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Weak functoriality of Cohen-Macaulay algebras


Author: Yves André
Journal: J. Amer. Math. Soc. 33 (2020), 363-380
MSC (2010): Primary 13D22, 13H05, 14G20
DOI: https://doi.org/10.1090/jams/937
Published electronically: January 6, 2020
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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.


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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

DOI: https://doi.org/10.1090/jams/937
Keywords: Direct summand conjecture, big Cohen-Macaulay algebra, perfectoid algebra
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
Article copyright: © Copyright 2020 American Mathematical Society