## 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.

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## Change history

### 08 August 2022

A Correction to this paper has been published: https://doi.org/10.1007/s00222-022-01147-4

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## Acknowledgements

Both authors are very grateful to Lars Hesselholt for suggesting this project, and for his tremendous help during preparation of this paper. Especially, he proposed to us using the descent along the base change map (1.1) in topological Hochschild homology. Thanks also to Benjamin Antieau, Dustin Clausen, Bjørn Ian Dundas, Jingbang Guo, Marc Levine, Thomas Nikolaus, Paul Arne Østvær, John Rognes and Longke Tang for valuable discussions and suggestions. Finally, we would like to thank anonymous referees for many helpful comments that helped us correct and improve earlier versions of this paper.

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R. Liu is partially supported by the National Natural Science Foundation of China under Agreement No. NSFC-11725101 and the Tencent Foundation. G. Wang is partially supported by the Shanghai Rising-Star Program under Agreement No. 20QA1401600 and Shanghai Pilot Program for Basic Research-FuDan University 21TQ1400100(21TQ002) and the National Natural Science Foundation of China under Agreement No. NSFC-11801082.

## Appendix A: A variant of Hochschild–Kostant–Rosenberg

### 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

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

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

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

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

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

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

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

where the second isomorphism follows from

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

and the isomorphism

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. \(\square \)

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Liu, R., Wang, G. Topological cyclic homology of local fields.
*Invent. math.* (2022). https://doi.org/10.1007/s00222-022-01134-9

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DOI: https://doi.org/10.1007/s00222-022-01134-9