F-purity and Frobenius splitting do not coincide in rigid analytic geometry -- Takumi Murayama, June 3, 2020.

Abstract: In 1962, Tate introduced rigid analytic spaces to formulate his uniformization theorem for elliptic curves over p-adic fields. Since then, rigid analytic spaces and their descendants (such as Berkovich spaces and perfectoid spaces) have been used extensively in algebraic and arithmetic geometry, and more recently in commutative algebra. In this talk, we focus on ring-theoretic aspects of rigid analytic spaces in positive characteristic. Using some ideas of Gabber, we show that the Frobenius map does not always split for rings of functions that converge on rigid-analytic polydiscs, even though these rings are excellent and regular. This resolves a long-standing open question raised by Hochster and others by showing that Hochster and Roberts's notion of F-purity is distinct from Mehta and Ramanathan's notion of Frobenius splitting. This is joint work with Rankeya Datta.