 
Summary: ON THE QUASIHEREDITARY PROPERTY FOR
STAGGERED SHEAVES
PRAMOD N. ACHAR
Abstract. Let G be an algebraic group over an algebraically closed field, act
ing on a variety X with finitely many orbits. Staggered sheaves are certain
complexes of Gequivariant coherent sheaves on X that seem to possess many
remarkable properties. In this paper, we construct "standard" and "costan
dard" objects in the category of staggered sheaves, and we prove that that
category has enough projectives and injectives.
1. Introduction
Let X be a variety over an algebraically closed field k, and let G be a linear
algebraic group over k acting on X with finitely many orbits. Staggered sheaves [A,
AT1] are the objects in the heart of certain tstructure on the bounded derived
category Db
G(X) of Gequivariant coherent sheaves on X. The category of staggered
sheaves, denoted M(X), enjoys a growing list of remarkable properties, analogous
in many ways to properties of adic mixed perverse sheaves [AT1, AT2]:
· Every object has finite length. Simple objects arise via an "IC" functor
and are parametrized by irreducible vector bundles on Gorbits.
· There is a wellbehaved notion of "purity" in Db
