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

Federico Binda, Tommy Lundemo, Alberto Merici, Doosung Park

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.

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

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 , and we establish a log version of the -variant of the Beilinson fiber square of Antieau--Mathew--Morrow--Nikolaus.
Paper Structure (30 sections, 25 theorems, 61 equations)

This paper contains 30 sections, 25 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Comparison with site-theoretic log prismatic cohomology
  3. A log variant of the Beilinson fiber square
  4. Acknowledgments
  5. Outline
  6. Recollections on log TC and prismatic cohomology
  7. The replete bar construction
  8. Animated log rings
  9. Log topological Hochschild homology
  10. Log topological cyclic homology
  11. The log cotangent complex
  12. The log quasisyntomic site
  13. Nygaard-complete log prismatic cohomology
  14. Saturated descent via the infinite root stack
  15. The saturated Čech nerve
  16. ...and 15 more sections

Key Result

Theorem 1.1

If $(A, M)$ is a quasisyntomic pre-log ring such that Then there is an equivalence of ${\Bbb E}_{\infty}$-rings relating Rognes' log ${\rm TC}$ and ${\rm TC}$ of the infinite root stack.

Theorems & Definitions (57)

  • Theorem 1.1: Section \ref{['sec:tcinftyroot']}
  • Theorem 1.3: Section \ref{['sec:sitecomp']}
  • Theorem 1.5: Section \ref{['sec:logbeilinson']}
  • Theorem 1.6
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Example 3.2
  • Definition 3.4
  • ...and 47 more