×

On the \(K\)-theory of local fields. (English) Zbl 1033.19002

[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
This fascinating paper offers one of the ultimate calculations of algebraic K-theory of fields through the topological cyclic homology of M. Bökstedt, W. C. Hsiang and I. Madsen [Invent. Math. 111, 465–539 (1993; Zbl 0804.55004)], and provides interesting algebraic interpretations of the calculation.
Let \(K\) be a complete discrete valuation field field of characteristic zero with perfect residue field \(k\) of characteristic \(p>2\). This paper verifies the Lichtenbaum–Quillen conjecture for \(K\): for \(s\geq 1\) and \(v\geq 1\) there are natural isomorphisms \[ \begin{aligned} K_{2s}(K,\mathbb Z/p^v)&\cong H^0(K,\mu^{\otimes s}_{p^v})\oplus H^2(K,\mu^{\otimes (s+1)}_{p^v})\\ K_{2s-1}(K,\mathbb Z/p^v)&\cong H^1(K,\mu^{\otimes s}_{p^v}). \end{aligned} \] When \(k\) is finite the Galois groups on the right can be effectively calculated, and in this case the authors also give the homotopy type of the algebraic K-theory space of \(K\) after \(p\)-completion. Away from the characteristic, the K-groups are accessible through rigidity results of Gabber and Suslin, and the calculation of Quillen of the K-theory of finite fields.
The result is obtained through trace methods. Let \(\mathcal C\) be a category with cofibrations and weak equivalences in the sense of Waldhausen. The cyclotomic trace is a natural transformation from algebraic K-theory to a variant of topological cyclic homology \(tr\colon K(\mathcal C)\to TC(\mathcal C;p)\) which in certain situations is close to being an equivalence. For the applications in the current paper the construction of B. I. Dundas and R. McCarthy [J. Pure Appl. Algebra 109, 231–294 (1996; Zbl 0856.19004)] suffices. In particular, if \(\mathcal C\) is the category \(\mathcal P_R\) of finitely generated projective modules over a ring \(R\), this construction agrees with the original, up to the equivalence \(TC(R) \simeq TC(\mathcal P_R)\).
Let \(A\) be the valuation ring of \(K\). The localization sequence in algebraic K-theory derives in this situation from applying Waldhausen’s \(S\)-construction to the sequence \[ C^b_z(\mathcal P_A)^q\subseteq C^b_z(\mathcal P_A)\to C^b_q(\mathcal P_A) \] of categories with cofibrations and weak equivalences. Here \(C^b_z(\mathcal P_A)\) is the category of bounded chain complexes in \(\mathcal P_A\) with injections as cofibrations and chain maps inducing isomorphism in homology as weak equivalences, \(C^b_z(\mathcal P_A)^q\) is the subcategory of chain complexes with torsion homology, and \(C^b_q(\mathcal P_A)\) equals \(C^b_z(\mathcal P_A)\) as a category, but with weak equivalences those chain maps inducing isomorphism in rational homology.
Extending techniques of Waldhausen, Thomason, McCarthy and the reviewer, the authors then show that the trace map induces a map of cofiber sequences \[ \begin{tikzcd} K(C^b_z(\mathcal P_A)^q)\ar[r]\ar[d,"\simeq" '] & K(C^b_z(\mathcal P_A))\ar[r]\ar[d,"\simeq"] & K(C^b_q(\mathcal P_A)) \ar[d]\\ TC(C^b_z(\mathcal P_A)^q;p)\ar[r] & TC(C^b_z(\mathcal P_A);p)\ar[r] & TC(C^b_q(\mathcal P_A);p) \,, \end{tikzcd} \] and that the first two vertical maps can be identified with the trace maps \(K(k)\to TC(k;p)\) and \(K(A)\to TC(A;p)\), which by earlier work of the authors [L. Hesselholt and I. Madsen, Topology 36, 29–101 (1997; Zbl 0866.55002)] are known to induce isomorphisms of homotopy groups with \(\mathbb Z/p^v\)-coefficients in non-negative degrees.
Hence the problem is reduced to analyzing \(TC(A| K;p)=TC(C^b_q(\mathcal P_A);p)\), which occupies the rest of the paper. Topological Hochschild homology \(T(-)\) comes with an action by the circle group \(\mathbb T\), and the authors let \(\text{TR}^{n+1}(-;p)\) be the fixed point spectrum \(T(-)^{C_{p^{n}}}\) where \(C_m\subseteq \mathbb T\) is the cyclic subgroup of order \(m\). Topological cyclic homology is then constructed as a homotopy limit of the \(\text{TR}^n\) under the “Restriction” and “Frobenius” maps.
The authors prove that \(\pi_*T(A| K)\) has a “log differential graded ring” structure in the sense of K. Kato [in: Algebraic analysis, geometry, and number theory, Proc. Conf. 1988, Baltimore/MD, 191–224 (1989; Zbl 0776.14004)]. This consists partly of the obvious map \(A\cap K^\times\subseteq A\to \pi_0T(A| K)\), but more interestingly of the following data. Consider the self-equivalences \(\text{Aut}(A)\cong A\cap K^\times\) in \(C^b_q(\mathcal P_A)\) of the chain complex with only non-zero entry the rank one module \(A\) placed in degree \(0\). Then the map \(\Sigma^\infty B \text{Aut}(A)_+\to K(C^b_q(\mathcal P_A))\) composed with the trace gives a map \(d\text{log}\colon A\cap K^\times\to \pi_1 T(A| K)\). These maps, together with Connes’ operator \(B\) (coming from the cyclic action) gives the desired structure on \(\pi_*T(A| K)\). Together with the abstract calculation of \(\pi_*T(A)\) by A. Lindenstrauss and I. Madsen [Trans. Am. Math. Soc. 352, 2179–2204 (2000; Zbl 0949.19003)] this allows the authors to identify \(\pi_*T(A| K)\) in terms of a universal log differential graded ring.
The log differential graded ring structure also lifts to \( \pi_*\text{TR}^{\bullet}(A| K)\). Together with the Restriction, Frobenius and “Verschiebung” maps this qualifies \(\pi_*\text{TR}^{\bullet}(A| K)\) as a “log Witt complex”. The authors show that there is a universal log Witt complex which they call the deRham-Witt complex with log poles \(W_{{\bullet}}\omega^*_{(A,A\cap K^\times)}\) extending the results of O. Hyodo and K. Kato [in: Périodes \(p\)-adiques, Astérisque 223, 221–268 (1994; Zbl 0852.14004)] from \(\mathbb F_p\)-algebras to \(\mathbb Z_{(p)}\)-algebras.
The main result of the paper is that there is a canonical isomorphism \[ W_{{\bullet}}\omega^*_{(A,A\cap K^\times)}\otimes S_{\mathbb Z/p^v}(\mu_{p^v})\cong \pi_*\text{TR}^{\bullet}(A| K;p,\mathbb Z/p^v). \] The identification of \(\pi_*\text{TR}^n(A| K;p)\) follows by the usual comparison with the Tate-spectrum \(\text{TR}^{n}(A| K;p)\to \hat{\mathbb H}(C_{p^n};T(A| K))\). This map is an equivalence (in positive degrees) for \(n=1\), and hence by S. Tsalidis [Topology 37, 913–934 (1998; Zbl 0922.19001)] for all \(n\). An analysis of the spectral sequence converging to the homotopy groups of the Tate spectrum then gives the desired homotopy groups.

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
11S70 \(K\)-theory of local fields
PDFBibTeX XMLCite
Full Text: DOI arXiv