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FORMALLY REGULAR RINGS AND DESCENT OF REGULARITY

Part of: Local theory Ring extensions and related topics Local rings and semilocal rings Commutative algebra: Homological methods

Published online by Cambridge University Press:  02 January 2024

Javier Majadas Open the ORCID record for Javier Majadas [Opens in a new window] ,
Samuel Alvite Open the ORCID record for Samuel Alvite [Opens in a new window]  and
Nerea G. Barral
Javier Majadas*
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain
Samuel Alvite
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain (samuelalvite@gmail.com)
Nerea G. Barral
Affiliation:
Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Santiago de Compostela, E15782 Santiago de Compostela, Spain (nereabarral@gmail.com)
*
j.majadas@usc.es
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Abstract

Valuation rings and perfectoid rings are examples of (usually non-Noetherian) rings that behave in some sense like regular rings. We give and study an extension of the concept of regular local rings to non-Noetherian rings so that it includes valuation and perfectoid rings and it is related to Grothendieck’s definition of formal smoothness as in the Noetherian case. For that, we have to take into account the topologies. We prove a descent theorem for regularity along flat homomorphisms (in fact for homomorphisms of finite flat dimension), extending some known results from the Noetherian to the non-Noetherian case, as well as generalizing some recent results in the non-Noetherian case, such as the descent of regularity from perfectoid rings by B. Bhatt, S. Iyengar and L. Ma.

