Detecting projectivity in sheaves associated to representations of infinitesimal groups

TL;DR

This paper investigates the relationship between sheaves on the projectivization of schemes associated with infinitesimal groups and the projectivity of modules, providing criteria, functorial properties, and examples, including connections to the BGG correspondence.

Contribution

It establishes a criterion linking the vanishing of a sheaf cohomology to module projectivity, characterizes the kernel of a functor as a thick triangulated subcategory, and offers new proofs and corrections in the theory.

Findings

H^[1](M) vanishes iff M is projective when P(G) is regular.

The functor H^[1] defines a kernel that is a thick triangulated subcategory.

Examples of G with regular P(G) and connections to BGG correspondence in characteristic 2.

Abstract

Let G be an infinitesimal group scheme of finite height r and V(G) the scheme which represents 1-parameter subgroups of G. We consider sheaves over the projectivization P(G) of V(G) constructed from a G-module M. We show that if P(G) is regular then the sheaf H^[1](M) is zero if and only if M is projective. In general, 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) 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.