Why derived categories

By contrast, modern treatments are coordinate free which is better much of time, although not all of the time. I tend to think of spectral sequences as writing things in coordinates; derived categories are coordinate free. This obviously a stretch. In hindsight, this seems to be my answer to the question Thinking and Explaining as well.

Let me spell this out. On the spectral sequence side, we get something too horrible to comtemplate. It's the notes from his lectures at the Snowbird summer school. I'm going to make a radical suggestion and hence I have marked this community wiki , that there is a rewarding way to study derived categories as the localizations of "categories with weak equivalences", or more specifically, homotopical categories in the sense of Dwyer-Hirschhorn-Kan-Smith.

That is, any functor preserving weak equivalences between homotopical categories descends uniquely to localizations by the universal property of localizations , so derived functors become the images of homotopical approximations of functors between the original categories where homotopical approximation means in some sense the closest weak-equivalence preserving functor from the left or the right. Indeed, using the simplicial localization of Dwyer-Kan hammock localization or its more modern variant "Grothendieck-style simplicial localization" detailed in sections 34 and 35 of Dwyer-Hirschhorn-Kan-Smith Homotopy Limit Functors on Model and Homotopical categories , we can embed the case of homotopical categories into the case of simplicially enriched categories.

The papers of Dwyer-Kan or the recollection of these techniques in Chapter 17 of Phil Hirschhorn's Model Categories and their Localizations give a way to compute the mapping space in terms of simplicial and cosimplicial resolutions. This formulation is interesting because it lets us use the well-developed techniques of homotopical algebra in a simplicially-enriched category as well for instance, there is a very mature theory of homotopy limits and colimits in this setting.

This also gives the correct data and bypasses the need of working with triangulated categories, for instance. For what it's worth, I learned most of what I know of derived categories from the last chapter of Weibel's book on homological algebra and in particular, doing the exercises there.

However, I also often looked at Hartshorne's Residues and Duality, Gelfand-Manin, and occasionally derived categories for the working mathematician. And, btw. Sign up to join this community. The best answers are voted up and rise to the top. A down-to-earth introduction to the uses of derived categories Ask Question. Asked 11 years, 1 month ago.

Active 3 years, 2 months ago. Viewed 8k times. Improve this question. Charles Staats. Derived categories are rather ubiquitous these days, do you have particular papers in mind you want to understand? For example, do you already care about things like perverse sheaves? Are you trying to study things like derived categories of coherent sheaves on algebraic varieties and what they tell you about the geometry of the variety? Are you simply trying to learn something about Grothendieck duality?

Older books on homological don't do derived categories, only the modern ones by Gelfand-Manin, Weibel Another book which is I quite like is Iversen's "Cohomology of sheaves". Recall that in Hartshorne's textbook on alg.

But is that the same map as the edge map in the Cech to derived-functor spectral sequence? Try to prove it;I found this very difficult to prove for myself when I first learned these things several pages of gigantic diagrams, etc. Once I learned derived categories, it became a 2-line argument. Show 6 more comments. Active Oldest Votes. Improve this answer. See Homology, Section The corresponding saturated multiplicative systems see Lemma The initial statements follow from Lemma The statement on kernels in 1 , 2 , 3 is a consequence of the definitions in each case.

Each of the functors is essentially surjective by Lemma To finish the proof we have to show the functors are fully faithful. We first do this for the bounded below version.

Hence by Lemma This finishes the proof that the functor in 1 is an equivalence. The proof of 2 is dual to the proof of 1.

To prove 3 we may use the result of 2. The argument given in the previous paragraph applies directly to show this where we consistently work with complexes which are already bounded above. Your email address will not be published.

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