Keywords

regular ringperfectoid ringformal smoothness

MSC classification

Primary: 14B10: Infinitesimal methods 13H05: Regular local rings 13B40: Étale and flat extensions; Henselization; Artin approximation
Secondary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 40
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Alvite, S., Barral, N. G. and Majadas, J., ‘On a theorem of Gulliksen on the homology of local rings’, J. Pure Appl. Algebra 227 (2023), Paper No. 107374.Google Scholar
André, M., ‘L’algèbre de Lie d’un anneau local’, in Sympos. Math., Roma 4, Teoria Numeri, Dic. 1968, e Algebra, Marzo 1969, (1970), 337–375.Google Scholar
André, M., ‘Algèbres graduées associées et algèbres symétriques plates’, Comment. Math. Helv. 49 (1974), 277–301.CrossRefGoogle Scholar
André, M., Homologie des Algèbres Commutatives (Springer, 1974).CrossRefGoogle Scholar
André, M., ‘Modules des différentielles en caractéristique p’. Manuscripta Math. 62(4) (1988), 477–502.CrossRefGoogle Scholar
André, M., ‘Homomorphismes réguliers en caractéristique p’, C. R. Acad. Sci. Paris Sér. I Math. 316(7) (1993), 643–646.Google Scholar
André, M., ‘Autre démonstration du théorème liant régularité et platitude en caractéristique p’, Manuscripta Math. 82(3–4) (1994), 363–379.CrossRefGoogle Scholar
André, M., ‘Examples of non-rigid cotangent complexes’, J. Algebra 186 (1996), 32–46.CrossRefGoogle Scholar
André, Y., Fiorot, L., ‘On the canonical, fpqc, and finite topologies on affine schemes. The state of the art’, Ann. Sc. Norm. Super. Pisa Cl. Sci. 23 (2022), 81–114.Google Scholar
Artin, M. and Mazur, B., Etale Homotopy, Lecture Notes in Mathematics, No. 100 (Springer-Verlag, 1969).CrossRefGoogle Scholar
Avramov, L. L., ‘Descente des déviations par homomorphismes locaux et génération des idéaux de dimension projective finie’, C. R. Acad. Sci. Paris Sér. I Math. 295(12) (1982), 665–668.Google Scholar
Bertin, J., Anneaux cohérents de dimension homologique finie, Publications mathématiques et informatique de Rennes, no. 4 (1972), Exposé no. 16. www.numdam.org/article/PSMIR_1972___4_A16_0.pdf.Google Scholar
Bhatt, B., Iyengar, S. B. and Ma, L., ‘Regular rings and perfect(oid) algebras’, Comm. Algebra 47(6) (2019), 2367–2383.CrossRefGoogle Scholar
Bhatt, B., Morrow, M. and Scholze, P., ‘Integral p-adic Hodge theory’, Publ. Math. Inst. Hautes Étud. Sci. 128 (2018), 219–397.CrossRefGoogle Scholar
Bhatt, B. and Scholze, P., ‘The pro-étale topology for schemes’, Astérisque No. 369 (2015), 99–201.Google Scholar
Bhatt, B. and Scholze, P., ‘Projectivity of the Witt vector affine Grassmannian’, Invent. Math. 209(2) (2017), 329–423.CrossRefGoogle Scholar
Blanco, A., Majadas, J. and Rodicio, A. G., ‘On the acyclicity of the Tate complex’, J. Pure Appl. Algebra 131(2) (1998), 125–132.CrossRefGoogle Scholar
Bourbaki, N., Algèbre, Chapitre 8 (Springer, 2012).CrossRefGoogle Scholar
Bourbaki, N., Algèbre Commutative, Chapitres 1 à 4 (Springer, 2006).CrossRefGoogle Scholar
Bourbaki, N., Algèbre Commutative, Chapitres 5 à 7 (Springer, 2006).CrossRefGoogle Scholar
Dumitrescu, T., ‘Regularity and finite flat dimension in characteristic $p>0$ ’, Comm. Algebra 24(10) (1996), 3387–3401.CrossRefGoogle Scholar
Gabber, O. and Ramero, L., Almost Ring Theory, Lecture Notes in Mathematics, 1800 (Springer-Verlag, Berlin, 2003).CrossRefGoogle Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. I (Springer, 1971).Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas I, Inst. Hautes Études Sci. Publ. Math. No. 20 (1964).Google Scholar
Grothendieck, A. and Dieudonné, J., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas II, Inst. Hautes Études Sci. Publ. Math. No. 24 (1965).Google Scholar
Grothendieck, A., ‘Cohomologie l-adique et fonctions L’, in Séminaire de Géométrie Algébrique du Bois-Marie 1965–1966 (SGA 5), Lecture Notes in Mathematics, vol. 589 (Springer-Verlag, 1977).Google Scholar
Ferrand, D., ‘Épimorphismes d’anneaux et algèbres séparables’, C. R. Acad. Sci. Paris Sér. A-B 265 (1967), A411–A414.Google Scholar
Gulliksen, T. H. and Levin, G., Homology of Local Rings, Queen’s Papers in Pure and Applied Mathematics, No. 20 (Queen’s University, Kingston, Ont. 1969).Google Scholar
de Jong, A. J. and Olander, N., ‘On weakly étale morphisms’, J. Pure Appl. Algebra 228 (2024), no. 3, Paper No. 107532, 9 pp.CrossRefGoogle Scholar
Kabele, T., ‘Regularity conditions in nonnoetherian rings’, Trans. Am. Math. Soc. 155 (1971), 363–374.CrossRefGoogle Scholar
Kunz, E., ‘Characterizations of regular local rings of characteristic p’, Amer. J. Math. 91 (1969), 772–784.CrossRefGoogle Scholar
Majadas, J., ‘Some homological criteria for regular, complete intersection and Gorenstein rings’, Proc. Edinb. Math. Soc. (2) 59(2) (2016), 473–481.CrossRefGoogle Scholar
Majadas, J. and Rodicio, A. G., ‘Commutative algebras of finite Hochschild homological dimension’, J. Algebra 149(1) (1992), 68–84.CrossRefGoogle Scholar
Majadas, J. and Rodicio, A. G., Smoothness, Regularity and Complete Intersection, London Math. Soc. Lect. Note Series 373 (Cambridge University Press, Cambridge, 2010).CrossRefGoogle Scholar
Northcott, D. G., ‘On the homology theory of general commutative rings’, J. Lond. Math. Soc. 36 (1961), 231–240.CrossRefGoogle Scholar
Pitteloud, Y., Le cinquième module d’homologie d’une algèbre commutative, Thèse n. 1007 (École polytechnique fédérale de Lausanne, 1992).Google Scholar
Planas-Vilanova, F., ‘Rings of weak dimension one and syzygetic ideals’, Proc. Amer. Math. Soc. 124(10) (1996), 3015–3017.CrossRefGoogle Scholar
Planas-Vilanova, F., ‘On the vanishing and non-rigidity of the André–Quillen (co)homology’, J. Pure Appl. Algebra 120(1) (1997), 67–75.CrossRefGoogle Scholar
Planas-Vilanova, F., ‘Noetherian rings of low global dimension and syzygetic prime ideals’, J. Pure Appl. Algebra 225(2) (2021), 106494.CrossRefGoogle Scholar
Quillen, D., ‘Homology of commutative rings’, Mimeographed notes, MIT 1967.Google Scholar
Radu, N., ‘Une classe d’anneaux noethériens’, Rev. Roumaine Math. Pures Appl. 37(1) (1992), 79–82.Google Scholar
Raynaud, M., Anneaux locaux henséliens. Lecture Notes in Mathematics, 169 (Springer 1970).CrossRefGoogle Scholar
Rodicio, A. G., ‘On a result of Avramov’, Manuscripta Math. 62(2) (1988), 181–185.CrossRefGoogle Scholar
Rodicio, A. G., ‘Smooth algebras and vanishing of Hochschild homology’, Comment. Math. Helv. 65(3) (1990), 474–477.CrossRefGoogle Scholar
Rodicio, A. G., ‘Flat exterior Tor algebras and cotangent complexes’, Comment. Math. Helv. 70(4) (1995), 546–557; Erratum ibid. 71 (1996) No. 2, 338.CrossRefGoogle Scholar
Sahandi, P., Sharif, T. and Yassemi, S., ‘Complete intersection flat dimension and the intersection theorem’, Algebra Colloq. 19(Special Issue No. 1) (2012), 1161–1166.CrossRefGoogle Scholar
Sather-Wagstaff, S., ‘Complete intersection dimensions and Foxby classes’, J. Pure Appl. Algebra 212(12) (2008), 2594–2611.CrossRefGoogle Scholar
Scholze, P., ‘Perfectoid spaces’, Publ. Math. Inst. Hautes Études Sci. 116 (2012), 245–313.CrossRefGoogle Scholar
Sjödin, G., ‘Hopf algebras and derivations’, J. Algebra 64 (1980), 218–229.CrossRefGoogle Scholar
Takahashi, R., ‘Upper complete intersection dimension relative to a local homomorphism’, Tokyo J. Math. 27(1) (2004), 209–219.CrossRefGoogle Scholar