Derived categories of coherent sheaves: suggested references? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:48:46Z http://mathoverflow.net/feeds/question/42463 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references Derived categories of coherent sheaves: suggested references? J Verma 2010-10-17T06:35:49Z 2010-10-18T12:24:54Z <p>I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least prerequisites. As I was looking through the papers of Bridgeland, I realized that much of the theorems are stated for Projective varieties (not schemes), I've just started learning Scheme theory in my Algebraic Geometry course,my background in schemes is not very good but I am fine with Sheaves. It would be better if you suggest some reference where everything is developed in terms of Projective varieties.</p> http://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references/42500#42500 Answer by ABayer for Derived categories of coherent sheaves: suggested references? ABayer 2010-10-17T13:52:44Z 2010-10-17T13:52:44Z <p>A good introduction to derived categories of coherent sheaves are Caldaru's notes <a href="http://front.math.ucdavis.edu/0501.5094" rel="nofollow">Derived categories of sheaves: a skimming</a>.</p> <p>As for Bridgeland's work, I would recommend reading his papers directly, using Huybrecht's book as a reference (as mentioned by Francesco above, "Fourier-Mukai transforms in algebraic geometry"). Specifically for Bridgeland stability conditions, I also have some short notes on <a href="http://www.math.uconn.edu/~bayer/" rel="nofollow">my homepage</a>, but again, you should also read his original papers.</p> http://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references/42536#42536 Answer by Piotr Achinger for Derived categories of coherent sheaves: suggested references? Piotr Achinger 2010-10-17T20:13:26Z 2010-10-17T20:13:26Z <p>Scheme theory won't be that important (as you said, most of the time you are concerned with smooth projective varieties over \$\mathbb C\$). However, you didn't mention about your experience in homological algebra. Huybrechts' book is a pleasure to read, but it can't serve as a textbook in homological algebra. I would recommend e.g. reading first first few chapters of Weibel's book (1, 2, 3, 5, 10). Gelfand-Manin "Methods of Homological Algebra" is also a good reference here.</p> <p>Moreover, there is a short introduction by R. P. Thomas <a href="http://arxiv.org/abs/math/0001045" rel="nofollow">"Derived categories for the working mathematician"</a>.</p> http://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references/42600#42600 Answer by Leo Alonso for Derived categories of coherent sheaves: suggested references? Leo Alonso 2010-10-18T08:57:32Z 2010-10-18T08:57:32Z <p>Let me suggest a perhaps longer program to master derived categories of coherent sheaves. In a comment you said "I don't know much Homological Algebra", so first you have to master the basics of abelian categories and ext and tor of modules. The first chapters of Weibel's "Homological Algebra" may be a useful reference.</p> <p>This said, I agree with Polizzi's suggestion of Huybrechts' book. The only issue I would point out is that the book concentrates on the <em>bounded</em> derived category of <em>coherent</em> sheaves. It is enough for its application to the classification of non-singular varieties. But if you have varieties with singularities in mind then things may get more complicated. First there is a difference between perfect complexes and bounded complexes of coherent sheaves. Second one sometimes need to use infinite methods, therefore one is forced to consider quasi-coherent sheaves and <em>unbounded</em> derived categories. For this beautiful theory, a good introduction is (as I have suggested here before) 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). </p> <p>This is not a shortest path, but in my experience getting into general methods is that they pay back later. Your mileage may vary.</p> http://mathoverflow.net/questions/42463/derived-categories-of-coherent-sheaves-suggested-references/42621#42621 Answer by Zoran Škoda for Derived categories of coherent sheaves: suggested references? Zoran Škoda 2010-10-18T12:24:54Z 2010-10-18T12:24:54Z <p>Kapustin-Orlov'a <em>survey</em> of derived categories of coherent sheaves is pretty good,</p> <ul> <li>A. N. Kapustin, D. O. Orlov, <em>Lectures on mirror symmetry, derived categories, and D-branes</em>, Uspehi Mat. Nauk <strong>59</strong> (2004), no. 5(359), 101--134; translation in Russian Math. Surveys <strong>59</strong> (2004), no. 5, 907--940, <a href="http://arxiv.org/abs/math/0308173" rel="nofollow">math.AG/0308173</a></li> </ul> <p>but more slow/elementary exposition starting with fundamentals of derived categories is in an earlier survey of Orlov </p> <ul> <li>D. O. Orlov, <em>Derived categories of coherent sheaves and equivalences between them</em>, Uspekhi Mat. Nauk, 2003, Vol. <strong>58</strong>, issue 3(351), pp. 89–172, <a href="http://www.mathnet.ru/php/getFT.phtml?jrnid=rm&amp;paperid=629&amp;volume=58&amp;year=2003&amp;issue=3&amp;fpage=89&amp;what=fullt&amp;option_lang=eng" rel="nofollow">Russian pdf</a>, English transl. in Russian Mathematical Surveys (2003),58(3):511, <a href="http://dx.doi.org/10.1070/RM2003v058n03ABEH000629" rel="nofollow">doi link</a>, <a href="http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf" rel="nofollow">pdf</a> at Orlov's webpage (not on arXiv!)</li> </ul> <p>There are also Orlov's handwritten slides in djvu from a 5-lecture course in Bonn</p> <ul> <li><a href="http://www.irb.hr/korisnici/zskoda/orlovMPIslides.djvu" rel="nofollow">djvu</a>, but the link is temporary</li> </ul> <p>For derived categories per se, apart from Gelfand-Manin methods book and Weibel's hoological algebra remember that a really good expositor is Bernhard Keller. E.g. his text </p> <ul> <li>Bernhard Keller, <em>Introduction to abelian and derived categories</em>, <a href="http://www.math.jussieu.fr/~keller/publ/cam.pdf" rel="nofollow">pdf</a> </li> </ul> <p>...and also his Handbook of Algebra entry on derived categories: <a href="http://www.math.jussieu.fr/~keller/publ/dcu.pdf" rel="nofollow">pdf</a></p>