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Cohomology theories in the moduli of ring stacks

Dhilan Lahoti, Deven Manam

TL;DR

The paper develops a cohesive framework for cohomology theories as ring stacks, using syntomic prismatization and a bootstrap from monoids to rings to prove that the syntomic moduli $A^{ ext{Syn}}$ fully embeds into the stack of $A$-algebra stacks, with the essential image described via underlying abelian monoid stacks. It extends these ideas to characteristic zero filtered de Rham, $oldsymbol{ ext{p}}$-adic étale, and Betti settings, each requiring its own technical toolkit (e.g., polyfiltered Cartier–Witt divisors, passable W-modules, descent theory, and transmutation). Central technical devices include a detailed analysis of affine $W$-module schemes, dualities, and the notion of polyfiltered divisors, which enable lifting monoid data to ring structures and controlling the image of the moduli problem. The work also discusses limits of the approach in the derived setting and outlines promising directions, including shtukas, analytic stacks, and refined invariants (e.g., Efimov’s refined TC) that connect to broader motivic and $p$-adic phenomena. Together, these results provide a modular blueprint for understanding cohomology theories as moduli problems of ring stacks and suggest rich future avenues in prismatic and analytic contexts.

Abstract

We show that the natural map from the syntomification of a ring $R$ to the stack of $R$-algebra stacks is fully faithful, answering a question of Drinfeld, and we describe its essential image in terms of underlying monoid stacks. We also give similar statements in the characteristic 0 filtered de Rham, $\ell = p$ étale, and Betti settings.

Cohomology theories in the moduli of ring stacks

TL;DR

The paper develops a cohesive framework for cohomology theories as ring stacks, using syntomic prismatization and a bootstrap from monoids to rings to prove that the syntomic moduli fully embeds into the stack of -algebra stacks, with the essential image described via underlying abelian monoid stacks. It extends these ideas to characteristic zero filtered de Rham, -adic étale, and Betti settings, each requiring its own technical toolkit (e.g., polyfiltered Cartier–Witt divisors, passable W-modules, descent theory, and transmutation). Central technical devices include a detailed analysis of affine -module schemes, dualities, and the notion of polyfiltered divisors, which enable lifting monoid data to ring structures and controlling the image of the moduli problem. The work also discusses limits of the approach in the derived setting and outlines promising directions, including shtukas, analytic stacks, and refined invariants (e.g., Efimov’s refined TC) that connect to broader motivic and -adic phenomena. Together, these results provide a modular blueprint for understanding cohomology theories as moduli problems of ring stacks and suggest rich future avenues in prismatic and analytic contexts.

Abstract

We show that the natural map from the syntomification of a ring to the stack of -algebra stacks is fully faithful, answering a question of Drinfeld, and we describe its essential image in terms of underlying monoid stacks. We also give similar statements in the characteristic 0 filtered de Rham, étale, and Betti settings.

Paper Structure

This paper contains 20 sections, 80 theorems, 143 equations.

Table of Contents

  1. Syntomic
  2. Prismatization
  3. W-module schemes
  4. Passable W-modules and polyfiltered Cartier--Witt divisors
  5. Covering M_m^N
  6. From monoids to rings
  7. From W^perf to W
  8. The main result
  9. De Rham
  10. Étale
  11. Betti
  12. Complements
  13. Outlook
  14. Pseudodescendable sheaves and divided powers
  15. Affine stacks
  16. ...and 5 more sections

Key Result

lemma 1

For a prism $(A, (d))$, with $\overline{A} \defeq A/d$, the natural square \xymatrix{ (\overline{A} / A)^\calN \ar[d] \ar[r] & \fakeheight{\overline{A}}{A}^\calN \ar[d] \\ \Spf A \times \bbZ_p^\calN \ar[r] & A^\calN }is a pullback.

Theorems & Definitions (214)

  • remark 1
  • remark 2
  • remark 3
  • remark 4
  • remark 5
  • lemma 1
  • proof
  • remark 6
  • remark 7
  • definition 1: The complete filtered prismatization
  • ...and 204 more