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Prismatic Steenrod operations and arithmetic duality on Brauer groups

Shachar Carmeli, Tony Feng

Abstract

We construct and analyze the "syntomic Steenrod algebra", which acts on the mod $p$ syntomic cohomology (also known as etale-motivic cohomology) of algebraic varieties in characteristic $p$. We then apply the resulting theory to resolve the last open cases of a 1966 Conjecture of Tate, concerning the existence of a symplectic form on the Brauer groups of smooth proper surfaces over finite fields. More generally, we exhibit symplectic structure on the higher Brauer groups of even dimensional varieties over finite fields. Although the applications are classical, our methods rely on recent advances in perfectoid geometry and prismatic cohomology, which we employ to define a theory of "spectral syntomic cohomology" with coefficients in motivic spectra. We then organize the resulting cohomology theories into a category of "spectral prismatic $F$-gauges", generalizing the prismatic $F$-gauges of Drinfeld and Bhatt--Lurie, for which we establish a ``spectral Serre duality'' extending classical coherent duality. These abstract constructions are leveraged to explicitly compute the syntomic Steenrod operations.

Prismatic Steenrod operations and arithmetic duality on Brauer groups

Abstract

We construct and analyze the "syntomic Steenrod algebra", which acts on the mod syntomic cohomology (also known as etale-motivic cohomology) of algebraic varieties in characteristic . We then apply the resulting theory to resolve the last open cases of a 1966 Conjecture of Tate, concerning the existence of a symplectic form on the Brauer groups of smooth proper surfaces over finite fields. More generally, we exhibit symplectic structure on the higher Brauer groups of even dimensional varieties over finite fields. Although the applications are classical, our methods rely on recent advances in perfectoid geometry and prismatic cohomology, which we employ to define a theory of "spectral syntomic cohomology" with coefficients in motivic spectra. We then organize the resulting cohomology theories into a category of "spectral prismatic -gauges", generalizing the prismatic -gauges of Drinfeld and Bhatt--Lurie, for which we establish a ``spectral Serre duality'' extending classical coherent duality. These abstract constructions are leveraged to explicitly compute the syntomic Steenrod operations.

Paper Structure

This paper contains 183 sections, 112 theorems, 587 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical motivations
  3. The order of the Brauer group
  4. Tate's Symplecticity Conjecture
  5. The syntomic Steenrod algebra
  6. Structure of the syntomic Steenrod algebra
  7. A hint of the construction
  8. Application to arithmetic duality
  9. $\mathbb{E}_\infty$ Steenrod operations
  10. Connection to the Milne--Artin--Tate pairing
  11. Calculation of Steenrod operations
  12. The comparison theorem
  13. Spectral prismatization
  14. Arithmetic duality and Steenrod equivariance
  15. Spectral prismatic $F$-gauges
  16. ...and 168 more sections

Key Result

Theorem 1.1.1

Let $X$ be a smooth, proper, geometrically connected surface over a finite field of characteristic $2$. Then the order of $\mathop{\mathrm{Br}}\nolimits(X)_{\mathop{\mathrm{nd}}\nolimits}$ is a perfect square.

Figures (2)

  • Figure 1: This roadmap depicts the long, winding journey to the symplectic arithmetic duality on higher Brauer groups.
  • Figure 2: A visualization of $(\mathop{\mathrm{Spec\,}}\nolimits k)^{\mathcal{N}}$. There are two (open) embeddings of $(\mathop{\mathrm{Spec\,}}\nolimits k)^{{\mathlarger{\mathbbl{\Delta}}}}$, one corresponding to Hodge--Tate cohomology and the other corresponding to de Rham cohomology; their intersection corresponds to Hodge cohomology. The two axes are glued via Frobenius to form $(\mathop{\mathrm{Spec\,}}\nolimits k)^{\mathop{\mathrm{Syn}}\nolimits}_{}$.

Theorems & Definitions (307)

  • Conjecture 1: Tate, Tate66
  • Conjecture 2: Tate, Tate66
  • Theorem 1.1.1
  • Conjecture 3: Tate's Symplecticity Conjecture, Tate66
  • Theorem 1.1.2
  • Remark 1: Higher dimensional generalizations
  • Remark 2
  • Theorem 1.2.1
  • Remark 3
  • Theorem 1.3.1
  • ...and 297 more