Non-unital algebraic $K$-theory and almost mathematics

TL;DR

This paper develops a non-unital algebraic K-theory framework that is an exact functor and demonstrates its decomposition properties, providing new insights into the Gersten conjecture in algebraic K-theory.

Contribution

It introduces a modified non-unital algebraic K-theory as an exact functor and proves its decomposition for almost unital algebras, advancing understanding of the Gersten conjecture.

Findings

Non-unital K-theory is an exact functor from non-unital algebras to spectra.

Decomposition of K-theory for almost unital algebras into ideal and residue algebra.

Implication of the Gersten property for the K-theory of the ideal.

Abstract

The Gersten conjecture is still an open problem of algebraic $K$-theory for mixed characteristic discrete valuation rings. In this paper, we establish non-unital algebraic $K$-theory which is modified to become an exact functor from the category of non-unital algebras to the stable $\infty$-category of spectra. We prove that for any almost unital algebra, the non-unital $K$-theory homotopically decomposes into the non-unital $K$-theory the corresponding ideal and the residue algebra, implying the Gersten property of non-unital $K$-theory of the the corresponding ideal.