Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T18:00:23.221Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Homological Criteria for Regular, Complete Intersection and Gorenstein Rings

Part of: Commutative algebra: Homological methods Local rings and semilocal rings

Published online by Cambridge University Press:  08 July 2015

Javier Majadas
Javier Majadas*
Affiliation:
Departamento de Álgebra, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain, ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field. In positive characteristic, the Frobenius endomorphism (and, more generally, any contracting endomorphism) can also be used for these characterizations. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.

Keywords

regular local ringfinite flat dimensioncomplete intersection

MSC classification

Primary: 13H05: Regular local rings 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Secondary: 13D03: (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13D05: Homological dimension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 473 - 481
Copyright
Copyright © Edinburgh Mathematical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. André, M., Homologie des algèbres commutatives, Grundlehren der mathematischen Wissenschaften, Volume 206 (Springer, 1974) (in French).Google Scholar
2. André, M., Modules des différentielles en caractéristique p, Manuscr. Math. 62(4) (1988), 477–502.Google Scholar
3. Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, Volume 94 (American Mathematical Society, Providence, RI, 1969).Google Scholar
4. Avramov, L. L., Flat morphisms of complete intersections, Dokl. Akad. Nauk SSSR 225(1) (1975), 11–14.Google Scholar
5. Avramov, L. L., Descente des déviations par homomorphismes locaux et génération des idéaux de dimension projective finie, C. R. Acad. Sci. Paris Sér. I 295(12) (1982), 665–668.Google Scholar
6. Avramov, L. L., Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology, Annals Math. (2) 150(2) (1999), 455–487.Google Scholar
7. Avramov, L. L., Homological dimensions and related invariants of modules over local rings, in Representations of algebra, Volumes I, II, pp. 1–39 (Beijing Normal University Press, Beijing, 2002).Google Scholar
8. Avramov, L. L. and Foxby, H.-B., Cohen-Macaulay properties of ring homomorphisms, Adv. Math. 133(1) (1998), 54–95.CrossRefGoogle Scholar
9. Avramov, L. L., Foxby, H.-B. and Herzog, B., Structure of local homomorphisms, J. Alg. 164(1) (1994), 124–145.Google Scholar
10. Avramov, L. L., Gasharov, V. N. and Peeva, I. V., Complete intersection dimension, Publ. Math. IHES 86 (1997), 67–114.CrossRefGoogle Scholar
11. Avramov, L. L., Iyengar, S. and Miller, C., Homology over local homomorphisms, Am. J. Math. 128(1) (2006), 23–90.Google Scholar
12. Avramov, L. L., Hochster, M., Iyengar, S. and Yao, Y., Homological invariants of modules over contracting endomorphisms, Math. Annalen 353(2) (2012), 275–291.CrossRefGoogle Scholar
13. Blanco, A. and Majadas, J., Sur les morphismes d’intersection compl`ete en caractéristique p, J. Alg. 208(1) (1998), 35–42.Google Scholar
14. Grothendieck, A., Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas, I, Publ. Math. IHES 20 (1964), 5–259 (in French).Google Scholar
15. Herzog, J., Charakteristik, Ringe der p und Frobeniusfunktoren, Math. Z. 140 (1974), 67–78.CrossRefGoogle Scholar
16. Iyengar, S. and Sather-Wagstaff, S., G-dimension over local homomorphisms: applications to the Frobenius endomorphism, Illinois J. Math. 48(1) (2004), 241–272.Google Scholar
17. Koh, J. and Lee, K., Some restrictions on the maps in minimal resolutions, J. Alg. 202(2) (1998), 671–689.Google Scholar
18. Kunz, E., Characterizations of regular local rings of characteristic p, Am. J. Math. 91 (1969), 772–784.CrossRefGoogle Scholar
19. Majadas, J., A descent theorem for formal smoothness, preprint, 2012 (arxiv.org/abs/1209.5055).Google Scholar
20. Majadas, J. and Rodicio, A. G., Smoothness, regularity and complete intersection, London Mathematical Society Lecture Note Series, Volume 373 (Cambridge University Press, 2010).Google Scholar
21. Rahmati, H., Contracting endomorphisms and Gorenstein modules, Arch. Math. 92(1) (2009), 26–34.CrossRefGoogle Scholar
22. Rodicio, A. G., On a result of Avramov, Manuscr. Math. 62(2) (1988), 181–185.Google Scholar
23. Sather-Wagstaff, S., Complete intersection dimensions and Foxby classes, J. Pure Appl. Alg. 212(12) (2008), 2594–2611.Google Scholar
24. Serre, J.-P., Sur la dimension homologique des anneaux et des modules noethériens, in Proc. of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955, pp. 175–189 (Science Council of Japan, Tokyo, 1956).Google Scholar
25. Takahashi, R., Upper complete intersection dimension relative to a local homomorphism, Tokyo J. Math. 27(1) (2004), 209–219.Google Scholar
26. Takahashi, R. and Yoshino, Y., Characterizing Cohen–Macaulay local rings by Frobenius maps, Proc. Am. Math. Soc. 132(11) (2004), 3177–3187.CrossRefGoogle Scholar
27. Veliche, O., Construction of modules with finite homological dimensions, J. Alg. 250(2) (2002), 427–449.CrossRefGoogle Scholar