Stratification and the comparison between homological and tensor triangular support

TL;DR

This paper compares homological and tensor triangular support in tensor triangulated categories, proving their equivalence under stratification and clarifying their differences with counterexamples.

Contribution

It establishes a bijection between the homological and tensor triangular spectra and extends Balmer's work on support coincidence in stratified categories.

Findings

Support functions coincide in stratified categories

Bijection between homological and tensor spectra is proven

Counterexamples illustrate differences between support notions

Abstract

We compare the homological support and tensor triangular support for `big' objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support.