Why quasi coherent sheaves?

Question

So i was wondering why one considers quasi-coherent sheaves in algebraic geometry. I have read a lot that they are closely linked to the geometric properties of the underlying space. This means that you can infer something about the geometry of a ringed space by looking at at the sheaf defined on it. I would love to have some examples in mind where this occurs and maybe there are other motivations on why to study them. Thanks!

Answer 1

Actually, quasicoherent sheaves are not necessarily the most obvious type of sheaf to study. If one studies the constant sheaf $\underline{A}$ of some Abelian group $A$ on a topological space $X$, then the sheaf cohomology $H^i(X,\underline{A})$ is isomorphic to the singular cohomology $H^i(X,A)$ (at least when the space is "nice"). With this in mind, a lot of algebraic topology can be recast in terms of constant sheaves. Unfortunately, for spaces with "bad" topology, this ceases to work. Indeed, for an irreducible topological spaces, every constant sheaf $\mathcal{F}$ is flasque and $H^i(X,\mathcal{F}) = 0$ for $i\ge 1$. In particular, constant sheaves on irreducible schemes or varieties (with the Zariski topology) are not interesting objects to study.

You can really compare and contrast the difference when you study complex manifolds. In this case, there are many manifolds that are diffeomorphic but not biholomorphic (e.g. complex tori of a given dimension). Consequently, singular cohomology (or any topological cohomology theory) cannot distinguish them, but sheaf cohomology with coefficients in a (quasi)coherent sheaf might. Indeed, there are examples of homeomorphic manifolds with different Hodge numbers: $$ h^{p,q}(X) :=\dim H^q(X,\Omega^p) $$ where $\Omega^p$ is the sheaf of holomorphic $p$-forms on $X$. So, these sheaf cohomology groups with coefficients in a coherent sheaf are sensitive to the underlying complex structure of the manifold. The exponential exact sequence $$0\to \underline{\Bbb{Z}}\to \mathcal{O}_X\to \mathcal{O}_X^*\to 0$$ relates the topological invariants $H^i(X,\mathbb{Z})$ to some holomorphic invariants $H^i(X,\mathcal{O}_X)$ and $H^i(X,\mathcal{O}_X^*)$ in the complex case.

Put a different way, the sheaf cohomology with coefficients in a (quasi)coherent sheaf is sensitive not only to the space $X$ itself, but also to the sheaf of rings $\mathcal{O}_X$ with which it is equipped. Since in the Zariski topology we would rather not think about the topological space itself (indeed: any two curves over a field $k$ are homeomorphic!) the more sensible thing to study is the invariant which sees finer structure.

P.S. It's also worth mentioning that the study of sheaves originated in the computation of existence of certain global meromorphic functions on Riemann surfaces. The sheaves of interest were the archetypal "coherent sheaves" : sheaves of meromorphic functions with prescribed poles and zeros!

Answer 2

Since this question is rather philosophical in nature, there is no single correct answer.

The statement that the category of quasi-coherent sheaves encodes geometric information about the scheme is correct, but rather high-level imho. Let me try to give a rather technical perspective.

As was pointed out or hinted at in the comments, the most notable technical feature is that the quasi-coherent sheaves are those which locally, on affine opens $\mathop{Spec}(A)$, correspond to the $A$-modules in a specific manner (and that manner is compatible with the way the scheme is glued from affine schemes). That is, morally, if a scheme is glued together from spectra of a bunch of rings, then any quasi-coherent sheaf is compatibly "glued together" (in an appropriate sense) from modules over those rings. In practice, this provides very much control over the individual quasi-coherent sheaves, while still being general enough to allow for pretty much every sheaf you would wish for when working with schemes. Rephrasing this a bit more conceptually, the condition of quasi-coherence is just ensuring the compatibility with the locally affine nature of a scheme.

For example: Usually the first important instance of the concept of quasi-coherence one encounters is that the quasi-coherent ideal sheaves in the structure sheaf bijectively correspond to the closed sub-schemes of the sheaf.

Answer 3

(Disclaimer: not an algebraic geometer.)

I would say quasicoherent sheaves are the right thing to look at for schemes (and their various generalisations), not necessarily all ringed spaces. The reason is not anything inherent in the usual definition of "quasicoherent" but instead in these facts:

1. Given a commutative ring $A$, taking global sections $\Gamma : \textbf{Qcoh} (\operatorname{Spec} A) \to \textbf{Mod} (A)$ is an equivalence of categories.

2. Moreover, the above equivalence is pseudofunctorial in $A$, in the sense that given a ring homomorphism $\phi : A \to B$, we have $$\require{AMScd} \begin{CD} \textbf{Qcoh} (\operatorname{Spec} A) @>{\Gamma}>> \textbf{Mod} (A) \\ @V{(\operatorname{Spec} \phi)^*}VV @VV{B \otimes_A {-}}V \\ \textbf{Qcoh} (\operatorname{Spec} B) @>>{\Gamma}> \textbf{Mod} (B) \end{CD}$$ commutative up to a coherent family of isomorphisms.

3. $\textbf{Qcoh} (-)$ maps colimits of "nice" diagrams of schemes to limits of categories.

The first two facts tell us that quasicoherent sheaves are the right thing to look at for affine schemes – at least to the extent that we believe that modules are the right thing to look at for rings. The third fact tells us that quasicoherent sheaves are local and can be glued together the same way schemes can be glued together. Since schemes are obtained by gluing affine schemes and quasicoherent sheaves are the right thing for affine schemes, we conclude quasicoherent sheaves are also the right thing for schemes in general.

Of course, being sheaves, quasicoherent sheaves automatically have geometric meaning – but this is much harder to extract for schemes than for, say, complex manifolds. I believe part of the reason is that even affine schemes have non-trivial geometry, but this is not well reflected by the Zariski topology on the prime spectrum, nor by the cohomology of quasicoherent sheaves on it. (For example, an affine scheme is always quasicompact, and the sheaf cohomology of any quasicoherent sheaf on an affine scheme is trivial!) By contrast, manifolds can be obtained by gluing together contractible pieces, which guarantees that the topology encodes meaningful geometry.

One way of making precise what would otherwise be just an analogy between algebraic geometry and complex analytic geometry is Serre's GAGA theorems: for smooth projective varieties $X$ over $\mathbb{C}$, coherent sheaves on $X$ incarnated as an algebraic variety have the same morphisms and cohomology as coherent sheaves on $X$ incarnated as a complex manifold.

Answer 4

To give something in the "infer something about the geometry of the ringed space" direction, let me just say that looking at all quasicoherent sheaves basically tells us everything we could possibly know.

Just like a commutative ring $R$ is uniquely determined by its category of modules $R-\text{Mod}$, so is a scheme $X$ uniquely determined by its category of quasicoherent sheaves $\text{QCoh}(X)$, and under possibly some mild hypotheses, its category of coherent sheaves $\text{Coh}(X)$, and even by $D^b_\text{Coh}(X)$, the category of bounded complexes with coherent cohomology up to quasiisomorphism, if it is really nice, for example, smooth and projective with an ample canonical or anticanonical bundle.