Published on *www.math.tifr.res.in* (http://www.math.tifr.res.in)

Nagoya University, Japan

September 3, 2009

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 $f$.

As an application, some results on Gorenstein and Cohen--Macaulay properties of the invariant subrings are obtained.