Cosupport for triangulated categories

TL;DR

This paper investigates the concept of cosupport in triangulated categories, providing new characterizations, computations, and comparisons with support, and introduces the notion of big cosupport to deepen understanding.

Contribution

It offers new characterizations and computations of cosupport, generalizes known results, and introduces the concept of big cosupport in triangulated categories.

Findings

Boundedness of local cohomology and homology functors studied

Characterizations of cosupport provided and generalized

Comparison between support and cosupport established

Abstract

The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $\mathrm{Spec}R$, called the big cosupport.