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The algebraic structure of twisted topological Hochschild homology

Danika Van Niel

TL;DR

The paper develops the algebraic structure of twisted topological Hochschild homology for cyclic-group equivariant settings, extending the Bökstedt spectral sequence to the twisted context under flatness. It establishes that $THH_{C_p}(R)$ is a commutative $R$-algebra in $C_p$-spectra and analyzes the associated twisted Bökstedt spectral sequence as a commutative $oxed{E}_*(R)$-algebra under suitable hypotheses, including a detailed description of module structures via Mackey-field coefficients. Concrete computations are carried out for $C_2$-twisted THH of the Real bordism spectrum $MU_{oldsymbol{ eal}}$ with a specific Mackey-field coefficient, and a general formula for $oxed{E}_*(THH_{C_p}(A))$ is given when $A$ is a cofibrant, associative ring $C_p$-spectrum and $E$ is a geometric $C_p$-spectrum. The work connects equivariant homotopy-theoretic inputs (Mackey functors, Green functors, and geometric fixed points) with computable algebraic structures, enabling new computations that relate twisted THH to classical THH via fixed-point reductions and orientation phenomena. Overall, the results provide a robust framework for evaluating twisted THH and its spectral sequences, with implications for equivariant K-theory and cyclotomic trace approaches in equivariant contexts.

Abstract

Topological Hochschild homology (THH) is an invariant of ring spectra developed by Bökstedt. In recent years many equivariant analogues to THH have emerged. One example is twisted THH which is an invariant of $C_n$-equivariant ring spectra developed by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell. In this paper, we study the algebraic structure of twisted THH, and perform some computations. Specifically, we compute $C_2$-twisted THH of the Real bordism spectrum and show that the $C_p$-twisted THH of geometric ring $C_p$-spectra reduces to a computation of classical THH. We extend the algebraic structure of twisted THH to the twisted Bökstedt spectral sequence of Adamyk, Gerhardt, Hess, Klang, and Kong. We show that, under appropriate flatness conditions and for $R$ a commutative ring $C_p$-spectrum, the $C_p$-twisted Bökstedt spectral sequence is a spectral sequence of commutative $\underline{E}_\star(R)$-algebras.

The algebraic structure of twisted topological Hochschild homology

TL;DR

The paper develops the algebraic structure of twisted topological Hochschild homology for cyclic-group equivariant settings, extending the Bökstedt spectral sequence to the twisted context under flatness. It establishes that is a commutative -algebra in -spectra and analyzes the associated twisted Bökstedt spectral sequence as a commutative -algebra under suitable hypotheses, including a detailed description of module structures via Mackey-field coefficients. Concrete computations are carried out for -twisted THH of the Real bordism spectrum with a specific Mackey-field coefficient, and a general formula for is given when is a cofibrant, associative ring -spectrum and is a geometric -spectrum. The work connects equivariant homotopy-theoretic inputs (Mackey functors, Green functors, and geometric fixed points) with computable algebraic structures, enabling new computations that relate twisted THH to classical THH via fixed-point reductions and orientation phenomena. Overall, the results provide a robust framework for evaluating twisted THH and its spectral sequences, with implications for equivariant K-theory and cyclotomic trace approaches in equivariant contexts.

Abstract

Topological Hochschild homology (THH) is an invariant of ring spectra developed by Bökstedt. In recent years many equivariant analogues to THH have emerged. One example is twisted THH which is an invariant of -equivariant ring spectra developed by Angeltveit, Blumberg, Gerhardt, Hill, Lawson, and Mandell. In this paper, we study the algebraic structure of twisted THH, and perform some computations. Specifically, we compute -twisted THH of the Real bordism spectrum and show that the -twisted THH of geometric ring -spectra reduces to a computation of classical THH. We extend the algebraic structure of twisted THH to the twisted Bökstedt spectral sequence of Adamyk, Gerhardt, Hess, Klang, and Kong. We show that, under appropriate flatness conditions and for a commutative ring -spectrum, the -twisted Bökstedt spectral sequence is a spectral sequence of commutative -algebras.
Paper Structure (21 sections, 41 theorems, 65 equations)

This paper contains 21 sections, 41 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Notation and conventions
  4. Acknowledgments
  5. Cp-Mackey functors
  6. Green functors
  7. Mackey fields
  8. Equivariant homotopy groups of Eilenberg-Mac Lane spectra
  9. Equivariant (topological) Hochschild homology
  10. Hochschild homology for Green functors
  11. Twisted topological Hochschild homology
  12. Computation of twisted THH of the Real bordism spectrum
  13. The trivial action
  14. Real oriented
  15. The computation
  16. ...and 6 more sections

Key Result

Theorem 1.1

For $\underline{F}$ as above the $RO(C_2)$-graded equivariant homology of $\mathop{\mathrm{THH}}\nolimits_{C_2}(MU_\mathbb{R})$ with coefficients in $\underline{F}$ is as an $H\underline{F}_{\star}$-module. Here $|\beta_i| = i\rho$ and $|z_i| = 1 + i\rho$.

Theorems & Definitions (107)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proposition 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5
  • ...and 97 more