Speaker: Mitsuyasu Hashimoto
Date: September 03, 2009
Affiliation: Nagoya University, Japan

Abstract: Some part of the work is joint with Masahiro Ohtani and Mitsuhiro Miyazaki. <p>Mumford defined a $G$-equivariant $O_X$-module ($(G,O_X)$-module) over a scheme $X$ with the action of an algebraic group $G$. We study this object, and try to generalize the theory of quasi-coherent sheaves (without a group action) to that of sheaves with a group action. In particular, we study the direct image, inverse image, and their derived versions. We also consider $Tor$, $Ext$, and the local cohomology. We construct the twisted inverse $f^!$ with respect to a morphism $f$. <p> As an application, some results on Gorenstein and Cohen--Macaulay properties of the invariant subrings are obtained.