Notes on quasi-coherent sheaves and coherent sheaves-1

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.

Sheaf Modules

Given a topological space X and a sheaf \mathfrak{O}_X 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 \mathfrak{O}_X-modules. In other words, an \mathfrak{O}_X-module is a sheaf \mathfrak{F} over X such that for any open subset U of X, \mathfrak{F}(U) is a module over \mathfrak{O}_X(U). We define morphisms between \mathfrak{O}_X-modules in an obvious way, thus we get a category Mod(\mathfrak{O}_X), and when there is no ambiguity, we write simply Mod(X).

It is not difficult to verify that Mod(X) 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 Mod(X) 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 x\in X, we consider the stalk \mathfrak{F}_x, which has a morphism \mathfrak{F}_x\rightarrow I_x where I_x is an injective module over \mathfrak{O}_{X,x}(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 j_x:\{x\}\rightarrow X for each point x\in X, thus if we view I_x also as the sheaf on the single point space \{x\}, then the push-forward map gives that j_{x,*}(I_x) is a sheaf on X. Now we take the products \mathfrak{I}=\prod_{x\in X}j_{x,*}(I_x), which is thus a sheaf over X. Next we show that \mathfrak{F} maps injectively into \mathfrak{I} and \mathfrak{I} is injective. Let’s first show that second point. Now that \mathfrak{I} is a product of these sheaves I_x, thus for any sheaf \mathfrak{G} over X, we have that Hom_{\mathfrak{O_X}}(\mathfrak{G},\mathfrak{I})=\prod_{x\in X}Hom_{\mathfrak{O}_X}(\mathfrak{G},j_{x,*}(I_x)). Yet one has that Hom_{\mathfrak{O}_X}(\mathfrak{G},j_{x,*}(I_x))=Hom_{\mathfrak{O}_{X,x}}(\mathfrak{G}_x,I_x) since I_x=\mathfrak{I}_x. Now we can in passing show the first point: take \mathfrak{G}=\mathfrak{F}, since for each x\in X, there is an injective map \mathfrak{F}_x\rightarrow\mathfrak{I}_x by construction, thus there is a morphism \mathfrak{F}\rightarrow\mathfrak{I}, which is thus also injective. Note that Hom_{\mathfrak{O_X}}(\mathfrak{G},\mathfrak{I})=\prod_{x\in X}Hom_{\mathfrak{O}_X}(\mathfrak{G},j_{x,*}(I_x)), the functor of taking stalks for all x\in X, \mathfrak{F}\mapsto\mathfrak{F}_x is exact(we emphasize on the word ‘all’), and at each stalk, the functor \mathfrak{G}_x\mapsto Hom_{\mathfrak{O}_{X,x}}(\mathfrak{G}_x,I_x) is also exact since I_x is injective, thus the functor Hom_{\mathfrak{O}_X}(\cdot, \mathfrak{I}) is exact, thus \mathfrak{I} is injective.

So we have that any Mod(X) has enough injectives, yet the problem of enough projectives is more subtle: not every Mod(X) 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, Hom_{\mathfrak{O}_X}(\mathfrak{G},j_{x,*}(I_x))=Hom_{\mathfrak{O}_{X,x}}(\mathfrak{G}_x,I_x). Note that j_x^{-1} is left adjoint to the functor j_{x,*}, 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, \Gamma(X,\cdot):Mod(X)\rightarrow Mod(\mathfrak{O}_X(X)), from Mod(X) to the category of \mathfrak{O}_X(X)-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 X=\mathbb{A}^1_k the affine line over a field k, and P,Q\in X two distinct closed points and U=X-\{P,Q\}, the open complement of them in X. Now we take the constant sheaf \mathfrak{O}_X=\mathbb{Z}_X over X. Now we set \mathbb{Z}_U=i_!(\mathbb{Z}_X|_U) where i:U\rightarrow X is the inclusion map and \mathbb{Z}_Y=j_*(\mathbb{Z}_X|_Y) where j:Y=\{P,Q\}\rightarrow X is also the inclusion. We have an exact sequence 0\rightarrow \mathbb{Z}_U\rightarrow\mathbb{Z}_X\rightarrow\mathbb{Z}_Y\rightarrow0. Yet consider what is \mathbb{Z}_Y now: by definition, \mathbb{Z}_Y is a sheaf on X such that for each open subset V\subset X, there is \mathbb{Z}_Y(V)=\mathbb{Z}_X|_Y(j^{-1}(V)). If P\in V,Q\not\in V, we have that \mathbb{Z}_Y(V)=\mathbb{Z}, similar for the case P\not\in V,Q\in V. If P,Q\in V, then \mathbb{Z}_Y(V)=\mathbb{Z}^2. Taking global section, we see that \mathbb{Z}_Y(X)=\mathbb{Z}^2, \mathbb{Z}_X(X)=\mathbb{Z}, \mathbb{Z}_U(X)=0, thus the global section sequence 0\rightarrow 0\rightarrow \mathbb{Z}\rightarrow\mathbb{Z}^2\rightarrow0 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 H^i(X,\mathfrak{F})=0 for i>0, where we write H^i(X,\mathfrak{F}) for the i-th right derived functor of the global section functor.

