Abstract
We prove a version of weakly functorial big Cohen-Macaulay algebras that suffices to establish Hochster-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 pseudo-rational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by Andr\'{e} and Bhatt.
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