Faithful perversities

TL;DR

This paper characterizes faithful highest weight hearts in algebraic triangulated categories and perverse sheaves, linking algebraic and topological properties, and provides bounds on global dimension and methods for computing hypercohomology.

Contribution

It introduces a new characterization of faithful highest weight hearts as serially faithful glued hearts and relates algebraic and topological conditions for perverse sheaves.

Findings

Faithful highest weight hearts are equivalent to serially faithful glued hearts.

Global dimension of faithful categories of perverse sheaves is bounded by the space dimension.

Hypercohomology can be computed from projective resolutions of the constant sheaf.

Abstract

We show that the faithful highest weight hearts in an algebraic triangulated category are the serially faithful glued hearts, equivalently the hearts containing a dual pair of full exceptional collections in the sense of Bodzenta--Bondal (arXiv:2601.22004). We then characterise faithful highest weight categories of perverse sheaves on topologically stratified spaces algebraically, in terms of the exactness of certain functors, and topologically, in terms of the vanishing of certain cohomology groups of pairwise links. We prove that the global dimension of a faithful category of perverse sheaves on a topologically stratified space $X$ with finitely many strata is bounded by the dimension of $X$. Finally, we show that in this setting the hypercohomology of a perverse sheaf can be computed from a projective resolution of the constant sheaf, and conversely that the multiplicities of the terms in a minimal projective resolution of the constant sheaf can be computed as intersection cohomology groups.