Henselian对的K -理论与拓扑循环同调

IF 3.5 1区 数学 Q1 MATHEMATICS
Dustin Clausen, Akhil Mathew, Matthew Morrow
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引用次数: 0

摘要

给出一个交换环的henselian对$(R, I)$,通过环切迹$K \到$ mathm {TC}$证明了相对$K$-理论和有限系数的相对拓扑循环同调。这产生了经典的Gabber-Gillet-Thomason-Suslin刚性定理(对于mod $n$系数,其中$n$在$R$中可逆)和McCarthy关于相对$K$理论的定理(当$I$为幂零时)的推广。对于一大类$p$-进环,我们推导出$p$-进环的环切迹是$p$-进环的K$-理论与拓扑循环同调的大程度等价。此外,我们还证明了有限系数的K -理论满足为F -有限模p -的完全诺瑟环的连续性。我们主要的新成分是具有有限系数的$\ mathm {TC}$的基本有限性质。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
$K$-theory and topological cyclic homology of Henselian pairs
Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic rings. In addition, we show that $K$-theory with finite coefficients satisfies continuity for complete noetherian rings which are $F$-finite modulo $p$. Our main new ingredient is a basic finiteness property of $\mathrm{TC}$ with finite coefficients.
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来源期刊
CiteScore
7.60
自引率
0.00%
发文量
14
审稿时长
>12 weeks
期刊介绍: All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics.
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