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Artin--Mazur formal groups and Milne duality via unipotent spectra

Shubhodip Mondal, Tasos Moulinos, Lucy Yang

Abstract

We introduce and develop the notion of "unipotent spectra." This is defined to be the stabilization of Toën's category of affine stacks, and is related to recent work of Mondal--Reinecke. Unipotent spectra give rise to unipotent stable homotopy groups and unipotent homology, which are new invariants for schemes valued in unipotent group schemes. As applications, we recover the Artin--Mazur formal groups associated to schemes without any vanishing assumptions. Further, we show that syntomic cohomology admits a natural refinement to a perfect unipotent spectrum. Finally, we extend Milne's work on arithmetic duality theorems to the category of perfect unipotent spectra and apply it to refine Poincaré duality in syntomic cohomology.

Artin--Mazur formal groups and Milne duality via unipotent spectra

Abstract

We introduce and develop the notion of "unipotent spectra." This is defined to be the stabilization of Toën's category of affine stacks, and is related to recent work of Mondal--Reinecke. Unipotent spectra give rise to unipotent stable homotopy groups and unipotent homology, which are new invariants for schemes valued in unipotent group schemes. As applications, we recover the Artin--Mazur formal groups associated to schemes without any vanishing assumptions. Further, we show that syntomic cohomology admits a natural refinement to a perfect unipotent spectrum. Finally, we extend Milne's work on arithmetic duality theorems to the category of perfect unipotent spectra and apply it to refine Poincaré duality in syntomic cohomology.

Paper Structure

This paper contains 24 sections, 95 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Motivation context
  3. Main theorems
  4. Outline
  5. Notation conventions
  6. Acknowledgements
  7. Unipotent spectra
  8. Generalities on unipotent spectra
  9. Symmetric monoidal structure on unipotent spectra
  10. Unipotent homology and a profiniteness theorem
  11. Recognition theorem for unipotent spectra
  12. Coniveau filtration on unipotent homology
  13. Graded pieces of the coniveau filtration and unipotent local homology
  14. Reformulation in terms of Beilinson $t$-structures
  15. Cohomology of Cohen--Macaulay schemes
  16. ...and 9 more sections

Key Result

Theorem 1.1.8

Let $n \ge 1$ be an integer. Let $X$ be a pointed proper scheme over an algebraically closed field $k$ of characteristic $p>0$ satisfying Let $\Phi^n(X)$ denote the $n$-th Artin--Mazur formal group defined in this context. Then if $n > 1$, $\Phi^n(X)$ is naturally isomorphic to the Cartier dual $\pi^{\mathrm{U}}_n(X)^\vee$ of the $n$-th unipotent homotopy group scheme of $X$. If $n=1$, $\Phi^n(X)

Theorems & Definitions (247)

  • Definition 1.1.1: Unipotent spectra
  • Remark 1.1.2
  • Remark 1.1.3: Unipotent stable homotopy type
  • Example 1.1.4
  • Example 1.1.5: \ref{['introex1']}
  • Example 1.1.6
  • Remark 1.1.7: Constructible sheaves and unipotent spectra
  • Theorem 1.1.8: soon
  • Theorem 1.2.1
  • Definition 1.2.2: Unipotent homology
  • ...and 237 more