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On topological coHochschild homology and cotensor products

Jiaxi Zha

TL;DR

The paper develops a dual perspective to topological Hochschild homology by studying coHochschild homology (coTHH) in ∞-categories of modules over a connective E∞-ring spectrum. It establishes a central cotensor formula, showing coTHH^{𝒞}(M;C) ≃ M ⬜_{C⊗C^{op}} C, and then proves that cotensor products become symmetric monoidal under suitable connectivity, enabling higher coalgebra structures on coTHH and Morita-Takeuchi invariance. The results extend to coaugmented E_k-coalgebras, yielding a hierarchy coTHH^R: CoAlg_{E_k}(Mod_R)_{R/}^{≥2} → CoAlg_{E_{k-1}}(Mod_R) and a terminal coaction property for cocommutative coalgebras with S^1-action, with applications to coTHH computations and spectral sequences. Overall, the work provides a robust ∞-categorical framework for dualizing THH phenomena and derives invariance and structural results for coTHH in the setting of coalgebras and comodules over connective ring spectra.

Abstract

In this work, we first study the cotensor product of comodules in the $\infty$-category $\mathrm{Mod}_R$ for a connected $\mathbb{E}_{\infty}$-ring spectrum $R$. We then apply these results to analyze higher coalgebra structures of topological coHochschild homology (coTHH) and establish its Morita-Takeuchi invariance, which are precisely dual to the corresponding properties of topological Hochschild homology.

On topological coHochschild homology and cotensor products

TL;DR

The paper develops a dual perspective to topological Hochschild homology by studying coHochschild homology (coTHH) in ∞-categories of modules over a connective E∞-ring spectrum. It establishes a central cotensor formula, showing coTHH^{𝒞}(M;C) ≃ M ⬜_{C⊗C^{op}} C, and then proves that cotensor products become symmetric monoidal under suitable connectivity, enabling higher coalgebra structures on coTHH and Morita-Takeuchi invariance. The results extend to coaugmented E_k-coalgebras, yielding a hierarchy coTHH^R: CoAlg_{E_k}(Mod_R)_{R/}^{≥2} → CoAlg_{E_{k-1}}(Mod_R) and a terminal coaction property for cocommutative coalgebras with S^1-action, with applications to coTHH computations and spectral sequences. Overall, the work provides a robust ∞-categorical framework for dualizing THH phenomena and derives invariance and structural results for coTHH in the setting of coalgebras and comodules over connective ring spectra.

Abstract

In this work, we first study the cotensor product of comodules in the -category for a connected -ring spectrum . We then apply these results to analyze higher coalgebra structures of topological coHochschild homology (coTHH) and establish its Morita-Takeuchi invariance, which are precisely dual to the corresponding properties of topological Hochschild homology.

Paper Structure

This paper contains 6 sections, 21 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Orgnization
  3. Conventions
  4. coHochschild homology and cotensor product
  5. The monoidality of the cotensor product
  6. Applications to coTHH

Key Result

Theorem 1.1

Let $\mathcal{C}$ be a symmetric monoidal $\infty$-category which admits small limits, and let $C$ be a coalgebra in $\mathcal{C}$ and $M$ be a $C$-bicomodule. Then there is an equivalence

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 43 more