Is D-module on flag variety of Lie algebra a scheme? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:31:07Z http://mathoverflow.net/feeds/question/11660 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/11660/is-d-module-on-flag-variety-of-lie-algebra-a-scheme Is D-module on flag variety of Lie algebra a scheme? Shizhuo Zhang 2010-01-13T14:10:46Z 2010-01-13T17:24:07Z <p>This question was motivated by the answers in <a href="http://mathoverflow.net/questions/11200/geometry-of-triangulated-category-and-d-modules-theory" rel="nofollow">D-module as quasi coherent sheaves on deRham stack</a>. What I am interested in is the case of D-module on flag variety of Lie algebra. So,in this case, if we realize the D-module as quasi coherent sheaves on De Rham stack X^DR. Then the stack X^DR is a scheme or not?</p> <p>Why do I ask this question? Because from the point of view of noncommutative algebraic geometry, if we realize D-module on flag variety of Lie algebra as a quasi coherent sheaves on <strong>"space"</strong>, then <strong>this space</strong> is actually a noncommutative separated scheme. </p> <p>So I guess, this DeRham stack will be a scheme in this special case</p> http://mathoverflow.net/questions/11660/is-d-module-on-flag-variety-of-lie-algebra-a-scheme/11671#11671 Answer by David Ben-Zvi for Is D-module on flag variety of Lie algebra a scheme? David Ben-Zvi 2010-01-13T17:24:07Z 2010-01-13T17:24:07Z <p>The de Rham space of a scheme is essentially never a scheme or algebraic space (unless I guess you're Spec of an Artin ring, in which case you'll get a discrete set of points). In particular this applies to the flag variety. I'm not sure which perspective of NC AG you're taking, but certainly if you define the field as the study of Grothendieck categories, or pretriangulated dg categories, etc then D-modules are a very nice (nonproper) noncommutative space. (An interesting comment on this is found at the end of Kontsevich's letter <a href="http://www.math.uchicago.edu/~mitya/langlands/kontsevichletter.txt" rel="nofollow">here</a>. Also from the point of view of "function theory" D-modules on a scheme are great, i.e. all functors are given by integral transforms, sheaves on a (fiber) product are tensor product of categories of sheaves, etc, see <a href="http://arxiv.org/abs/0904.1247" rel="nofollow">here</a>.) But I don't think this says anything about the de Rham stack in a classical commutative sense..</p>