Timeline for A down-to-earth introduction to the uses of derived categories
Current License: CC BY-SA 2.5
22 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 13, 2018 at 13:34 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Sep 13, 2018 at 12:42 | answer | added | user97900 | timeline score: 2 | |
| Nov 20, 2010 at 18:07 | vote | accept | Charles Staats | ||
| Sep 22, 2010 at 12:19 | answer | added | Donu Arapura | timeline score: 30 | |
| Sep 22, 2010 at 10:31 | answer | added | Harry Gindi | timeline score: 8 | |
| Sep 22, 2010 at 9:50 | answer | added | Leo Alonso | timeline score: 4 | |
| Sep 21, 2010 at 22:29 | comment | added | BCnrd | 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. | |
| Sep 21, 2010 at 22:25 | comment | added | BCnrd | 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. | |
| Sep 21, 2010 at 21:22 | comment | added | Karl Schwede | Charles, if you are mostly looking for ways to avoid thinking about spectral sequences, derived categories are very convenient. You want to then find a book which covers composition of derived functors in an approachable way (and many of the books described here do that). | |
| Sep 21, 2010 at 19:08 | comment | added | Jim Humphreys | The comments and answers here cover the ground well, I think, but it has to be kept in mind that derived categories mainly provide a language. This is used widely both in algebraic geometry and in various areas of representation theory (some purely algebraic). So different sources will have different emphases when it comes to motivation or applications. | |
| Sep 21, 2010 at 18:52 | comment | added | Donu Arapura | 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". | |
| Sep 21, 2010 at 18:26 | answer | added | Sasha | timeline score: 30 | |
| Sep 21, 2010 at 17:32 | comment | added | Charles Staats | I want to learn how to do things using derived categories, that can be done without using derived categories (but perhaps with more of a headache in the latter case). In particular, there is a proof in Altman-Kleiman, Introduction to Grothendieck Duality Theory, using spectral sequences, that I could not follow, and that my advisor said was more easily understood using derived categories. I've also seen a comment elsewhere on MathOverflow that it was easier to work with compositions of derived functors (especially if one is left-derived and the other is right-derived) using derived categories. | |
| Sep 21, 2010 at 16:15 | answer | added | Karl Schwede | timeline score: 6 | |
| Sep 21, 2010 at 16:12 | comment | added | Karl Schwede | 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? | |
| Sep 21, 2010 at 16:06 | answer | added | Mariano Suárez-Álvarez | timeline score: 10 | |
| Sep 21, 2010 at 16:03 | comment | added | Charles Staats | So far as I can tell, "Derived Categories for the Working Mathematician" is a nice, down-to-earth description of what derived categories are, and why they are "natural" things to consider. What I'm looking for is an explanation of how to use derived categories as tools. Not that DCWM has no bearing on this, but the emphasis is different. | |
| Sep 21, 2010 at 15:45 | answer | added | JSE | timeline score: 19 | |
| Sep 21, 2010 at 15:07 | comment | added | David E Speyer | If you are looking for something more in depth than that, perhaps the later chapters of Gelfand and Manin, "Methods of Homological Algebra". | |
| Sep 21, 2010 at 15:05 | comment | added | David E Speyer | I liked "Derived Categories for the Working Mathematician" arxiv.org/abs/math/0001045 . | |
| Sep 21, 2010 at 14:52 | comment | added | Mariano Suárez-Álvarez | Most textbooks on homological algebra fit your description :) | |
| Sep 21, 2010 at 14:51 | history | asked | Charles Staats | CC BY-SA 2.5 |