Equivariant operations in topological Hochschild homology

TL;DR

The paper addresses the pronounced asymmetry between $THH(R)$ and $THC(R)$ by constructing an $S^1$-equivariant twisted associative model $THHC(R)_{S^1}$ that is equivalent to $THH(THC(R))$ and acts on $THH(R)_{S^1}$. It develops the necessary foundations via the multiplicative norm and the unframed cactus operad to handle $S^1$-twisting, then demonstrates a concrete calculation in the basic case $R=H\mathbb{F}_p$, yielding explicit descriptions of $THH(THC(H\mathbb{F}_p))^{\mathbb{Z}/p^{r-1}}_*$ and $TR(THC(H\mathbb{F}_p))_*$. The results express these groups in terms of $F(H\mathbb{Z}/p^r,H\mathbb{Z}/p^r)_*$, divided power algebras $\Gamma_{\mathbb{Z}/p^r}(\rho)$, and generators with precise degrees, reflecting a structured, operation-theoretic picture of higher THH. The work suggests connections to operadic dualities and p-adic cohomology theories, and provides a template for further calculations in more complex cases. Overall, it advances understanding of how THH and THC interact under equivariant twisting and lays groundwork for broader applications in algebraic K-theory and prismatic contexts.

Abstract

We observe a new equivariant relationship between topological Hochschild homology and cohomology. We also calculate the topological Hochschild homology of the topological Hochschild cohomology of a finite prime field, which can be viewed as a certain ring of structured operations in this case.

Figures (4)

• Figure 1: An example of a cactus datum
• Figure 2: The cactus graph corresponding to a cactus datum
• Figure 3: Cactus composition: Example 1
• Figure 4: Cactus composition: Example 2

Theorems & Definitions (13)

• Theorem 1
• Theorem 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof