Abstract
We refine a well-known theorem of Auslander and Reiten about the extension closedness of nth syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre’s condition (S n ) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen–Macaulay local ring with canonical module to an arbitrary local ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araya, T., Iima, K.-i.: Locally Gorensteinness over Cohen-Macaulay rings, Preprint. arXiv:1408.3796v1 (2014)
Auslander, M, Bridger, M: Stable module theory, Memoirs of the American Mathematical Society, No. 94. American Mathematical Society, Providence (1969)
Auslander, M, Reiten, I: Applications of contravariantly finite subcategories. Adv Math 86(1), 111–152 (1991)
Auslander, M, Reiten, I: Homologically finite subcategories. In: Representations of algebras and related topics. London Math. Soc. Lecture Note Ser., 168, Cambridge Univ. Press, Cambridge, 1992, pp 1–42, Kyoto (1990)
Auslander, M, Reiten, I: Syzygy modules for Noetherian rings. J Algebra 183 (1), 167–185 (1996)
Bruns, W, Herzog, J: Cohen–Macaulay rings, Revised edition, Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Evans, EG, Griffith, P: Syzygies, London mathematical society lecture note series, vol. 106. Cambridge University Press, Cambridge (1985)
Fossum, R, Foxby, H-B, Griffith, P, Reiten, I: Minimal injective resolutions with applications to dualizing modules and Gorenstein modules. Inst Hautes Études Sci Publ Math 45, 193–215 (1975)
Hartshorne, R: Complete intersections and connectedness. Amer J Math 84, 497–508 (1962)
Leuschke, G, Wiegand, R: Ascent of finite Cohen-Macaulay type. J Algebra 228(2), 674–681 (2000)
Mac Lane, S: Categories for the working mathematician, Second edition, Graduate Texts in Mathematics, 5 (1998)
Author information
Authors and Affiliations
Corresponding author
Additional information
Presented by Peter Littelmann.
The first author was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400051. The second author was partly supported by JSPS Grant-in-Aid for Scientific Research (C) 25400038.
Rights and permissions
About this article
Cite this article
Goto, S., Takahashi, R. Extension Closedness of Syzygies and Local Gorensteinness of Commutative Rings. Algebr Represent Theor 19, 511–521 (2016). https://doi.org/10.1007/s10468-015-9585-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-015-9585-0