Detecting projectivity in sheaves associated to representations of infinitesimal groups

Let G be an infinitesimal group scheme of finite height r   and V(G)V(G) the scheme which represents 1-parameter subgroups of G  . We consider sheaves over the projective support variety P(G)P(G) constructed from a G-module M  . We show that if P(G)P(G) is regular then the sheaf H[1](M)H[1](M) is zero if and only if M   is projective. In general, H[1]H[1] defines a functor from the stable module category and we prove that its kernel is a thick triangulated subcategory. Finally, we give examples of G   such that P(G)P(G) is regular and indicate, in characteristic 2, the connection to the BGG correspondence. Along the way we will provide new proofs of some known results and correct some errors in the literature.