On the local cohomology and support for triangulated categories

Abstract

Recently a notion of support and a construction of local cohomology functors for [TR5] compactly generated triangulated categories were introduced and studied by Benson, Iyengar, and Krause. Following their idea, we assign to any object of the category a new subset of $Spec\left(R\right)$, again called the (big) support. We study this support and show that it satisfies axioms such as exactness, orthogonality, and separation. Using this support, we study the behavior of the local cohomology functors and show that these triangulated functors respect boundedness. Then we restrict our study to the categories generated by only one compact object. This condition enables us to get some nice results. Our results show that one can get a satisfactory version of the local cohomology theory in the setting of triangulated categories, compatible with the known results for the local cohomology for complexes of modules.