Motivic real topological Hochschild spectrum

Title: Motivic Real Topological Hochschild Spectrum
Author: Doosung Park
Source: arXiv:2305.04150v3 [math.KT]

Problem Statement

Topological Hochschild homology (THH) and its variants, such as topological cyclic homology (TC), are central to the computation of algebraic K-theory via the cyclotomic trace. However, THH lacks $\mathbb{A}^1$-invariance, even for regular schemes, which prevents the direct application of classical motivic homotopy theory (initiated by Morel and Voevodsky) to THH. While logarithmic motivic homotopy theory (logSH) has been developed to accommodate non-$\mathbb{A}^1$-invariant theories like THH and TC by replacing the interval $\mathbb{A}^1$ with the pair $(\mathbb{P}^n, \mathbb{P}^{n-1})$, the "real" analogues of these theories—Real Topological Hochschild homology (THR) and Real Topological Cyclic homology (TCR)—had not yet been systematically integrated into this framework.

The paper addresses the gap in defining a motivic real topological Hochschild spectrum that lives in a $\mathbb{Z}/2$-equivariant logarithmic motivic category. Specifically, it seeks to define THR for log schemes with involutions, establish its invariance properties necessary for motivic representability, and construct the associated motivic spectra.

Methodology

The author employs a strategy of extending existing definitions to the logarithmic and equivariant settings, followed by proving specific invariance properties required for motivic representability.

1. Dihedral Replete Bar Constructions:
   The paper begins by refining Rognes' replete bar construction for log rings. The author introduces dihedral replete bar constructions ($B_{drep}$) and dihedral nerves ($N_{di}$) for commutative monoids with involutions. This involves utilizing the theory of real simplicial sets and dihedral sets (crossed simplicial groups) to handle the $\mathbb{Z}/2$-equivariant structure.

2. Definition of THR and TCR:

   • THR: Defined for log rings with involutions as a coproduct in the $\infty$-category of normed algebras ($NAlg_{\mathbb{Z}/2}$):
     $$THR(A, P) := THR(A) \wedge_{S[B_{di}P]} S[B_{drep}P]$$
   • Extension to Schemes: THR is extended to separated log schemes with involutions by defining it as a sheafification in the strict isovariant étale topology.
   • TCR: Following the work of Quigley and Shah, the author defines real cyclotomic spectra and constructs TCR as the real topological cyclic homology of the resulting THR objects.

3. Motivic Invariance and Descent:
   The core technical effort involves proving that the defined THR satisfies:

   • Isovariant étale descent.
   • Invariance under passing to the associated log structure.
   • $(\mathbb{P}^n, \mathbb{P}^{n-1})$-invariance: This is the crucial step for motivic representability. The author computes $THR(\mathbb{P}^n, \mathbb{P}^{n-1})$ explicitly using cubes in $\infty$-categories to demonstrate that the map $THR(X) \to THR(X \times (\mathbb{P}^n, \mathbb{P}^{n-1}))$ is an equivalence.

4. Construction of Motivic Spectra:
   Using the periodicity of THR with respect to $\mathbb{P}^1 \wedge \mathbb{P}^\sigma$ (where $\mathbb{P}^\sigma$ is $\mathbb{P}^1$ with the involution $[x:y] \mapsto [y:x]$), the author constructs the $\infty$-category of prelogarithmic motivic $\mathbb{Z}/2$-spectra ($prelogSH^{\mathbb{Z}/2}(S)$). Within this category, the motivic real topological Hochschild spectrum ($THR$) and motivic real topological cyclic spectrum ($TCR$) are defined as $(\mathbb{P}^1 \wedge \mathbb{P}^\sigma)$-spectra.

Key Contributions and Results

• Definition of THR for Log Schemes: The paper provides a rigorous definition of THR for separated log schemes with involutions, utilizing dihedral replete bar constructions to handle the interaction between the log structure and the involution.
• Motivic Representability: The primary result is the proof that THR is $(\mathbb{P}^n, \mathbb{P}^{n-1})$-invariant for all $n \geq 1$. This allows THR to be represented as an object in the logarithmic motivic category.
• Localization and Blow-up Sequences:
  • Theorem 1.1: Establishes fiber sequences for THR and TCR associated with closed immersions of smooth schemes, involving the Thom space of the normal bundle and the blow-up. This serves as the localization sequence for THR.
  • Theorem 1.2: Proves that the squares induced by blow-ups along smooth centers are cartesian for both THR and TCR.
• Equivariant Motivic Spectra: The author constructs the $\infty$-category $prelogSH^{\mathbb{Z}/2}(S)$ and defines the periodic spectra $THR$ and $TCR$ within it.
• Relationship to Classical Theories:
  • Proposition 6.17: Shows that the "forgetful" functor $i^*$ (which restricts to schemes with trivial involutions) maps the motivic real spectra to the classical motivic spectra: $i^*THR \simeq THH$ and $i^*TCR \simeq TC$.
  • Corollary 4.9: Establishes that the fixed points of the motivic real spectrum recover the classical motivic THH: $THR(i^\sharp X)^{\mathbb{Z}/2} \simeq THH(X)$.

Significance

The paper successfully integrates Real Topological Hochschild homology into the framework of logarithmic motivic homotopy theory. By establishing the necessary invariance properties, it enables the application of motivic methods (such as purity, Gysin sequences, and blow-up formulas) to real K-theory and hermitian K-theory via the real trace method.

The work lays the groundwork for a "real refinement" of the Dundas-Goodwillie-McCarthy theorem, as hinted at in the introduction by the forthcoming work of Harpaz, Nikolaus, and Shah. It provides the technical infrastructure (the motivic spectra $THR$ and $TCR$) required to compute real and hermitian K-theory using motivic techniques, analogous to how THH and TC are used for algebraic K-theory. The paper does not claim to solve the computation of specific K-groups but rather provides the representable objects and structural theorems (localization, blow-up invariance) that make such computations feasible within the motivic framework.