On the \(K\)-theory of finite algebras over Witt vectors of perfect fields
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 McCarthy [to appear in Acta Math.]. Note that the integral version of \(TC\) in the paper under review is not the same as the one introduced by Goodwillie, and which is needed to prove the integral equivalence of the relative cyclotomic trace; but the two functors agree after profinite completion. In the second part of the paper, the authors consider the following situation. Let \(k\) be a perfect field of characteristic \(p>0\), let \(W(k)\) be the ring of \(p\)-typical Witt vectors and let \(A\) be a \(W(k)\)-algebra which is finitely generated as a \(W(k)\)-module. The authors calculate \(TC(k)\), and after taking the connective cover and \(p\)-completing, the cyclotomic trace is an equivalence \[ K(k)^\wedge_p@>\sim>>TC(k)[0,\infty)^\wedge_p \simeq H(\mathbb{Z}_p). \] Using arguments parallel to the analysis of Suslin and Yufryakov they find that \(K(A)^\wedge_p @>\sim>>\text{holim}_\leftarrow K(A/p^i)^\wedge_p\) is an equivalence, and they prove the corresponding statement for \(TC\). From this and the theorem of McCarthy, they show the conjecture of the last author, that \[ K(A)^\wedge_p@>\sim>>TC(A)^\wedge_p \] is an equivalence; the non-obvious part being the start of the induction that the cyclotomic trace induces an equivalence for \(A/p\). As an application they calculate \(TC(k[t]/(t^2))\), and so also \(K(k[t]/(t^2))\). In a later paper, the authors calculate \(K(k[t]/(t^n))\) for all \(n\).
scientific article; zbMATH DE number 938856