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

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?

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I liked "Derived Categories for the Working Mathematician" arxiv.org/abs/math/0001045 . – David Speyer Sep 21 '10 at 15:05
My feeling is that it depends on what context you want to use them in. 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? – Karl Schwede Sep 21 '10 at 16:12
I have to respectfully disagree with Mariano's claim. 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". – Donu Arapura Sep 21 '10 at 18:52
Dear Charles: A most excellent example is the Kunneth formula in the non-flat case, since in principle it involves infinitely many Tor-terms (beyond deceptively simple cases like beginning life over a Dedekind domain that has projective dimension 1). At a fancy level, one can scarcely formulate non-derived Kunneth for passing from one artinian coefficient ring to another without a mess (as comes up in $\ell$-adic cohomology), and at a more dramatic level just take a look at the nightmare of spectral sequences in EGA III$_2$ compared against the same topic in Weibel's book. – BCnrd Sep 21 '10 at 22:25
Dear Charles: Here's a nifty example for you to try for yourself. Recall that in Hartshorne's textbook on alg. geom., he constructs the map from Cech cohomology to derived functor cohomology "by hand", mapping a Cech resolution to an injective resolution. 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.; maybe I was missing something obvious). Once I learned derived categories, it became a 2-line argument. – BCnrd Sep 21 '10 at 22:29

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

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"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.

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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 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.

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Really liked this answer, Donu. Let me ask a question. I find that often when I really want to sit down and compute some cohomology group of some physical thing, I really do want to write down a spectral sequence. Is that "analogous" in our sense to the fact that, sometimes, if you want to compute some linear-algebra object in some completely concrete way, it is not such a bad thing to write everything down in coordinates? E.G. is this sort of thing what you have in mind by your italicized "although not all the time?" – JSE Sep 23 '10 at 1:57
Yes, that's exactly the sort of thing I had in mind. – Donu Arapura Sep 23 '10 at 17:17

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.

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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.

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I also find this a very enlightening view point, but just for the record: Ho(co)lims in cocomplete triangulated categories are MUCH easier to compute by completing the right map to an exact triangle than by going via a simplicial (or any other) enrichment... – Peter Arndt Sep 22 '10 at 12:28
@Peter: I can't disagree with you on that. I just think that this viewpoint gives a much prettier conceptual idea of what's going on. This approach makes it clear that triangulated categories are nothing more than a computationally useful approximation of the underlying picture. – Harry Gindi Sep 22 '10 at 12:43

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.

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I think Weibel's book is the right source if you want to learn to use derived categories (rather than just learn how they are used). If you want to see what they are used for, then it really depends on what applications you are interested in, what your background is, etc. – Arend Bayer Sep 21 '10 at 22:42

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).

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I wouldn't recommend this book for a beginner. The theory developed is so deep that one can be drowned therein! – Sasha Sep 22 '10 at 11:37
Sasha, you may be right. Some of us had access to the early versions which were thinner and so less intimidating. One advantage, of Lipman's approach is that he works in the unbounded derived category. So awkward problems with composing left and right derived functors disappear. – Donu Arapura Sep 22 '10 at 12:55