  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.
