Logarithmic TC via the Infinite Root Stack and the Beilinson Fiber Square

TL;DR

This paper develops a unified framework for logarithmic cohomology theories using the infinite root stack, connecting various theories and establishing new structural results like a log Beilinson fiber square.

Contribution

It introduces a novel approach to express diverse logarithmic cohomology theories via the infinite root stack and proves a log version of the Beilinson fiber square.

Findings

Nygaard-completion matches site-theoretic log prismatic cohomology

Log TC and prismatic cohomology are expressed through the infinite root stack

Established a log Beilinson fiber square analogous to the classical case

Abstract

We apply our previous results on ``saturated descent'' to express a wide range of logarithmic cohomology theories in terms of the infinite root stack. Examples include the log cotangent complex, Rognes' log topological cyclic homology, and Nygaard-complete log prismatic cohomology. As applications, we show that the Nygaard-completion of the site-theoretic log prismatic cohomology coincides with the definition arising from log ${\rm TC}$, and we establish a log version of the ${\rm TC}$-variant of the Beilinson fiber square of Antieau--Mathew--Morrow--Nikolaus.