Sheaf Description of G-Bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:45:47Z http://mathoverflow.net/feeds/question/2414 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles Sheaf Description of G-Bundles Charles Siegel 2009-10-25T02:52:32Z 2009-10-27T17:20:30Z <p>Now, among algebraic geometers, at least, it is well known that there is an equivalence between locally free O_X-modules of rank n and vector bundles of rank n. So, equivalently, principal GL(n,C)-bundles are given by locally free sheaves of rank n.</p> <p>So...what about other groups? I guess that SL(n,C) bundles are then locally free sheaves of rank n with top exterior power trivial, but can we phrase everything in terms of the properties of a sheaf and a group?</p> <p>My guess is that in this context, if we can do it, we'll end up with something that's not quite locally free sheaves of rank n for GL(n,C), but which will be equivalent.</p> <p>Note: I'm aware that we could just say something like "the sheaf of local sections of a G-bundle" but I'm looking for something intrinsic, a set of properties of the sheaf without reference to the geometric bundle, which can be reconstructed from the sheaf description.</p> http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles/2423#2423 Answer by Ben Webster for Sheaf Description of G-Bundles Ben Webster 2009-10-25T03:39:55Z 2009-10-25T20:47:58Z <p>If G is an affine algebraic group, a G-bundle is the same as a monoidal functor from G-reps to coherent sheaves. The map one way is take associated bundle, the other involves reconstructing the structure sheaf of the G-bundle from the associated ones. Roughly, you think of the functions on the group as a ring ind-object in the category of representations, and take the corresponding ring object in quasi-coherent sheaves. The Spec of this sheaf of rings is the G-bundle.</p> <p>For GL(n), you'e lucky, since its category has a simple description: it's (basically) the free monoidal category with a single generator of dimension n. Other groups are a little more complicated, but not much worse.</p> http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles/2425#2425 Answer by Anton Geraschenko for Sheaf Description of G-Bundles Anton Geraschenko 2009-10-25T04:01:24Z 2009-10-25T04:37:16Z <p>Principal GL(n)-bundles <strong>are</strong> equivalent to locally free sheaves of rank n, but not in the way you're describing. The easiest way to see this is to note that principal G-bundles have fibers that look like G (in the case of GL(n), these would be n&sup2;-dimensional), but locally free sheaves of rank n have n-dimensional fibers.</p> <p>The way you get a locally free sheaf of rank n from a GL(n)-torsor P is by twisting the trivial rank n bundle O<sup>n</sup> (which has a natural GL(n)-action) by the torsor. Explicitly, the locally free sheaf is F=O<sup>n</sup>x<sup>GL(n)</sup>P, whose (scheme-theoretic) points are (v,p), where v is a point of the trivial bundle and p is a point of P, subject to the relation that (v&sdot;g,p)&sim;(v,g&sdot;p). It happens that the map is bijective. Given a locally free sheaf F of rank n, the automorphism sheaf Aut(F) is a GL(n) torsor (and this procedure is inverse to the one I described).</p> <p>Similarly, if you have a group G <strong>and a representation</strong> V, then you can associate to any G-torsor a locally free sheaf of rank dim(V). I don't know of a characterization of which locally free sheaves of rank dim(V) arise in this way.</p> <p>Operations with the locally free sheaf (like taking top exterior power) should just correspond to doing that operation with the representation V, so I think you're right that in the case of SL(n) you get exactly those locally free sheaves whose top exterior power is trivial (since SL(n) has no non-trivial 1-dimensional representations).</p> http://mathoverflow.net/questions/2414/sheaf-description-of-g-bundles/2831#2831 Answer by Charles Siegel for Sheaf Description of G-Bundles Charles Siegel 2009-10-27T15:59:14Z 2009-10-27T17:20:30Z <p>Adding it so that it's easily found. The thing I was looking for, which is generally not written out except in the case of vector bundles, is that the sheaf of sections of an F-bundle with fiber F is a sheaf of sets that is locally isomorphic in the etale topology to the sheaf hom(-,F) ranging over small enough open subsets of X.</p>