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Equivariant Witt Complexes and Twisted Topological Hochschild Homology

Anna Marie Bohmann, Teena Gerhardt, Cameron Krulewski, Sarah Petersen, Lucy Yang

TL;DR

This work extends the classical relationship between THH and Witt vectors to an equivariant setting by introducing and axiomatizing equivariant Witt complexes. The authors define C_n-equivariant Witt complexes over a C_n-Tambara functor and prove that the graded Green functors given by the equivariant homotopy groups of twisted THH, THH_{C_n}( R), assemble into such a structure when p∤n. They establish a natural map from equivariant Witt vectors to the zeroth equivariant THH and construct multiplicative lifts that recover the classical p^k-th power behavior in the non-equivariant limit (n=1). The results connect equivariant algebraic structures (Mackey and Tambara functors) with equivariant homotopy-theoretic invariants, paving the way for a robust equivariant de Rham–Witt framework in future work.

Abstract

The topological Hochschild homology of a ring (or ring spectrum) $R$ is an $S^1$-spectrum, and the fixed points of THH($R$) for subgroups $C_n\subset S^1$ have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work of Angeltveit, Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.

Equivariant Witt Complexes and Twisted Topological Hochschild Homology

TL;DR

This work extends the classical relationship between THH and Witt vectors to an equivariant setting by introducing and axiomatizing equivariant Witt complexes. The authors define C_n-equivariant Witt complexes over a C_n-Tambara functor and prove that the graded Green functors given by the equivariant homotopy groups of twisted THH, THH_{C_n}( R), assemble into such a structure when p∤n. They establish a natural map from equivariant Witt vectors to the zeroth equivariant THH and construct multiplicative lifts that recover the classical p^k-th power behavior in the non-equivariant limit (n=1). The results connect equivariant algebraic structures (Mackey and Tambara functors) with equivariant homotopy-theoretic invariants, paving the way for a robust equivariant de Rham–Witt framework in future work.

Abstract

The topological Hochschild homology of a ring (or ring spectrum) is an -spectrum, and the fixed points of THH() for subgroups have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work of Angeltveit, Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.
Paper Structure (13 sections, 13 theorems, 158 equations)

This paper contains 13 sections, 13 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Acknowledgements
  4. Topological Hochschild homology and Witt vectors
  5. Topological Hochschild homology
  6. Witt vectors and Witt complexes
  7. Equivariant Hochschild theories
  8. Twisted topological Hochschild homology
  9. Equivariant algebra
  10. Hochschild homology for Green functors
  11. Equivariant Witt vectors
  12. Equivariant Witt complexes
  13. Twisted topological Hochschild homology and equivariant Witt complexes

Key Result

Theorem 1.1

Let $n$ be a positive integer and $p$ an odd prime not dividing $n$. For $\underline{R}$ a $C_n$-Tambara functor such that $\underline{R}(C_n/C_m)$ is a $\mathbb{Z}_{(p)}$-algebra for each $C_m\subset C_n$, the graded Green functors $\{\underline{\pi}_*^{C_{p^mn}} \mathop{\mathrm{THH}}\nolimits_{C_n

Theorems & Definitions (85)

  • Theorem 1.1
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 75 more