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Trace methods for coHochschild homology

Sarah Klanderman, Maximilien Péroux

TL;DR

This work extends Dennis trace-type methods from rings to coalgebras by introducing coalgebraic K-theories $K^c(C)$ and $G^c(C)$ and by defining level-zero traces to $\mathrm{coHH}_0(C)$, realized as a bicategorical shadow. It builds both discrete and derived frameworks, establishing that $\mathrm{coHH}$ is a shadow on the bicategory of bicomodules and is invariant under Morita–Takeuchi equivalence in the derived setting, including a cobar-model for the derived cotensor product. Two concrete trace refinements are developed: the Hattori–Stallings cotrace and the colinear Hattori–Stallings trace, connecting corank/rank data to $\mathrm{coHH}_0(C)$ and its dual. Collectively, the paper provides a principled coalgebraic analogue of the Dennis trace, with potential implications for understanding ring K-theory via coalgebraic constructions and for extending traces to higher levels $K_n^c(C)\to (\mathrm{coHH}_n(C))^*$.

Abstract

Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic $K$-theory via the Dennis trace. At level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this paper, we introduce new algebraic $K$-theories of coalgebras and obtain coalgebraic refinements of the Hattori-Stallings trace that connect these algebraic $K$-theories to coHochschild homology (the invariant analogous to Hochschild homology but for coalgebras). We employ bicategorical methods of Ponto to show that coHochschild homology is a shadow. Consequently, we obtain that coHochschild homology is Morita-Takeuchi invariant.

Trace methods for coHochschild homology

TL;DR

This work extends Dennis trace-type methods from rings to coalgebras by introducing coalgebraic K-theories and and by defining level-zero traces to , realized as a bicategorical shadow. It builds both discrete and derived frameworks, establishing that is a shadow on the bicategory of bicomodules and is invariant under Morita–Takeuchi equivalence in the derived setting, including a cobar-model for the derived cotensor product. Two concrete trace refinements are developed: the Hattori–Stallings cotrace and the colinear Hattori–Stallings trace, connecting corank/rank data to and its dual. Collectively, the paper provides a principled coalgebraic analogue of the Dennis trace, with potential implications for understanding ring K-theory via coalgebraic constructions and for extending traces to higher levels .

Abstract

Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic -theory via the Dennis trace. At level zero, the Dennis trace is induced by the Hattori-Stallings trace. In this paper, we introduce new algebraic -theories of coalgebras and obtain coalgebraic refinements of the Hattori-Stallings trace that connect these algebraic -theories to coHochschild homology (the invariant analogous to Hochschild homology but for coalgebras). We employ bicategorical methods of Ponto to show that coHochschild homology is a shadow. Consequently, we obtain that coHochschild homology is Morita-Takeuchi invariant.
Paper Structure (11 sections, 43 theorems, 139 equations)

This paper contains 11 sections, 43 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Coalgebraic variations of the Hattori--Stallings trace
  3. The Hattori--Stallings trace
  4. The Hattori--Stallings cotrace
  5. The colinear Hattori--Stallings trace
  6. Bicategories of bicomodules
  7. CoHochschild homology as a shadow
  8. Dualizable bicomodules
  9. Bicategorical traces
  10. Morita--Takeuchi invariance
  11. Homotopy theory of connective bicomodules

Key Result

Theorem 1.2

Let $\mathbbm{k}$ be a commutative ring with global dimension zero. The zeroth coHochschild homology $\mathrm{coHH}_0$ defines a shadow on the bicategory of $\mathbbm{k}$-coalgebras and bicomodules with their relative cotensor products.

Theorems & Definitions (141)

  • Theorem 1.2: Theorem \ref{['thm: coHH zero is a shadow']}
  • Definition 1.4
  • Theorem 1.5: Theorem \ref{['thm: the corank as a theorem']}
  • Definition 1.6
  • Theorem 1.7: Theorem \ref{['thm: the second trace on K-theory']}
  • Theorem 1.8: Theorem \ref{['theorem: coHH is shadow (derived)']}
  • Theorem 1.9: Theorem \ref{['thm: coHH is morita invariant']}
  • Definition 1.10
  • Definition 1.11
  • Definition 1.12
  • ...and 131 more