Degeneration Theorems of Connes and Feigin--Tsygan Type in Mixed Characteristic, with q-Analogues

Technical Summary: Degeneration Theorems of Connes and Feigin–Tsygan Type in Mixed Characteristic, with Q-Analogues

Problem Statement
The paper addresses the degeneration of spectral sequences associated with topological cyclic homology and periodic cyclic homology in mixed characteristic settings. Classically, for smooth algebras over a $\mathbb{Q}$-algebra, the Connes–Feigin–Tsygan theorem identifies periodic cyclic homology ($HP$) with the 2-periodization of the de Rham complex. In characteristic $p$, this picture is refined by the Hochschild–Kostant–Rosenberg (HKR) filtration and the Deligne–Illusie theorem, which guarantee degeneration under dimension constraints (e.g., $\dim X < p$).

The central problem is to establish mixed-characteristic analogues of these degeneration theorems. Specifically, the author investigates whether the spectral sequences relating crystalline cohomology to topological periodic homology ($TP$), and de Rham cohomology to periodic cyclic homology ($HP$), are split degenerate for smooth proper varieties over rings of integers in $p$-adic fields. A split degeneration implies that the spectral sequence collapses at the $E_2$ page and that the resulting filtration on the target homotopy groups splits, yielding an isomorphism between the homotopy groups and the direct sum of the cohomology groups.

Methodology
The paper employs a synthesis of non-commutative $p$-adic Hodge theory, Breuil–Kisin module theory, and topological K-theory.

1. Non-Commutative $p$-adic Hodge Theory: The author utilizes the framework developed by Bhatt–Morrow–Scholze and previous work by the author. This involves comparing topological invariants ($TP$, $TC^-$) of schemes over $p$-adic rings with their étale and crystalline counterparts via $K(1)$-local K-theory.
2. Breuil–Kisin Modules: A significant portion of the technical machinery is dedicated to the structure theory of Breuil–Kisin modules. The paper establishes a variant of the low-ramification structure theorem (Theorem 4.4) that accommodates finite free Breuil–Kisin modules alongside torsion ones. This is crucial for handling the filtrations arising from the BMS (Bhatt–Morrow–Scholze) spectral sequences.
3. Spectral Sequence Analysis: The author adapts Li's criteria for filtered complexes over discrete valuation rings (DVRs) to the spectral setting. The core strategy involves proving that the length of the torsion submodules of the total homotopy groups equals the sum of the lengths of the torsion submodules of the graded pieces. This equality, combined with the degeneration of the spectral sequence after inverting $p$, implies split degeneration.
4. Topological K-Theory and Chern Characters: To handle the $q$-de Rham analogue, the paper proves an integral refinement of the Chern character decomposition for topological K-theory. It shows that for a finite CW complex of dimension $d$, the Chern character becomes an isomorphism after inverting $\lfloor (d+1)/2 \rfloor!$. This controls the integrality range required for the topological $q$-de Rham result.

Key Contributions and Results

The paper proves three main theorems regarding split degeneration under specific dimension and ramification bounds. Let $W = W(k)$ be the Witt vectors of a perfect field $k$ of characteristic $p > 0$, $K$ a finite extension of $\text{Frac}(W)$ with ramification index $e$, and $X$ a smooth proper variety over the ring of integers $\mathcal{O}_K$.

1. Crystalline-to-TP Degeneration (Theorem 1.1 & 3.9):
   The crystalline-to-$TP$ spectral sequence is split degenerate if:

   • $\dim X < p - 1$ when $\mathcal{O}_K = W$ (unramified).
   • $e(2 \dim X - 1) < p - 1$ when $e \ge 2$ (ramified).
     This yields an isomorphism of $W$-modules:
     $$\pi_j TP(X_k; \mathbb{Z}_p) \simeq \bigoplus_{i \in \mathbb{Z}} H^{-j+2i}_{\text{cry}}(X_k/W)$$
     The bound here is slightly sharper in the ramified case compared to the de Rham case because the proof relies directly on crystalline comparison and topological K-theoretic splitting criteria, bypassing the Breuil–Kisin splitting argument used for the other cases.

2. De Rham-to-HP Degeneration (Theorem 1.1 & 5.14):
   The de Rham-to-$HP$ spectral sequence is split degenerate if $2e \dim(X/\mathcal{O}_K) < p - 1$. This implies:
   $$HP_j(X/\mathcal{O}_K; \mathbb{Z}_p) \simeq \bigoplus_{i \in \mathbb{Z}} H^{-j+2i}_{\text{dR}}(X/\mathcal{O}_K)$$
   The proof proceeds by establishing the split degeneration of the Breuil–Kisin-to-$TC^-$ spectral sequence (Theorem 5.13) and then specializing via maps to $\mathcal{O}_K$ and $A_{\inf}$.

3. $A_{\inf}$-to-TP Degeneration (Theorem 1.3 & 5.15):
   Under the same condition $2e \dim(X/\mathcal{O}_K) < p - 1$, the $A_{\inf}$-to-$TP$ spectral sequence is split degenerate:
   $$TP_j(X; \mathbb{Z}_p) \simeq \bigoplus_{i \in \mathbb{Z}} H^{-j+2i}_{A_{\inf}}(X)$$

4. Topological $q$-de Rham Analogue (Theorem 1.3 & 5.18):
   For a smooth proper variety $X$ over $\mathbb{Z}[1/N]$ (where $N$ is even), the paper proves that after inverting $(2d+1)!$ (where $d = \dim X$), the $S^1$-equivariant even filtration on $TP(X \otimes ku / ku)$ is split degenerate. This yields an isomorphism with $q$-de Rham cohomology:
   $$\pi_j TP(X \otimes ku / ku)[(2d+1)!^{-1}] \simeq \bigoplus_{i \in \mathbb{Z}} H^{-j+2i}_{q\text{-dR}}(X/\mathbb{Z})[(2d+1)!^{-1}]$$
   This result relies on the integral Chern character decomposition (Theorem 2.2) and a descent argument comparing the $q$-de Rham filtration to the Breuil–Kisin filtration over $\ell$-adic completions for primes $\ell > 2d+1$.

Significance and Claims
The paper claims to establish the first mixed-characteristic analogues of the Connes–Feigin–Tsygan degeneration theorems with explicit bounds on dimension and ramification. By proving split degeneration (rather than just degeneration), the author ensures that the homotopy groups of the topological invariants decompose as direct sums of the corresponding cohomology groups, preserving the module structure over the base ring.

The work bridges commutative and non-commutative $p$-adic Hodge theory. As noted in the acknowledgments, the motivation stems from the question of whether commutative $p$-adic Hodge theory can be recovered from its non-commutative counterpart. The results demonstrate that for smooth proper varieties, the "motivic" filtrations on topological cyclic homology (constructed by Bhatt–Morrow–Scholze) degenerate in a way that recovers classical crystalline, de Rham, and $A_{\inf}$ cohomologies, provided the dimension is sufficiently small relative to the characteristic $p$ and the ramification index $e$.

The inclusion of the $q$-de Rham analogue extends these results to a topological setting involving $KU$-theory and $q$-deformations, showing that the structural properties of these spectral sequences persist even after inverting a specific factorial to control denominators arising from the Chern character.