Spectral sequence - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-06-20T10:06:04Z http://mathoverflow.net/feeds/question/26023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26023/spectral-sequence Spectral sequence szts 2010-05-26T14:46:46Z 2010-06-23T09:03:19Z <p>what is Koszul resolution? what is its role played in the computation of spectral sequence?</p> http://mathoverflow.net/questions/26023/spectral-sequence/26026#26026 Answer by Charles Matthews for Spectral sequence Charles Matthews 2010-05-26T14:56:49Z 2010-05-26T14:56:49Z <p>The <em>Koszul complex</em> is defined at <a href="http://en.wikipedia.org/wiki/Koszul_complex" rel="nofollow">http://en.wikipedia.org/wiki/Koszul_complex</a> . In certain cases, one of which is explained there, the Koszul complex is a resolution (typically a free resolution, see <a href="http://en.wikipedia.org/wiki/Free_resolution#Projective_resolutions" rel="nofollow">http://en.wikipedia.org/wiki/Free_resolution#Projective_resolutions</a>). </p> <p>There is a complicated history of the Koszul complex, but really it began with Lie algebra cohomology, before becoming (also) a tool generally used in commutative algebra. Any free or projective resolution might appear in a spectral sequence argument: that query is not very specific. Maybe you want some standard argument from Lie algebra cohomology?</p> http://mathoverflow.net/questions/26023/spectral-sequence/26741#26741 Answer by Sean Tilson for Spectral sequence Sean Tilson 2010-06-01T17:03:46Z 2010-06-01T17:03:46Z <p>When doing things related to free resolutions you may want the actual computation or you may want something highly structured. For the former you use the koszul resolution, it is nice and little and small. If you want a lot of structure you use the Bar resolution. This is sort of a philosophical things, so when you actually want to compute something you use the Koszul resolution since it is pretty small and you know that making this choice wont affect your answer.</p>