Weak functoriality of Cohen-Macaulay algebras
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- by Yves André
- J. Amer. Math. Soc. 33 (2020), 363-380
- 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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Bibliographic 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