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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Weak functoriality of Cohen-Macaulay algebras
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by Yves André HTML | PDF
J. Amer. Math. Soc. 33 (2020), 363-380 Request permission

Abstract:

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of “homological conjectures” in commutative algebra; namely, for any local homomorphism $R\to R’$ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay $R$-algebra and some Cohen-Macaulay $R’$-algebra.

When $R$ contains a field, this is already known. When $R$ is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz’s refined treatment of weak functoriality of Cohen-Macaulay algebras in characteristic $p$; in fact, developing a “tilting argument” due to K. Shimomoto, we combine the perfectoid techniques of the author’s earlier work with Dietz’s result.

References
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Additional Information
  • Yves André
  • Affiliation: Institut de Mathématiques de Jussieu, Sorbonne-Université, 4 place Jussieu, 75005 Paris, France
  • Email: yves.andre@imj-prg.fr
  • Received by editor(s): January 31, 2018
  • Received by editor(s) in revised form: April 12, 2018, June 20, 2018, June 21, 2018, November 24, 2018, January 4, 2019, August 6, 2019, August 11, 2019, and August 15, 2019
  • Published electronically: January 6, 2020
  • © Copyright 2020 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 33 (2020), 363-380
  • MSC (2010): Primary 13D22, 13H05, 14G20
  • DOI: https://doi.org/10.1090/jams/937
  • MathSciNet review: 4073864