I have worked through most of Hartshorne's Algebraic Geometry text, and I'd like a recommendation for a book or set of lecture notes, which go beyond in particular chapter 3 on cohomology.
To be specific, I'd like a reference which includes using spectral sequences, but motivated by and in the context of algebraic geometry.
I have Rotman's book on homological algebra which goes way beyond Hartshorne in terms of cohomology, but there is an even greater emphasis on category theory (which I don't want), and it isn't an algebraic geometry text.
Do you know any complex algebraic geometry yet? Hodge theory beyond the basics (which admittedly rely on hard analysis) makes fairly liberal use of spectral sequences. You can see a little bit in Griffiths & Harris (right before the chapter on algebraic surfaces) and quite a bit more in Claire Voisin's Hodge Theory and Complex Algebraic Geometry. Both books essentially start from scratch, so you can use them to fill in any background you don't have; while I haven't spent much time with it, Voisin's book is much easier on the eyes (having been published in the last few decades) and does not have the legendary reputation for typos that is as much a part of G&H's reputation as its extraordinary content.