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.

Adams spectral sequences and Franke's algebraicity conjecture

“…The study of quotients of the Freyd envelope has recently gained traction in the setting when C is rigid monoidal; starting with the work of Balmer, Krause and Stevenson who call certain such quotients the homological residue fields [7], [5] [8]. Our work clarifies the relationship between these residue fields and the Adams spectral sequence.…”

Stratification and the comparison between homological and tensor triangular support

“…It is known that the family need not be jointly conservative (see [BCHS23,Example 14.26]). In light of Theorem 1.8, the converse of Theorem 1.3 would follow from Balmer's "Nerves of Steel" Conjecture that the homological and tensor triangular spectra always coincide; see [BHS21a].…”