Skip to main content

Topological cyclic homology of local fields

A Publisher Correction to this article was published on 08 August 2022

This article has been updated

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.

This is a preview of subscription content, access via your institution.

Change history

Notes

  1. Using Proposition 6.43, the argument of Lemma 6.36 (hence Remark 6.38) adapts to \(\tilde{E'}_{1,j,*}({\mathrm {TP}}({\mathcal {O}}_{K'});{{\mathbb {F}}_p})\) for all \(j\in {\mathbb {Z}}\).

References

  1. Antieau, B.: Periodic cyclic homology and derived de Rham cohomology. Ann. K-Theory 4(3), 505–519 (2019)

    MathSciNet  Article  Google Scholar 

  2. Bhatt, B.: \(p\)-adic derived de Rham cohomology. arXiv:1204.6560v1 (2012)

  3. Bhatt, B., Lurie, J.: The prismatization of p-adic formal schemes. arXiv:2201.06124v1 (2022)

  4. Bhatt, B., Morrow, M., Scholze, P.: Topological Hochschild homology and integral \(p\)-adic Hodge theory. Publ. Math. IHÉS 129, 199–310 (2019)

    MathSciNet  Article  Google Scholar 

  5. Bökstedt, M., Madsen, I.: Topological cyclic homology of the integers. Astérisque 226, 57–145 (1995)

    MATH  Google Scholar 

  6. Drinfeld, V.: Prismatization. arXiv:2005.04746v2 (2020)

  7. Hesselholt, L., Madsen, I.: On the \(K\)-theory of local fields. Ann. Math. 158, 1–113 (2003)

    MathSciNet  Article  Google Scholar 

  8. Krause, A., Nikolaus, T.: Bökstedt periodicity and quotients of DVR’s. arXiv:1907.03477v1 (2019)

  9. Lindenstrauss, A., Madsen, I.: Topological Hochschild homology of of number rings. Trans. Am. Math. Soc. 352, 2179–2204 (2000)

    MathSciNet  Article  Google Scholar 

  10. Liu, R., Wang, G.: Topological cyclic homology of locally complete intersections (in preparation)

  11. Loday, J.-L.: Cyclic Homology, Grundlehren der mathematischen Wissenschaften, vol. 301. Springer, Berlin (1992)

    Google Scholar 

  12. Lurie, J.: Higher Topos Theory, Annals of Mathematics Studies, vol. 170. Princeton University Press, Princeton (2009)

    Google Scholar 

  13. Lurie, J.: Elliptic Cohomology II: Orientations. http://people.math.harvard.edu/~lurie/papers/Elliptic-II.pdf

  14. Mathew, A., Naumann, N., Noel, J.: Nilpotence and descent in equivariant stable homotopy theory. Adv. Math. 305, 994–1084 (2017)

    MathSciNet  Article  Google Scholar 

  15. Mitchell, S.A.: The algebraic \(K\)-theory spectrum of a 2-adic local field. K-Theory 25, 1–37 (2002)

    MathSciNet  Article  Google Scholar 

  16. Mitchell, S.A.: K(1)-Local Homotopy, Iwasawa Theory and Algebraic \(K\)-Theory, Handbook of K-Theory, vol. 2, pp. 955–1010. Springer, Berlin (2005)

    MATH  Google Scholar 

  17. Nikolaus, T., Scholze, P.: On topological cyclic homology. Acta Math. 221, 203–409 (2018)

    MathSciNet  Article  Google Scholar 

  18. Rognes, J., Weibel, C.: Étale descent for two-primary algebraic \(K\)-theory of totally imaginary number rings. K-Theory 16, 101–104 (1999)

    MathSciNet  Article  Google Scholar 

  19. Rognes, J.: Topological cyclic homology of the integers at two. J. Pure Appl. Algebra 134, 219–286 (1999)

    MathSciNet  Article  Google Scholar 

  20. Thomason, R.W.: Algebraic \(K\)-theory and étale cohomology. Ann. Sci. Éc. Norm. Sup. 13, 437–552 (1985)

    Article  Google Scholar 

  21. Tsalidis, S.: On the topological cyclic homology of the integers. Am. J. Math. 119, 103–125 (1997)

    MathSciNet  Article  Google Scholar 

  22. Voevodsky, V.: On motivic cohomology with \({\mathbb{Z}}/\ell \)-coefficients. Ann. Math. 174, 401–438 (2011)

    MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Guozhen Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

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

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. \(\square \)

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

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