$p$-adic six functors on diamonds

Abstract: Motivated by $p$-adic Poincaré duality in rigid geometry, we develop a $p$-adic six functor formalism on rigid varieties, or more generally for diamonds. This is achieved by defining a category of "quasi-coherent $\mathcal{O}_X^+/p$-modules" on a diamond $X$ and then using the recent development of a quasi-coherent 6-functor formalism on schemes by Clausen-Scholze to obtain a similar 6-functor formalism on diamonds. One easily deduces the desired p-adic Poincaré duality on a smooth proper rigid variety $X$ in mixed characteristic, noting that by Scholze's primitive comparison theorem, $\mathbb F_p$-cohomology on $X$ can be computed via cohomology of the sheaf $O_X^+/p$. Of course, our p-adic 6-functor formalism allows for many more potential applications; for example, we expect to gain new insights in the $p$-adic Langlands program.

commutative algebraalgebraic geometrynumber theory

Audience: researchers in the topic

Recent Advances in Modern p-Adic Geometry (RAMpAGe)

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