On the \(K\)-theory of finite algebras over Witt vectors of perfect fields
type: Item
instance of: scholarly article
title: On the \(K\)-theory of finite algebras over Witt vectors of perfect fields
zbMATH Open document ID: 0866.55002
DOI: 10.1016/0040-9383(96)00003-1
author: Lars Hesselholt
author: Ib Madsen
published in: Topology
publication date: 1996-12-03T00:00:00Z
review text: Topological cyclic homology (\(TC\)) first appeared in [\textit{M. Bökstedt, W. C. Hsiang} and the second author, Invent. Math. 111, 465-539 (1993; Zbl 0804.55004)], where it was used to prove an algebraic K-theory version of the Novikov conjecture. This paper is partly an exposition of some formal properties of \(TC\), and partly a proof of a conjecture of the second author (see below). Let \(L\) be a ring up to homotopy, or more precisely an FSP (functor with smash products). In analogy with the Hochschild homology of an ordinary ring, Bökstedt defined the topological Hochschild homology \(THH(L)\). This is an \(S^1\) spectrum, and topological cyclic homology is constructed from fixed point information about the topological Hochschild homology. The authors make a point of the fact that \(THH(L)\) is what they call a cyclotomic spectrum, which in the present situation is reflected in the fundamental cofibration sequence \[ THH(L)_{hC_{p^n}}\to THH(L)^{C_{p^n}}@>R>> THH(L)^{C_{p^{n-1}}} \] where \(C_q\) is the cyclic group of order \(q\), \(THH(L)_{hC_q}\) is the homotopy orbit spectrum, and \(THH(L)^{C_q}\) the actual fixed point spectrum. An observation with notational significance proven in this paper, is that if \(A\) is a commutative ring, then \(\pi_0THH(A)^{C_{p^n}}\cong W_{n+1}(A)\), the \(p\)-typical Witt vectors of length \(n+1\); and the inclusion of fixed points (resp. the map \(R\) in the fundamental cofibration sequence) induces the Frobenius map (resp. restriction map) under this isomorphism. The importance of topological cyclic homology lies in the fact that the cyclotomic trace from algebraic K-theory \(K@>\text{trc}>>TC\) often preserves important information. For instance, if \(A\to B\) is a map of FSPs inducing a surjection \(\pi_0(A)\to \pi_0(B)\) with nilpotent kernel, then the cyclotomic trace induces an equivalence between relative K-theory and relative \(TC\). This paper uses the profinite version of this fact for ordinary rings, proven by M…
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 19D55
zbMATH DE Number: 938856
zbMATH Keywords: topological cyclic homology
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: Novikov conjecture
zbMATH Keywords: Hochschild homology
zbMATH Keywords: cyclotomic spectrum
zbMATH Keywords: cyclotomic trace
MaRDI profile type: Publication
full work available at URL: 0040-9383(96)00003-1
OpenAlex ID: W2073256203
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description: scientific article; zbMATH DE number 938856