Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2001.10425

Purity in chromatically localized algebraic $K$-theory

Markus Land·Akhil Mathew·Lennart Meier·Georg Tamme
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\geq 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \dots \oplus T(n)$-localization of $R$. This complements a classical result of Waldhausen in rational $K$-theory. Combining our result with work of Clausen--Mathew--Naumann--Noel, one finds that $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $\mathrm{TC}(\tau_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$.

Explore related subjects

math.KTmath.AT

Keep this discovery

BibTeXRIS

Markus Land, Akhil Mathew, Lennart Meier, Georg Tamme. 2020-01-28. Purity in chromatically localized algebraic $K$-theory. https://doi.org/10.1090/jams/1043

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

KEEP EXPLORING

Related papers

arXiv2026

The Oka principle for \'etale Chow groups

The celebrated theorems of Shilov, Arens--Royden, and Forster give direct descriptions of the first three integral cohomology groups of the Gelfand spectrum of a commutative complex Banach algebra. In his 1974 ICM address, Taylor asked whether the higher cohomology groups admit descriptions in terms of the underlying ring. We give a solution to this question in even degrees: The \'etale (aka Lichtenbaum) Chow group in every codimension is canonically isomorphic to the corresponding even integral cohomology group of the Gelfand spectrum.

math.KT↗
arXiv2026

General linear and Steinberg groups over the Leavitt algebra $L_{\mathbb F_2}(1,2)$

Let $R=L_{\F_2}(1,2)$. We prove that $\GL_r(R)$ is integrally acyclic for every $r\geq1$ and that the canonical map $\St_r(R)\to\GL_r(R)$ is an isomorphism for every $r\geq3$. The homology calculation combines simultaneous extensions of ordered frames with scalar actions of the multiplicative groups of finite fields on their stabilizers. The presentation associated with the same frame complex defines a surjective section of the Steinberg map. An explicit finite presentation of $R^\times$ then follows from the theorem of Krsti\'c and McCool. We formulate separate criteria for acyclicity and for the Steinberg comparison over other rings.

math.KT↗
arXiv2026

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

math.KT↗