Some words for a new series

One cannot hope to go deep in algebraic geometry without knowledge about commutative algebra and homological algebra. This is a difficult point, but it also makes algebraic geometry more attractive. Recently, I am reading about the theory of quasi-coherent sheaves, and in the case of affine schemes, all things begin with modules. For examples, the category of quasi-coherent sheaf on an affine scheme is equivalent to the category of modules over a ring, and a locally free sheaf on an affine scheme is equivalent to a finitely generated projective module over a ring. A sheaf has its natural geometric intuition, but these kinds of sheaves also reflect a deep connection to algebraic objects.

My first objective is to understand about representable functors, but for the example, to understand about Grassmanian functor, we need much more consideration on quasi-coherent sheaf. Understanding things in details make the subject more interesting, and I will try to type down everything, even about projective/injective modules, which I have never studied seriously before.