Topological cyclic homology of local fields

Abstract

We introduce a new approach to determining the structure of topological cyclic homology by means of a descent spectral sequence. We carry out the computation for a p-adic local field with ${{\mathbb {F}}_p}$-coefficients, including the case $p=2$ which was only covered by motivic methods except in the totally unramified case.

Appendix A: A variant of Hochschild–Kostant–Rosenberg

The goal of this appendix is to prove the following theorem.

Theorem A.1

Let R be a commutative ring over ${\mathbb {Z}}_p$, and let I be a locally complete intersection ideal of R. Let $A=R/I$. Suppose that R is I-separated and A is p-torsion free. Then as filtered rings, the periodic cyclic homology $\mathrm {HP}_0(A/R)$ is canonically isomorphic to the completion of $D_R(I)$ with respect to the Nygaard filtration. Moreover, the Tate spectral sequence for $\mathrm {HP}_0(A/R)$ collapses at the $E^2$-term. Consequently, there is a canonical isomorphism of graded rings

$$\begin{aligned} \mathrm {HH}_*(A/R) \cong \Gamma _A( I/I^2). \end{aligned}$$

In the following, all tensor products are taken in the derived category of R-modules.

Proof

We first assume $I=(a)$ for a non-zero divisor $a\in R$. Let $D=R[x]/(x^2)$ be the commutative DG-algebra over R with $|x|=1$ and $d(x) = a$. Then D is a flat resolution of A over R. Thus $\mathrm {HH}(A/R)$ can be computed by the normalized Hochschild complex (cf. [11, §1.1.4]) as follows. Let ${\bar{D}} = D/R$. Set the double complex $C_*(D)$ as

$$\begin{aligned} \dots \rightarrow D\otimes {\bar{D}}^{\otimes 2} \xrightarrow {b} D\otimes {\bar{D}}\xrightarrow {b} D. \end{aligned}$$

The boundary map $b:D\otimes {\bar{D}}^{\otimes n}\rightarrow D\otimes {\bar{D}}^{\otimes n-1}$ is given by the formula $\sum _{i=0}^n (-1)^i {\bar{d}}_i$ with $d_i:D^{\otimes n+1}\rightarrow D^{\otimes n}$, where

$$\begin{aligned}\begin{gathered} d_0 = m\otimes \mathrm {id} \otimes \dots \otimes \mathrm {id}, \\ d_1 =\mathrm {id} \otimes m \otimes \dots \otimes \mathrm {id}, \\ \dots \\ d_n = (m\otimes \mathrm {id} \otimes \dots \otimes \mathrm {id})\circ t'; \end{gathered}\end{aligned}$$

here $m:D\otimes D\rightarrow D$ is the multiplication and $t': D^{\otimes n+1}\rightarrow D^{\otimes n+1}$ is the cyclic permutation operator sending the last factor to the first.

A short computation shows that $b(1\otimes x^{\otimes n}) = 1\otimes x^{\otimes n-1}+(-1)^{n+n-1}1\otimes x^{\otimes n-1}=0$. It follows that for $n\ge 0$, $\mathrm {HH}_{2n+1}(A/R) = 0$ and $\mathrm {HH}_{2n}(A/R) \cong A$, where the latter is generated by the element represented by the cycle $1\otimes x^{\otimes n}$. In particular, the graded ring $\mathrm {HH}_{*}(A/R)$ is p-torsion free as A is p-torsion free.

Next we determine $\mathrm {HP}_*(A/R)$. By [11, §2.1.9], this may be computed by the double complex

(A.2)

Here $B:D\otimes {\bar{D}}^{\otimes n} \rightarrow D\otimes {\bar{D}}^{\otimes n+1}$ is given by the formular $s \circ N$, where

$$\begin{aligned} N= 1+t+\dots +t^n \end{aligned}$$

with $t=(-1)^nt'$ and $s=u\otimes \mathrm {id}:D^{\otimes n+1}\rightarrow D^{\otimes n+2}$ with $u:R\rightarrow D$ being the unit map. Note that the spectral sequence associated to the double complex (A.2) is the Tate spectral sequence for $\mathrm {HP}(A/R)$, which collapses at the $E^2$-term because everything is concentrated in even degrees. It follows that the associated graded algebra of $\mathrm {HP}_0(A/R)$ with respect to the Nygaard filtration is isomorphic to $\mathrm {HH}_{*}(A/R)$. This yields that $\mathrm {HP}_0(A/R)$ is p-torsion free.

