A down-to-earth introduction to the uses of derived categories - MathOverflow

A down-to-earth introduction to the uses of derived categories

2010-09-21T14:51:01Z

When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about spectral sequences in order to use them.

Does anyone know of a similar source for derived categories? Something that concentrates on showing how these things are used, without developing the entire theory, or necessarily even giving complete, rigorous definitions?

Answer by JSE for A down-to-earth introduction to the uses of derived categories

2010-09-21T15:45:11Z

"Derived Categories of Sheaves: A Skimming," by my colleague Andrei Caldararu, might do the trick. It's the notes from his lectures at the Snowbird summer school.

Answer by Mariano Suárez-Alvarez for A down-to-earth introduction to the uses of derived categories

2010-09-21T16:06:28Z

In Bernhard Keller's list of preprints and publications you will find characteristically nice expositions on derived categories (with the point of view of someone interested in representations, mostly, I'd say): Derived categories and their uses, from the first volume of the Handbook of algebra, and Introduction to abelian and derived categories, the notes from his lectures in a school at ICTP (Trieste 2006), &c.

Answer by Karl Schwede for A down-to-earth introduction to the uses of derived categories

2010-09-21T16:15:05Z

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.

My goal at the time was to understand better things like Grothendieck duality.

Answer by Sasha for A down-to-earth introduction to the uses of derived categories

2010-09-21T18:26:32Z

I would suggest "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts.

Answer by Leo Alonso for A down-to-earth introduction to the uses of derived categories

2010-09-22T09:50:26Z

For someone interested in algebraic geometry, the first chapters of Lipman's "Notes on Derived Functors and Grothendieck Duality", (in Foundations of Grothendieck Duality for Diagrams of Schemes, Lecture Notes in Mathematics, no. 1960, Springer, 2009) is a must. (And, btw. it is way better to read chapter 4 in this work for understanding Grothendieck duality than Harthorne's notes).

Answer by Harry Gindi for A down-to-earth introduction to the uses of derived categories

2010-09-22T10:31:27Z

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. This turns our ordinary derived categories into categories enriched in weak homotopy types of CW complexes such that $\pi_0Map_C(X,Y):=Hom_{Ho(Set_\Delta)}(\Delta^0, Map_C(X,Y))\cong Hom_{W^{-1}C}(X,Y)$, where $W^{-1}C$ is the derived category. 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. To use the language of the nLab (which is often scarier-sounding than it actually is), this is the $(\infty,1)$-categorical approach to derived categories.

(Note that in the above, we would let $C$ be an appropriate category of chain complexes (of sheaves)).

If I remember correctly, this approach is roughly discussed in SGA 4 exposé 2.

Answer by Donu Arapura for A down-to-earth introduction to the uses of derived categories

2010-09-22T12:19:38Z

I won't claim to be fluent in the language of derived categories, but I understand it and can make myself understood. For most people, that's the right level of proficiency. Since you already have plenty of references, let me instead share few thoughts about the relation between the old and new languages. This is an imperfect analogy, but in the older differential geometry literature, everything was written in coordinates leading to messy formulas ("the debauch of indices"). 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 http://mathoverflow.net/questions/38639/thinking-and-explaining as well.)

Let me spell this out. Given left exact functors $F:A\to B$ and $G:B\to C$ between abelian categories, under the usual assumptions, we get the Grothendieck spectral sequence $$E_2^{pq} = R^pF (R^qG M) \Rightarrow R^{p+q}F\circ G M$$ By constrast, in the derived category world we see a composition law $$\mathbb{R} F\circ \mathbb{R} G\cong \mathbb{R}F\circ G$$ For 3 or more functors, the last formula generalizes in the obvious way. On the spectral sequence side, we get something too horrible to comtemplate. Well no, let me comtemplate it: $$E_2^{pqr\ldots} = R^pF (R^qG (R^rH\ldots))$$ $$d_2^{2,-1,0,\ldots}: E_2^{pqr\ldots} \to E_2^{p+2,q-1,r,\ldots}$$ $$d_2^{0,2,-1\ldots}\ldots$$ $$\ldots$$ Don't get me wrong, spectral sequences are still useful, but not here.