On pro-cdh descent on derived schemes

TL;DR

This paper establishes a pro-cdh descent theorem for derived schemes, extending Grothendieck's formal functions theorem to broader cohomological invariants, with applications to algebraic K-theory and a generalization of Weibel's conjecture.

Contribution

It proves a new pro-cdh descent result for derived schemes, applicable to various cohomological invariants, and generalizes Weibel's conjecture to derived, non-Noetherian contexts.

Findings

Pro-cdh descent holds for algebraic K-theory, Hochschild and cyclic homology, and the cotangent complex.

Derived schemes of valuative dimension d have vanishing K-theory in degrees less than -d.

Generalizes Weibel's conjecture to quasi-compact, quasi-separated derived schemes.

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. In this article, we prove an analogous descent result, called ``pro-cdh descent'', for certain cohomological invariants of arbitrary quasi-compact, quasi-separated 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 quasi-compact, quasi-separated derived schemes $X$ of valuative dimension $d$. This generalises Weibel's conjecture, which was originally stated for Noetherian (non-derived) $X$ of Krull dimension $d$, and proved in this form in 2018 by Kerz, Strunk, and the third author.