Note

Canonical Descent and Formal Pro-Excision for Spectral Schemes

Author: Wei Yang · Added: August 21, 2026

Article date: August 21, 2026.

Abstract

We prove that topological Hochschild homology, regarded as a cyclotomic spectrum, and integral topological cyclic homology satisfy Cech descent for the canonical topology on connective spectral rings. The proof begins with canonical descent for finite symmetric powers of the spectral cotangent complex with bounded-below coefficients. It then passes through the complete circle-equivariant Hochschild-Kostant-Rosenberg filtration; linear connectivity of the filtration tails supplies continuity for finite-group Tate constructions, and naturality of the unfiltered cyclotomic Frobenius then upgrades the resulting comparison to cyclotomic descent.

As an application, we establish formal pro-excision for proper modifications of connective quasi-compact and quasi-separated spectral schemes. Let $Y \to X$ be proper and locally almost of finite presentation, and suppose that it is an equivalence over a quasi-compact open subscheme $U \subset X$ . For both integral topological cyclic homology and nonconnective algebraic $K$ -theory, the square obtained by completing $X$ along $|X| \setminus U$ and $Y$ along its inverse image is weakly cartesian in pro-spectra. The proof combines the finite case, comparison with animated formal completion over a classical base, derived pro-cdh descent, proper formal functions on derived loop spaces, and 1-connective pro-devissage.

We also determine the scope of filtered descent. The complete Hodge filtration on Hodge-completed infinitesimal cohomology, the complete circle-equivariant Hochschild-Kostant-Rosenberg filtration on topological Hochschild homology and its finite-group Tate variants, and the $p$ -adic filtration on derived $p$ -complete topological cyclic homology satisfy canonical descent. By contrast, for every prime $p$ , infinitely many positive associated graded pieces of the ambient Hahn-Raksit-Wilson even filtration fail to be sheaves for the canonical topology. On chromatically quasisyntomic rings these graded pieces are the Hahn-Raksit-Wilson syntomic complexes. The obstruction is chromatic: the $p$ -complete sphere is a retract of $\operatorname{TC}(\mathbb{S})^{\wedge}_p$ , whereas the multiplicative cyclotomic trace and Mitchell’s theorem imply that $\operatorname{TC}(H\mathbb{Z})^{\wedge}_p$ is $K(2)$ -acyclic.