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Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

Ayelet Lindenstrauss, Birgit Richter

TL;DR

The work unifies reflexive homology and involutive Hochschild homology with equivariant Loday constructions for the group $C_2$, showing that under $2$-invertibility and flatness of the underlying module, these theories coincide with the homotopy groups of the $C_2$-equivariant Loday construction on the fixed-point Tambara functor evaluated at the trivial orbit. It extends to a relative setting over a commutative ground ring $k$ and elucidates the role of the flip circle $S^{\sigma}$ in producing a bar-model for the constructions. The paper also analyzes edge cases where $2$ is not invertible, showing failure of the identifications for $\underline{\mathbb{F}_2}^c$ and $\underline{\mathbb{Z}}^c$, and finally extends the results to associative rings with anti-involution via a modified norm structure. Together, these results provide a coherent framework linking algebraic and equivariant homologies through Loday-type constructions and their fixed-point data, with implications for Real/Hochschild-type theories and potential extensions to higher dihedral contexts.

Abstract

For associative rings with anti-involution several homology theories exists, for instance reflexive homology as studied by Graves and involutive Hochschild homology defined by Fernàndez-València and Giansiracusa. We prove that the corresponding homology groups can be identified with the homotopy groups of an equivariant Loday construction of the one-point compactification of the sign-representation evaluated at the trivial orbit, if we assume that $2$ is invertible and if the underlying abelian group of the ring is flat. We also show a relative version where we consider an associative $k$-algebra with an anti-involution where $k$ is an arbitrary ground ring.

Reflexive homology and involutive Hochschild homology as equivariant Loday constructions

TL;DR

The work unifies reflexive homology and involutive Hochschild homology with equivariant Loday constructions for the group , showing that under -invertibility and flatness of the underlying module, these theories coincide with the homotopy groups of the -equivariant Loday construction on the fixed-point Tambara functor evaluated at the trivial orbit. It extends to a relative setting over a commutative ground ring and elucidates the role of the flip circle in producing a bar-model for the constructions. The paper also analyzes edge cases where is not invertible, showing failure of the identifications for and , and finally extends the results to associative rings with anti-involution via a modified norm structure. Together, these results provide a coherent framework linking algebraic and equivariant homologies through Loday-type constructions and their fixed-point data, with implications for Real/Hochschild-type theories and potential extensions to higher dihedral contexts.

Abstract

For associative rings with anti-involution several homology theories exists, for instance reflexive homology as studied by Graves and involutive Hochschild homology defined by Fernàndez-València and Giansiracusa. We prove that the corresponding homology groups can be identified with the homotopy groups of an equivariant Loday construction of the one-point compactification of the sign-representation evaluated at the trivial orbit, if we assume that is invertible and if the underlying abelian group of the ring is flat. We also show a relative version where we consider an associative -algebra with an anti-involution where is an arbitrary ground ring.
Paper Structure (11 sections, 23 theorems, 94 equations)

This paper contains 11 sections, 23 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Equivariant Loday constructions
  3. Basic results about fixed point Tambara functors
  4. Working relative to a commutative ground ring
  5. Identifying $\mathcal{L}_{S^\sigma }^{C_2}(\underline R^\mathrm{fix})$
  6. Relating $\mathcal{L}_{S^\sigma}^{C_2}({\underline{R}})$ to reflexive homology
  7. Involutive Hochschild homology as a Loday construction
  8. The cases ${\underline{\mathbb{F}_2}}^c$ and ${\underline{\mathbb{Z}}}^c$
  9. Comparison for ${\underline{\mathbb{F}_2}}^c$
  10. Comparison for ${\underline{\mathbb{Z}}}^c$
  11. The case of rings and algebras with anti-involution

Key Result

Lemma 3.2

Assume that $M$ and $N$ are two abelian groups with involution and assume that $2$ is invertible in $M$ or in $N$. Then there is an equivalence of $C_2$-Mackey functors which is natural in $M$ and $N$. Here $C_2$ acts on $M\otimes N$ by the diagonal action. If, in addition, $M$ and $N$ are both commutative rings with involution, then $\underline M^\mathrm{fix}$, $\underline N^\mathrm{fix}$, and $

Theorems & Definitions (51)

  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • ...and 41 more