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Have a better understanding of where spectral sequences arise in different examples, e.g. base change of Picard groups.
Is there always a filtration behind spectral sequences?
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Some tricky issues I don't understand, but I also don't think they are that important:
- Naturality of the boundary map for derived functors seems annoying to prove. See Wikipedia: "Derived functors and the long exact sequences are "natural" in several technical senses." The difficulty seems to be that a map of short exact sequences of chain complexes might not induce a commutative diagram on cohomology if we only assume it to commute up to chain homotopy; see MSE: Naturality of connecting morphisms found by googling "commutes up to chain homotopy". Eisenbud sort of sketchily leaves it to the reader; Weibel and Sharifi notes do this with a bit of work; not sure if this is worth looking at.
- Consider Cech-to-derived-functor-sheaf-cohomology map, Hartshorne Lemma III.4.4 (functoriality is believable, because we can probably just be lazy with commuting squares up to homotopy... but I don't easily see the naturality of the boundary map in long exact sequences). Compare annoying Hartshorne Theorem III.4.5 ( on affines vanish (or just the fact that global sections functor is exact on affines when the leftmost term is quasicoherent) + Leray acyclicity theorem) and Exercise III.4.4 with hands-on Qing Liu Chapter 5 Proposition 2.15. It seems like the latter lets us work with Cech cohomology for all practical purposes (cf. Vakil treatment) without thinking about derived functor ("correct") approach at all. The comparison is comforting to know but maybe not that important in the end.
- Why do spectral sequences help us organize the comparison of cohomologies? See Wikipedia: Čech-to-derived functor spectral sequence.