Vanishing of Nil-terms and negative K-theory for additive categories

TL;DR

This paper generalizes the concept of regular coherence from rings to additive categories, demonstrating that negative K-groups and Nil-groups vanish for regular coherent additive categories, with applications to algebraic K-theory of p-adic groups.

Contribution

It extends regular coherence to additive categories and proves the vanishing of negative K-theory and Nil-groups in this broader context.

Findings

Negative K-groups vanish for regular coherent additive categories

Nil-groups vanish for regular coherent additive categories

Application to algebraic K-theory of p-adic group Hecke algebras

Abstract

We extend the notion of regular coherence from rings to additive categories and show that well-known consequences of regular coherence for rings also apply to additive categories. For instance the negative K-groups and all twisted Nil-groups vanish for an additive category if it is regular coherent. This will be applied to nested sequences of additive categories, motivated by our ongoing project to determine the algebraic K-theory of the Hecke algebra of a reductive p-adic group.