Hostname: page-component-7d684dbfc8-tvhzr Total loading time: 0 Render date: 2023-09-24T13:18:58.230Z Has data issue: false Feature Flags: { "corePageComponentGetUserInfoFromSharedSession": true, "coreDisableEcommerce": false, "coreDisableSocialShare": false, "coreDisableEcommerceForArticlePurchase": false, "coreDisableEcommerceForBookPurchase": false, "coreDisableEcommerceForElementPurchase": false, "coreUseNewShare": true, "useRatesEcommerce": true } hasContentIssue false

Direct Summands of Syzygy Modules of the Residue Class Field

Published online by Cambridge University Press:  11 January 2016

Ryo Takahashi*
Department of Mathematics, School of Science and Technology, Meiji University, Kawasaki 214-8571, Japan,
Department of Mathematical Sciences, Faculty of Science, Shinshu University, Matsumoto 390-8621, Japan,
Rights & Permissions [Opens in a new window]


Core share and HTML view are not possible as this article does not have html content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let R be a commutative Noetherian local ring. This paper deals with the problem asking whether R is Gorenstein if the nth syzygy module of the residue class field of R has a non-trivial direct summand of finite G-dimension for some n. It is proved that if n is at most two then it is true, and moreover, the structure of the ring R is determined essentially uniquely.


Research Article
Copyright © Editorial Board of Nagoya Mathematical Journal 2008


[1] Auslander, M., Anneaux de Gorenstein, et torsion en alg`ebre commutative, Sèminaire d’Alg`ebre Commutative dirigè par Pierre Samuel, 1966/67. Texte rèdigè, d’apr`es des exposès de Maurice Auslander, Marquerite Mangeney, Christian Peskine et Lucien Szpiro. Ecole Normale Supèrieure de Jeunes Filles, Secrètariat mathèmatique, Paris, 1967.Google Scholar
[2] Auslander, M., Minimal Cohen-Macaulay approximations, unpublished paper.Google Scholar
[3] Auslander, M. and Bridger, M., Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969.Google Scholar
[4] Auslander, M., Ding, S. and Solberg, Ø., Liftings and weak liftings of modules, J. Algebra, 156 (1993), 273317.CrossRefGoogle Scholar
[5] Avramov, L. L., Infinite free resolutions, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math., 166, Birkhauser, Basel, 1998, pp. 1118.Google Scholar
[6] Avramov, L. L., Homological dimensions and related invariants of modules over local rings, Representations of Algebras, ICRA IX (Beijing, 2000), vol. I, Beijing Normal Univ. Press, 2002, pp. 139.Google Scholar
[7] Bruns, W. and Herzog, J., Cohen-Macaulay rings, revised edition, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, Cambridge, 1998.Google Scholar
[8] Christensen, L. W., Gorenstein dimensions, Lecture Notes in Mathematics, 1747, Springer-Verlag, Berlin, 2000.Google Scholar
[9] Dutta, S. P., Syzygies and homological conjectures, Commutative algebra (Berkeley, CA, 1987), Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989, pp. 139156.CrossRefGoogle Scholar
[10] Herzog, J., Ringe mit nur endlich vielen Isomorphieklassen von maximalen, unzerleg-baren Cohen-Macaulay-Moduln, Math. Ann., 233 (1978), no. 1, 2134.CrossRefGoogle Scholar
[11] Martsinkovsky, A., A remarkable property of the (co) syzygy modules of the residue field of a nonregular local ring, J. Pure Appl. Algebra, 110 (1996), no. 1, 913.CrossRefGoogle Scholar
[12] Nagata, M., Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers, John Wiley & Sons, New York-London, 1962.Google Scholar
[13] Takahashi, R., On the category of modules of Gorenstein dimension zero II, J. Algebra, 278 (2004), no. 1, 402410.CrossRefGoogle Scholar
[14] Tate, J., Homology of Noetherian rings and local rings, Illinois J. Math., 1 (1957), 1427.Google Scholar
[15] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.Google Scholar
[16] Yoshino, Y., Cohen-Macaulay approximations (Japanese), Proceedings of the 4th Symposium on Representation Theory of Algebras, Izu, Japan, 1993, pp. 119138.Google Scholar
[17] Yoshino, Y. and Kawamoto, T., The fundamental module of a normal local domain of dimension 2, Trans. Amer. Math. Soc., 309 (1988), no. 1, 425431.Google Scholar