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

## TL;DR

This paper proves a version of big Cohen-Macaulay algebras in mixed characteristic, establishing the vanishing conjecture for Tor maps and showing that direct summands of regular rings are pseudo-rational, using perfectoid spaces.

## Contribution

It introduces a weakly functorial big Cohen-Macaulay algebra framework in mixed characteristic, enabling new results on vanishing conjectures and singularity properties.

## Key findings

- Proves the vanishing conjecture for maps of Tor in mixed characteristic.
- Shows direct summands of regular rings are pseudo-rational in mixed characteristic.
- Utilizes perfectoid spaces inspired by recent breakthroughs on the direct summand conjecture.

## 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.