Quasi-coherent sheaves

Quasi-coherent sheaves partially answers this question. Now let’s give the definition:

Suppose \mathfrak{F} is an \mathfrak{O}_X-module, we say that \mathfrak{F} is generated by its global sections at x\in X if the canonical morphism \mathfrak{F}(X)\bigotimes_{\mathfrak{O}_X(X)}\mathfrak{O}_{X,x}\rightarrow \mathfrak{F}_x is surjective. We say that \mathfrak{F} is generated by its global sections if it is so at each point x\in X.

It is not hard to see that \mathfrak{F} is generated by its global sections if and only if there is an index set I and a surjective morphism \mathfrak{O}_X^{\bigoplus |I|}=\mathfrak{O}_X^{(I)}\rightarrow\mathfrak{F}\rightarrow0.

We say that \mathfrak{F} is quasi-coherent if for each x\in X, there is an open neighborhood x\in U\subset X such that there is an exact sequence \mathfrak{O}_X^{(J)}|_U\rightarrow\mathfrak{O}_X^{(I)}|_U\rightarrow \mathfrak{F}|_U\rightarrow0.

So this means that a quasi-coherent sheaf \mathfrak{F} locally admits free resolutions(of exact sequence of length 2).

Quasi-coherent sheaves behave quite nicely over an affine scheme. This can be seen from the result below.

Let’s fix a commutative ring R and set X=Spec(R). Now if M is a module over R, then we can give a natural \mathfrak{O}_X-module M^{\sim} constructed as follows: for each affine open subset D(r) of X with r\in R, we set M^{\sim}(D(r))=M_r=M\bigotimes_RR_r. For general open subsets, we take the direct limit. It is easy to see that M^{\sim}(X)=M, and (M^{\sim})_{\mathfrak{p}}=M_{\mathfrak{p}} for any \mathfrak{p}\in X. Clearly, we have that (\bigoplus_iM_i)^{\sim}\simeq \bigoplus_iM^{\sim}_i where each M_i is an R-module. One surprising result is that

A sequence of R-modules L\rightarrow M\rightarrow N is exact if and only if L^{\sim}\rightarrow M^{\sim}\rightarrow N^{\sim} is so. This is really an important proposition, and let’s give a proof of it.

Suppose that L\rightarrow M\rightarrow N is exact, then we know that R_{\mathfrak{p}} is flat, thus the sequence L\bigotimes_RR_{\mathfrak{p}}\rightarrow M\bigotimes_RR_{\mathfrak{p}}\rightarrow N\bigotimes_RR_{\mathfrak{p}} is again exact, yet the latter is just (L^{\sim})_{\mathfrak{p}}\rightarrow(M^{\sim})_{\mathfrak{p}}\rightarrow(N^{\sim})_{\mathfrak{p}}, which shows that L^{\sim}\rightarrow M^{\sim}\rightarrow N^{\sim} is indeed exact. Conversely, suppose that L^{\sim}\rightarrow M^{\sim}\rightarrow N^{\sim} is exact, then at stalk level, we have that (L^{\sim})_{\mathfrak{p}}\rightarrow(M^{\sim})_{\mathfrak{p}}\rightarrow(N^{\sim})_{\mathfrak{p}} is also exact, so is L\bigotimes_RR_{\mathfrak{p}}\rightarrow M\bigotimes_RR_{\mathfrak{p}}\rightarrow N\bigotimes_RR_{\mathfrak{p}}. Now consider the original sequence L\xrightarrow{f} M\xrightarrow{g} N where g\circ f=0 since the sheaf sequence is exact. We have that (Ker(g)/Im(f))_{\mathfrak{p}}=0 for all \mathfrak{p}\in Spec(R). This shows that Ker(g)/Im(f)=0, thus L\rightarrow M\rightarrow N 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 M is an R-module, then it admits a free presentation, R^{(J)}\rightarrow R^{(I)}\rightarrow M\rightarrow 0 is an exact sequence. And therefore (R^{(J)})_r\rightarrow (R^{(I)})_r\rightarrow M_r\rightarrow 0 is again exact since R_r is flat module over R for each r\in R. So, this means that for each \mathfrak{p}\in X, there is an open neighborhood \mathfrak{p}\in D(r)\subset X such that (R^{(J)})^{\sim}|_{D(r)}\rightarrow (R^{(I)})^{\sim}|_{D(r)}\rightarrow M|_{D(r)}\rightarrow 0 is exact since M|_{D(r)}=M_r, similar for the other two.

