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!
Why quasi coherent sheaves
algebraic-geometryquasicoherent-sheaves
Best Answer
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!