Flat Model Structures for Nonunital Algebras and Higher K-Theory

TL;DR

This paper establishes a flat model structure for complexes of h-unitary modules over nonunital rings, enabling Morita-invariant higher K-theory development compatible with tensor products.

Contribution

It introduces a Quillen flat model structure for nonunital algebras, facilitating homological algebra and higher K-theory in this context.

Findings

Provides a Morita-invariant homological framework.

Develops a higher K-theory compatible with tensor products.

Establishes a Quillen flat model structure for nonunital rings.

Abstract

We prove the existence of a Quillen Flat Model Structure in the category of unbounded complexes of h-unitary modules over a nonunital ring (or a $k$-algebra, with $k$ a field). This model structure provides a natural framework where a Morita-invariant homological algebra for these nonunital rings can be developed. And it is compatible with the usual tensor product of complexes. The Waldhausen category associated to its cofibrations allows to develop a Morita invariant excisive higher $K$-theory for nonunital algebras.