I have some background in vector bundles in the context of differential geometry and I have seen how vector fields form a module over smooth functions on a smooth manifold.
Recently I came across quasi coherent sheaf in the context of $O_X$-modules over a scheme $(X,O_X)$. My teacher introduced it as sheaf of modules that locally look like associated sheaf wrt to some module M (although most online resources seem to define it using some exact sequence of modules which I do not quite get).
Now coming to my questions :
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In many online resources I am seeing a recurrent comment "Quasi coherent sheaves are generalizations of vector bundles in context of Algebraic Geometry". What does this mean?
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How does that exact sequence definition come into play?
Best Answer
Vector bundles are nice to think about, but they have problems: it is not true that the kernel and cokernel of a map of vector bundles is necessarily a vector bundle. Consider for example the ideal sheaf of the origin inside $\Bbb A^1_k$: this is a vector bundle (it's the sheaf associated to the free module $xk[x]$), and it injects in to another vector bundle $\mathcal{O}_{\Bbb A^1_k}$ (the sheaf associated to the free module $k[x]$), but the cokernel is the structure sheaf of the origin (the sheaf associated to the non-free $k[x]$-module $k$).
We would like to be in a situation where we work in an abelian category: in particular, we want to be able to take kernels and cokernels and still have them be in our category. Quasi-coherent sheaves provide one such category where we can do this, and in some sense it's the smallest possible one (the precise sense is that it's the smallest cocomplete abelian category containing vector bundles aka locally free sheaves).
This links in nicely with the "definition using exact sequences" you mention in part 2. To be precise, this definition is that locally, every quasicoherent sheaf $\mathcal{F}$ can be represented as the cokernel of a morphism of free sheaves: for every point $x\in X$ there is an open neighborhood $U\subset X$ with an exact sequence $$\mathcal{O}_X|_U^{\oplus I} \to \mathcal{O}_X|_U^{\oplus J}\to \mathcal{F}|_U\to 0$$ for some sets $I,J$.
Most reasonable notions in algebraic geometry are in some sense "local" - this means that if we want to verify that some property holds, we ought to be able to check that it holds in a neighborhood of every point. This definition of a quasi-coherent sheaf provides us the correct way to do that, and this definition is equivalent to the one above (any sheaf in the smallest cocomplete abelian category containing locally free sheaves fulfills the above definition as locally a cokernel of free sheaves and vice-versa). For some more involved discussion, you may wish to consult Vakil's FOAG, section 13.1.9, beginning on page 374, as well as this MO question, and/or this MSE question.