Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.08378

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

Yuki Kato
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.KTmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuki Kato. 2023-02-27. Non-unital algebraic $K$-theory and almost mathematics. https://arxiv.org/abs/2207.08378

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Finite-coefficient Gersten injectivity fails in ramified mixed characteristic

Let $V$ be a complete discrete valuation ring of mixed characteristic $(0,3)$ in which $3$ is a uniformizer, and put $A=V[[x,y]]/(3+x^2-y^3)$. We construct a nonzero class $a\in K_2(A;\mathbf Z/3)$ whose restriction to the fraction field of $A$ is zero. Thus Gersten injectivity for algebraic $K$-theory with $\mathbf Z/3$-coefficients fails for a two-dimensional ramified regular local ring. We also indicate the expected analogous construction for every odd prime. This counterexample does not contradict the integral Gersten conjecture but it rules out a naive reduction to finite coefficients.

math.KT↗
arXiv2026

$E$-theory is compactly assembled

We show that the equivariant $E$-theory category $\mathrm{E}_{\mathrm{sep}}^{G}$ for separable $C^{*}$-algebras is a compactly assembled stable $\infty$-category. We derive this result as a consequence of the shape theory for $C^{*}$-algebras developed by Blackadar and Dardarlat and a new construction of $\mathrm{E}_{\mathrm{sep}}^{G}$. As an application we investigate a topological enrichment of the homotopy category of a compactly assembled $\infty$-category in general and argue that the results of Carrión and Schafhauser on the enrichment of the classical $E$-theory category can be derived by specialization.

math.KT↗
arXiv2026

Scissors automorphism groups I: Homological stability and K-theory

In any category with a reasonable notion of cover, each object has a group of scissors automorphisms. We prove that under mild conditions, the homology of this group is independent of the object, and can be expressed in terms of the scissors congruence K-theory spectrum defined by Zakharevich. We therefore obtain both a group-theoretic interpretation of Zakharevich's higher scissors congruence K-theory, as well as a method to compute the homology of scissors automorphism groups. In the classical case of scissors congruence of polytopes, this leads to calculations and structural results on the homology of the scissors automorphism group. In two sequels, we show that more generally our approach leads to homology calculations for various families of groups appearing in group theory and dynamics, recovering results of Szymik-Wahl, Li, and Tanner.

math.KT↗