 
Summary: PURITY AND DECOMPOSITION THEOREMS FOR
STAGGERED SHEAVES
PRAMOD N. ACHAR AND DAVID TREUMANN
Abstract. Two major results in the theory of adic mixed constructible
sheaves are the purity theorem (every simple perverse sheaf is pure) and the
decomposition theorem (every pure object in the derived category is a direct
sum of shifts of simple perverse sheaves). In this paper, we prove analogues
of these results for coherent sheaves. Specificially, we work with staggered
sheaves, which form the heart of a certain tstructure on the derived category
of equivariant coherent sheaves. We prove, under some reasonable hypotheses,
that every simple staggered sheaf is pure, and that every pure complex of
coherent sheaves is a direct sum of shifts of simple staggered sheaves.
1. Introduction
Let Z be a variety over a finite field Fq, and let Db
m(Z) denote the bounded
derived category of adic mixed constructible sheaves on Z. Recall that the weights
of an object F Db
m(Z) are certain integers defined in terms of the eigenvalues of
the Frobenius morphism on the stalks at F at Fqpoints of Z. An object is said
to be pure of weight w Z if both it and its Verdier dual have weights w. The
