On pro-cdh descent on 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.