On pro-cdh descent on derived schemes

Kelly, Shane; Saito, Shuji; Tamme, Georg

doi:10.2140/gt.2026.30.337

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On pro-cdh descent on derived schemes

Shane Kelly, Shuji Saito and Georg Tamme

Geometry & Topology 30 (2026) 337–372

DOI: 10.2140/gt.2026.30.337

Abstract

Grothendieck’s formal functions theorem states that the coherent cohomology of a Noetherian scheme can be recovered from that of a blowup, and the infinitesimal thickenings of the center and of the exceptional divisor of the blowup. We prove an analogous descent result, called “pro-cdh descent”, for certain cohomological invariants of arbitrary quasicompact, quasiseparated derived schemes. Our results in particular apply to algebraic $K$ -theory, topological Hochschild and cyclic homology, and the cotangent complex.

As an application, we deduce that $K_{n} (X) = 0$ when $n < - d$ for quasicompact, quasiseparated derived schemes $X$ of valuative dimension $d$ . This generalizes Weibel’s conjecture, which was originally stated for Noetherian (nonderived) $X$ of Krull dimension $d$ , and proved in this form in 2018 by Kerz, Strunk, and the third author.

Keywords

algebraic K-theory, pro-cdh descent, derived schemes, localizing invariants, cotangent complex, infinitesimal thickenings

Mathematical Subject Classification

Primary: 14A30, 19E08

References

Publication

Received: 5 November 2024

Revised: 10 May 2025

Accepted: 17 May 2025

Published: 19 January 2026

Proposed: Haynes R Miller

Seconded: Mark Behrens, Marc Levine

Authors

Shane Kelly
Graduate School of Mathematical Sciences University of Tokyo Tokyo Japan

Shuji Saito
Graduate School of Mathematical Sciences University of Tokyo Tokyo Japan

Georg Tamme
Institut für Mathematik Johannes Gutenberg University Mainz Mainz Germany

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Volume 30, issue 1 (2026)