Homological residue fields as comodules over coalgebras

TL;DR

This paper demonstrates that homological residue fields in tensor triangulated categories can be explicitly described as categories of comodules over coalgebras, revealing their natural geometric and algebraic significance.

Contribution

It provides explicit descriptions of homological residue fields as comodules over coalgebras in various mathematical contexts, connecting abstract theory with concrete models.

Findings

Homological residue fields are categories of comodules over coalgebras.

These fields encode tangent data at points on the spectrum.

The results unify perspectives across algebra, geometry, and topology.

Abstract

We explicitly present homological residue fields for tensor triangulated categories as categories of comodules in a number of examples across algebra, geometry, and topology. Our results indicate that, despite their abstract nature, they are very natural objects and encode tangent data at the corresponding point on the spectrum.