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<p>Hochschild homology is a classical invariant of rings that plays an important role because of its connection to algebraic <i>K</i>-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 <i>K</i>-theories of coalgebras and obtain coalgebraic refinements of the Hattori–Stallings trace that connect these algebraic <i>K</i>-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.</p>

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Trace methods for coHochschild homology

  • 期刊论文

  • 发表日期：

    2025年3月14日

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  • Sarah Klanderman,
  • Maximilien Péroux

摘要

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.