>>11697029Well, I'd say finding out properties of the ring to be studying the ring in itself. For example, Noetherian, artinian, coherent, local, flat, UFD, PID, ED, DVR, its ring extensions and prime properties, integral, finiteness, etc are all things that I'd want to find out about a ring, in some setting or another.
Right now, I'm reading about quasicoherent modules over a scheme, and I have a feeling that a similar sort of conclusion is going to come out of it, that is, studying qc modules over a scheme is a way to studying the scheme itself . But already, over an affine scheme, studying qc modules is like studying modules over a ring, so I have trouble seeing what exactly am I going to learn from looking at them over a scheme if I can't even see what's the benefit over an affine one.
