Vol. 12, No. 7, 2018

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

Raymond Heitmann and Linquan Ma

Vol. 12 (2018), No. 7, 1659–1674
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.

Keywords
big Cohen–Macaulay algebras, direct summand conjecture, perfectoid space, vanishing conjecture for maps of Tor, pseudorational singularities
Mathematical Subject Classification 2010
Primary: 13D22
Secondary: 13H05
Milestones
Received: 18 May 2017
Revised: 7 February 2018
Accepted: 13 March 2018
Published: 27 October 2018
Authors
Raymond Heitmann
University of Texas, Austin
Austin, TX
United States
Linquan Ma
Purdue University
West Lafayette, IN
United States