Stratification and the comparison between homological and tensor triangular support

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.

18 pages, comments welcome. v2 - updated to include referee comments. Version to appear in The Quarterly Journal of Mathematics

References in corpus (1)

• Support and vanishing for non-Noetherian rings and tensor triangulated categories

Cited by in corpus (6)

• The spectrum of excisive functors
• The shift-homological spectrum and parametrising kernels of rank functions
• Homological stratification and descent
• Towards the Nerves of Steel Conjecture
• Telescope conjecture via homological residue fields with applications to schemes
• Big categories, big spectra