Big Cohen–Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic

Abstract

We prove a version of weakly functorial big Cohen–Macaulay algebras that suffices to establish Hochster and Huneke’s vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot’s theorem that direct summands of regular rings are pseudorational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by André and Bhatt.