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

Heitmann, Raymond; Ma, Linquan

doi:10.2140/ant.2018.12.1659

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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

DOI: 10.2140/ant.2018.12.1659

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

Vol. 12, No. 7, 2018