Tensor-triangular fields: Ruminations

TL;DR

This paper explores the concept of fields within tensor-triangular geometry, providing examples, discussing approaches, and proposing homological tensor-functors to Grothendieck categories due to unresolved residue tt-fields.

Contribution

It introduces homological tensor-functors to Grothendieck categories as a new approach to tensor-triangular fields, addressing open problems in the area.

Findings

Examples of tensor-triangular fields are gathered.

Open problems regarding residue tt-fields are highlighted.

Homological tensor-functors to Grothendieck categories are constructed.

Abstract

We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable homological tensor-functors to Grothendieck categories.