Syntomic cohomology and real topological cyclic homology

Technical Summary: Syntomic Cohomology and Real Topological Cyclic Homology

Problem Statement
The paper addresses the computational gap in hermitian and real K-theories, particularly in contexts where 2 is not invertible. While the cyclotomic trace and the Dundas–Goodwillie–McCarthy theorem have established topological cyclic homology (TC) as a powerful tool for computing algebraic K-theory, analogous tools for real K-theory ($KR$) and hermitian K-theory have remained underdeveloped. Recent advances in syntomic cohomology and prismatic cohomology (Bhatt, Morrow, Scholze) have provided motivic filtrations on $TC$ with graded pieces identified as syntomic cohomology $\mathbb{Z}_p(i)$. However, a parallel structural understanding for real topological cyclic homology ($TCR$) and real topological Hochschild homology ($THR$) was lacking. The paper aims to bridge this gap by establishing a relationship between syntomic cohomology and real topological cyclic homology, thereby enabling the application of syntomic techniques to real and hermitian K-theories.

Methodology
The author employs a combination of equivariant stable homotopy theory, motivic filtrations, and descent theory. The core methodological steps include:

1. Motivic Filtrations on Real Spectra: The paper defines motivic filtrations on $THR$, $TCR^-$, $TPR$, and $TCR$. This involves constructing natural complete exhaustive multiplicative filtrations on these spectra for quasisyntomic rings.
2. Strongly Even Spectra and Slice Filtrations: A crucial technical component is the introduction of the "regular slice filtration" (based on Ullman's work) in the category of $\mathbb{Z}/2$-spectra ($Sp^{\mathbb{Z}/2}$). The author proves that for quasiregular semiperfectoid rings $S$, $THR(S; \mathbb{Z}_p)$ is "strongly even." This property implies that the slice filtration behaves predictably, with non-vanishing slices only in specific degrees, allowing for the identification of graded pieces.
3. Sheaf Properties and Descent: The paper establishes that the presheaves of real Hochschild homology and its companions are quasisyntomic sheaves. This allows the reduction of computations from general quasisyntomic rings to the more tractable case of quasiregular semiperfectoid rings via descent.
4. Completeness of Filtrations: The author proves the completeness of the real Hochschild–Kostant–Rosenberg (HKR) filtration under specific conditions (e.g., surjectivity of the relative Frobenius), ensuring that the spectral sequences associated with these filtrations converge strongly.
5. Spectral Sequence Computations: Using the established filtrations, the paper constructs RO($\mathbb{Z}/2$)-graded multiplicative spectral sequences to compute the homotopy groups of $TCR$ for specific rings, such as truncated polynomial algebras $k[x]/x^e$ over perfect fields of characteristic 2.

Key Contributions and Results

• Theorem 1.1 (Main Structural Result): For a quasisyntomic ring $A$ with trivial involution, there exist natural complete multiplicative filtrations on $THR(A; \mathbb{Z}_p)$, $TCR^-(A; \mathbb{Z}_p)$, $TPR(A; \mathbb{Z}_p)$, and $TCR(A; \mathbb{Z}_p)$. The $n$-th graded pieces are identified as equivariant suspensions of syntomic cohomology and related prismatic objects:

  • $gr_n THR(A; \mathbb{Z}_p) \simeq \Sigma^{n+n\sigma} \iota N_{<n} \widehat{\Delta}_A\{n\}$
  • $gr_n TCR^-(A; \mathbb{Z}_p) \simeq \Sigma^{n+n\sigma} \iota N_{\ge n} \widehat{\Delta}_A\{n\}$
  • $gr_n TPR(A; \mathbb{Z}_p) \simeq \Sigma^{n+n\sigma} \iota \widehat{\Delta}_A\{n\}$
  • $gr_n TCR(A; \mathbb{Z}_p) \simeq \Sigma^{n+n\sigma} \iota \mathbb{Z}_p(n)(A)$
    Here, $\iota$ is the left adjoint to the fixed-point functor, $\widehat{\Delta}$ denotes Nygaard-completed prismatic cohomology, and $\mathbb{Z}_p(n)$ is syntomic cohomology.

• Theorem 1.2 (Strong Evenness): For a quasiregular semiperfectoid ring $S$ with trivial involution, $THR(S; \mathbb{Z}_p)$ is strongly even. This implies natural equivalences relating the slices of $THR$ to the homotopy groups of $THH$, specifically $P^{2n}_{2n} THR(S; \mathbb{Z}_p) \simeq \Sigma^{n+n\sigma} H\pi_{2n} THH(S; \mathbb{Z}_p)$ and vanishing odd slices.

• Theorem 1.3 (Sheaf Property): The presheaves $THR(-; \mathbb{Z}_p)$, $TCR^-(-; \mathbb{Z}_p)$, and $TPR(-; \mathbb{Z}_p)$ on the opposite category of quasisyntomic rings are quasisyntomic sheaves.

• Theorem 1.4 (Explicit Computation): The paper provides a detailed computation of the RO($\mathbb{Z}/2$)-graded homotopy groups $\pi^{ \mathbb{Z}/2}_{s,w} TCR(k[x]/x^e; \mathbb{Z}_2)$ for a perfect field $k$ of characteristic 2. The result is expressed as a direct sum of modules involving truncated Witt vectors, Verschiebung operators, and specific generators ($\tau, \rho, \gamma, x_t, y$) with defined degrees.

• Corollary 10.4: The paper computes the equivariant slices of $\Sigma^{2\tau \ge 1} KR(\mathcal{O}_K/\omega^n; \mathbb{Z}_p)$ for finite extensions of $\mathbb{Q}_p$, utilizing previous computations of syntomic cohomology by Antieau, Krause, and Nikolaus.

Significance and Claims
The paper claims to provide the foundational structural link between syntomic cohomology and real topological cyclic homology. By establishing motivic filtrations on $THR$ and $TCR$ with graded pieces identified as syntomic cohomology (and its variants), the work enables the transfer of computational tools from the realm of algebraic K-theory (via $TC$) to real and hermitian K-theories.

The author notes that these results are conditional on a "real refinement of the Dundas–Goodwillie–McCarthy theorem," a work in progress by Harpaz, Nikolaus, and Shah. If this refinement holds, the computed homotopy groups of $TCR$ (specifically the non-negative part $\tau_{\ge 0} TCR$) would directly yield the homotopy groups of real K-theory $KR$. The paper does not claim to prove the Dundas–Goodwillie–McCarthy refinement itself but rather demonstrates how such a theorem would render the provided computations effective for $KR$.

The significance lies in moving beyond the few known computations for hermitian and real K-theories (often restricted to cases where 2 is invertible) to a general framework applicable to rings where 2 is not invertible, utilizing the power of prismatic and syntomic cohomology.