We claim that the natural map $R\rightarrow \mathrm {HP}_0(A/R)$ extends uniquely to a map

$$\begin{aligned} D_I(R)\rightarrow \mathrm {HP}_0(A/R) \end{aligned}$$

(A.3)

as filtered rings. The uniqueness follows from the fact that $\mathrm {HP}_0(A/R)$ is p-torsion free. For the existence, first note that the image of $a^n$ is represented by the same element in the above double complex. Since $d(a^{n-1}x) = a^n$, b is trivial, $B(a^{n-1}x) = a^{n-1}\otimes x$, we get that $a^n$ is homologous to $a^{n-1}\otimes x$ up to a sign. By induction, we deduce that for $0\le m\le n$, $a^n$ is homologous to $m!a^{n-m}\otimes x^{m}$ up to a sign. In particular, $a^n$ is homologous to $n!\otimes x^{\otimes n}$ up to a sign. This proves the claim.

To proceed, first note that R is p-torsion free by the assumption that it is I-separated and A is p-torsion free. This implies that the associated graded algebra of $D_I(R)$ is isomorphic to $\Gamma _A(I/I^2)$. Now taking the associated graded of the natural map $D_I(R)\rightarrow \mathrm {HP}_0(A/R)$, we get the map

$$\begin{aligned} \Gamma _A(I/I^2)\cong A\langle {\bar{a}}\rangle \rightarrow \mathrm {HH}_*(A/R). \end{aligned}$$

(A.4)

By what we have proved, one easily checks that ${\bar{a}}^{[n]}$ maps to the element represented by the cycle $1\otimes x^{\otimes n}$. This implies that (A.4) is an isomorphism of graded rings. It follows that (A.3) becomes an isomorphism after taking completion with respect to the Nygaard filtration on the source.

Now suppose that I is generated by a regular sequence $a_1,\dots ,a_n$. By the fact that R is I-separated and A is p-torsion free, we first deduce that $A_i=R/(a_i)$ is p-torsion free for all i. Using the previous case and the natural isomorphism $A\cong A_1\otimes \dots \otimes A_n$, we get the isomorphism

$$\begin{aligned} \mathrm {HH}_{*}(A/R) \cong \mathrm {HH}_{*}(A_1/R)\otimes \dots \otimes \mathrm {HH}_{*}(A_n/R)\cong \Gamma _A(I/I^2), \end{aligned}$$

where the second isomorphism follows from

$$\begin{aligned} \Gamma _{A_1}((a_1)/(a_1)^2)\otimes \dots \otimes \Gamma _{A_n}((a_n)/(a_n)^2)\cong \Gamma _A(I/I^2). \end{aligned}$$

This implies that $\mathrm {HH}_{*}(A/R)$ is p-torsion free and the Tate spectral sequence for $\mathrm {HP}(A/R)$ collapses at the $E^2$-term. We thus get that $\mathrm {HP}_0(A/R)$ is p-torsion free. Moreover, using the natural map

$$\begin{aligned} \mathrm {HP}_0(A_1/R)\otimes \dots \otimes \mathrm {HP}_0(A_n/R) \rightarrow \mathrm {HP}_0(A/R) \end{aligned}$$

and the isomorphism

$$\begin{aligned} D_{(a_1)}(R)\otimes \dots \otimes D_{(a_n)}(R)\cong D_I(R), \end{aligned}$$

we obtain the map $D_I(R)\rightarrow \mathrm {HP}_0(A/R)$, which uniquely extends the natural map $R\rightarrow \mathrm {HP}_0(A/R)$. By the same argument as in the previous case, we deduce that it becomes an isomorphism after taking completion on the source.

For general I, by Zariski descent and the previous case, we first have that $\mathrm {HH}_{*}(A/R)$ is concentrated on even degrees. It follows that the Tate spectral sequence for $\mathrm {HP}(A/R)$ collapses at the $E^2$-term. Similarly, we get that $\mathrm {HP}_0(A/R)$ is p-torsion free and $R\rightarrow \mathrm {HP}_0(A/R)$ uniquely extends to $D_I(R)\rightarrow \mathrm {HP}_0(A/R)$. Moreover, by Zariski descent and the previous case, the associated graded map $\Gamma _A(I/I^2)\rightarrow \mathrm {HH}_{*}(A/R)$ is an isomorphism. This concludes the proof. $\square$