References on derived categories for schemes

Question

I want to study derived categories to show the theorems in this page (the non-noetherian case of the theorems in Hartshorne III.12), and the theorems about spectral sequences (I've heard that the Grothendieck spectral sequence theorem is old theory, and it is better to use derived category than to use spectral sequences).

But I'm not familiar with the theories. So please suggest me some references.
Is the stack project good for those who study derived categories for the first time?

Any help will be much appreciated!

Answer 1

I think the best notes are those from Murfet. These notes are well organised and he combined many results from the standard literature to produce them.

Answer 2

A few weeks ago I tried learning something about derived categories as well.

Here are some references that I found particularly useful: I really enjoyed reading about derived categories in Methods of homological algebra by Gelfand-Manin. They have a very insightful introduction where they try to motivate why derived categories are useful and what's the basic idea behind them.

Also they construct it step-by-step. First the "naive" approach by just localising the category of complexes at all quasi-isomorphisms and then the "right" approach by first "modding out" homotopy.

This was very helpful to me, because they in particular explain whats "wrong" with the naive approach and why it is done the second way.

Derived categories for the working mathematician by Thomas is a nice introductory as well.

I was looking for other results, but there is Fourier-Mukai transforms in algebraic geometry that also nicely explains derived categories and triangulated structures on them etc., but later on tries to actually use it to study (bounded) derived categories of (coherent) sheaves of modules on schemes.

I'm not sure if the things you want to learn are discussed there though, as I didn't look for them. Could be worthwile to look into though.