Nilpotence theorems via homological residue fields

Abstract

We prove nilpotence theorems in tensor-triangulated categories using suitable Gabriel quotients of the module category, and discuss examples.

Tensor-triangular fields: ruminations

“…On one hand it seems to hint at new vistas in tt-geometry and works uniformly. On the other it is a sufficient framework for proving an extremely general tensor-nilpotence theorem [Bal17] which can be used for computations.…”

Cohomology for Drinfeld doubles of some infinitesimal group schemes

“…(2) Does the Drinfeld double of a general infinitesimal group scheme G admit enough (universal) π-points, in the sense of Corollary A. 16 Of course, question (3) has to do with one's (in)ability to use π-point support in certain tensor triangular investigations, as in Section 8 and [11,9,8,7] for example.…”

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.…”

“…We combine this with a Goerss-Hopkins obstruction theory argument to complete the proof of the conjecture in §7. 5.…”

“…In the special case where C is monoidal, certain such localizing subcategories have been studied in the work of Balmer, Krause and Stevenson, who called the corresponding quotients the homological residue fields of C [7], [5]. These can be often described as comodules over comonads or coalgebras as in the work of Balmer, Cameron and Stevenson [6], [30], see also Remark 6.45 below.…”