Cohomological length functions

TL;DR

This paper explores integer-valued length functions on triangulated categories, linking them to cohomological functors, analyzing their properties, and providing explicit calculations for perfect complexes over specific rings.

Contribution

It establishes a correspondence between length functions and cohomological functors, and studies the topological space of irreducible cohomological functions.

Findings

Irreducible cohomological functions form a topological space.

Explicit calculations for perfect complexes over specific rings.

Properties of the space of cohomological functions are analyzed.

Abstract

We study certain integer valued length functions on triangulated categories and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The irreducible cohomological functions form a topological space. We discuss its basic properties and include explicit calculations for the category of perfect complexes over some specific rings.