Prismatic Steenrod operations and arithmetic duality on Brauer groups

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.