In fact, we can show that quasi-coherent sheaves locally look like this, that is:

Proposition: let X be a scheme(not necessarily affine), and \mathfrak{F} be an \mathfrak{O}_X-module. Then \mathfrak{F} is quasi-coherent if and only for each affine open subset U\subset X, there is \mathfrak{F}(U)^{\sim}\simeq \mathfrak{F}|_U.

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 X be a scheme which is Noetherian or separated and quasi-compact, \mathfrak{F} be a quasi-coherent sheaf over X, then for each f\in \mathfrak{O}_X(X), the canonical morphism

\mathfrak{F}(X)_f=\mathfrak{F}(X)\bigotimes_{\mathfrak{O}_X(X)}\mathfrak{O}_X(X)_f\rightarrow \mathfrak{F}(X_f)

is an isomorphism, where X_f=\{x\in X|f_x\in \mathfrak{O}_{X,x}^*\}.

Note that since \mathfrak{F} is quasi-coherent, for each x\in X, there is an affine open subset x\in U\subset X such that there is a free representation \mathfrak{O}_X^{(J)}|_U\rightarrow \mathfrak{O}_X^{(I)}|_U\xrightarrow{f} \mathfrak{F}|_U\rightarrow 0. Taking global sections, we set M=f(U), then we have that \mathfrak{O}_X^{(J)}|_U\rightarrow \mathfrak{O}_X^{(I)}|_U\xrightarrow{f} M^{\sim}\rightarrow 0 is exact, which implies that M^{\sim}\simeq \mathfrak{F}|_U, thus we get that M\simeq \mathfrak{F}(U). In other words, for each x\in X, there is an affine open subset x\in U\subset X such that \mathfrak{F}|_U\simeq \mathfrak{F}(U)^{\sim}. By assumption, X is quasi-compact, thus we can find a finitely many such U_i covering X. Moreover, we set V_i=U_i\bigcap X_f. Let’s spell out what V_i is. Recall that X_f consists of points x\in X such that f_x is invertible in \mathfrak{O}_{X,x}, in other words, if x is viewed as a prime ideal in R_i where U_i=Spec(R_i), then f|_{U_i}\not\in x. So we see that V_i=D(f|_{U_i}). Now for each i, we have maps \mathfrak{F}(U_i)_f=\mathfrak{F}(U)\bigotimes_{\mathfrak{O}_X(U_i)}\mathfrak{O}_X(U_i)_f\rightarrow \mathfrak{F}(V_i)=\mathfrak{F}(D(f|_{U_i})), this is an isomorphism since \mathfrak{F}|_{U_i}\simeq \mathfrak{F}(U_i)^{\sim}. Now using the routine commutative diagram,

pic-my-blog-notes on quasi-coherent sheaves and coherent sheaves

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 U_i\bigcap U_j is again affine, which indeed is the case since X 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 U is an affine open subset of X, then for each f\in \mathfrak{O}_X(U), we have that \mathfrak{F}(U)_f\simeq \mathfrak{F}(D(f)) by the above lemma, thus \mathfrak{F}(U)^{\sim}\simeq \mathfrak{F}|_U. 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 X be an affine scheme, and 0\rightarrow \mathfrak{F}\rightarrow\mathfrak{G}\rightarrow\mathfrak{H}\rightarrow 0 be an exact sequence of sheaves over X. Suppose that \mathfrak{F} is quasi-coherent, then the sequence

0\rightarrow \mathfrak{F}(X)\rightarrow\mathfrak{G}(X)\rightarrow\mathfrak{H}(X)\rightarrow 0

is exact.

We will show this result in the next post, which will use Cech cohomology.

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