way, thus we get a category , and when there is no ambiguity, we write simply .
It is not difficult to verify that is an abelian category. So there is a natural question arising: does this category admit enough projective or injective objects? Very interestingly, Grothendieck showed in his famous Tohoku paper ‘Sur quelques points d’algèbre homologique’ that indeed has enough injectives. Nowadays the proof of this result becomes fair standard, and let’s copy such one from Hartshorne’s book:
For each , we consider the stalk , which has a morphism where is an injective module over (interestingly, the existence of enough injectives in a category of modules over a ring is more difficult to prove than that of enough projectives). Now that there is a canonical injection for each point , thus if we view also as the sheaf on the single point space , then the push-forward map gives that is a sheaf on . Now we take the products , which is thus a sheaf over . Next we show that maps injectively into and is injective. Let’s first show that second point. Now that is a product of these sheaves , thus for any sheaf over , we have that . Yet one has that since . Now we can in passing show the first point: take , since for each , there is an injective map by construction, thus there is a morphism , which is thus also injective. Note that , the functor of taking stalks for all , is exact(we emphasize on the word ‘all’), and at each stalk, the functor is also exact since is injective, thus the functor is exact, thus is injective.
So we have that any has enough injectives, yet the problem of enough projectives is more subtle: not every has enough projectives. Perhaps it is better to give some counterexamples(these are taken from the answers in this post). Moreover, it is easy to see that above argument does not apply to the projective case, the essential obstacle is that, . Note that is left adjoint to the functor , which gives the above identity. So the existence of enough injectives comes essentially from the point that the stalk functor at one point is left adjoint to the extension by zero functor. And that is why we can use this argument to show the existence of enough projectives.
Now there is a natural functor, the global section functor, , from to the category of -modules.
Unfortunately, this functor is not an exact functor(in some sense, this is also formate for us, since otherwise there would not be such a rich theory of sheaf cohomology). Suppose that the affine line over a field , and two distinct closed points and , the open complement of them in . Now we take the constant sheaf over . Now we set where is the inclusion map and where is also the inclusion. We have an exact sequence . Yet consider what is now: by definition, is a sheaf on such that for each open subset , there is . If , we have that , similar for the case . If , then . Taking global section, we see that , , , thus the global section sequence is no longer exact.
So now we may wonder what conditions we should pose on the underlying topological space or on the sheaves to ensure that for , where we write for the -th right derived functor of the global section functor.
Quasi-coherent sheaves partially answers this question. Now let’s give the definition:
Suppose is an -module, we say that is generated by its global sections at if the canonical morphism is surjective. We say that is generated by its global sections if it is so at each point .
It is not hard to see that is generated by its global sections if and only if there is an index set and a surjective morphism .
We say that is quasi-coherent if for each , there is an open neighborhood such that there is an exact sequence .
So this means that a quasi-coherent sheaf locally admits free resolutions(of exact sequence of length ).
Quasi-coherent sheaves behave quite nicely over an affine scheme. This can be seen from the result below.
Let’s fix a commutative ring and set . Now if is a module over , then we can give a natural -module constructed as follows: for each affine open subset of with , we set . For general open subsets, we take the direct limit. It is easy to see that , and for any . Clearly, we have that where each is an -module. One surprising result is that
A sequence of -modules is exact if and only if is so. This is really an important proposition, and let’s give a proof of it.
Suppose that is exact, then we know that is flat, thus the sequence is again exact, yet the latter is just , which shows that is indeed exact. Conversely, suppose that is exact, then at stalk level, we have that is also exact, so is . Now consider the original sequence where since the sheaf sequence is exact. We have that for all . This shows that , thus is indeed exact.
So in the case of affine schemes, for this type of sheaves, we see that the global section functor is exact.
Then what is the relation of this kind of sheaves with quasi-coherent shaves that we have just introduced? Well, note that if is an -module, then it admits a free presentation, is an exact sequence. And therefore is again exact since is flat module over for each . So, this means that for each , there is an open neighborhood such that is exact since , similar for the other two.
In fact, we can show that quasi-coherent sheaves locally look like this, that is:
Proposition: let be a scheme(not necessarily affine), and be an -module. Then is quasi-coherent if and only for each affine open subset , there is .
This is really exciting news: quasi-coherent sheaves locally look all like sheaf modules constructed as above, which, as we have seen, have very nice properties.
Let’s also try to give a proof of this result. For this, we need a lemma:
Lemma: let be a scheme which is Noetherian or separated and quasi-compact, be a quasi-coherent sheaf over , then for each , the canonical morphism
is an isomorphism, where .
Note that since is quasi-coherent, for each , there is an affine open subset such that there is a free representation . Taking global sections, we set , then we have that is exact, which implies that , thus we get that . In other words, for each , there is an affine open subset such that . By assumption, is quasi-compact, thus we can find a finitely many such covering . Moreover, we set . Let’s spell out what is. Recall that consists of points such that is invertible in , in other words, if is viewed as a prime ideal in where , then . So we see that . Now for each , we have maps , this is an isomorphism since . Now using the routine commutative diagram,
The horizontal sequences are exact according to the definition of scheaves, and the first, the third and the fourth vertical arrows are isomorphisms(we should ensure that is again affine, which indeed is the case since is Noetherian or separated, so we used the hypothesis in an essential way), and thus we get that the second vertical arrow is also an isomorphism, which finishes the proof of this lemma.
Return to our original proposition. Suppose that is an affine open subset of , then for each , we have that by the above lemma, thus . The converse is clear.
At last, let’s give a proposition which reveals the exactness of the global section functor in a larger extent:
Proposition: let be an affine scheme, and be an exact sequence of sheaves over . Suppose that is quasi-coherent, then the sequence
We will show this result in the next post, which will use Cech cohomology.