When is $R \ltimes I$ an almost Gorenstein local ring?
HTML articles powered by AMS MathViewer
- by Shiro Goto and Shinya Kumashiro PDF
- Proc. Amer. Math. Soc. 146 (2018), 1431-1437 Request permission
Abstract:
Let $(R, \mathfrak {m})$ be a Gorenstein local ring of dimension $d > 0$ and let $I$ be an ideal of $R$ such that $(0) \ne I \subsetneq R$ and $R/I$ is a Cohen-Macaulay ring of dimension $d$. There is given a complete answer to the question of when the idealization $A = R \ltimes I$ of $I$ over $R$ is an almost Gorenstein local ring.References
- Valentina Barucci and Ralf Fröberg, One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), no. 2, 418–442. MR 1435367, DOI 10.1006/jabr.1996.6837
- Joseph P. Brennan, Jürgen Herzog, and Bernd Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), no. 2, 181–203. MR 947472, DOI 10.7146/math.scand.a-12198
- T. D. M. Chau, S. Goto, S. Kumashiro, and N. Matsuoka, Sally modules of canonical ideals in dimension one and $2$-AGL rings, Preprint 2017.
- Shiro Goto and Futoshi Hayasaka, Finite homological dimension and primes associated to integrally closed ideals, Proc. Amer. Math. Soc. 130 (2002), no. 11, 3159–3164. MR 1912992, DOI 10.1090/S0002-9939-02-06436-5
- Shiro Goto, Naoyuki Matsuoka, and Tran Thi Phuong, Almost Gorenstein rings, J. Algebra 379 (2013), 355–381. MR 3019262, DOI 10.1016/j.jalgebra.2013.01.025
- Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida, The almost Gorenstein Rees algebras of parameters, J. Algebra 452 (2016), 263–278. MR 3461066, DOI 10.1016/j.jalgebra.2015.12.022
- Shiro Goto, Naoyuki Matsuoka, Naoki Taniguchi, and Ken-ichi Yoshida, The almost Gorenstein Rees algebras over two-dimensional regular local rings, J. Pure Appl. Algebra 220 (2016), no. 10, 3425–3436. MR 3497969, DOI 10.1016/j.jpaa.2016.04.007
- K. Yoshida, S. Goto, N. Taniguchi, and N. Matsuoka, Almost Gorenstein Rees algebras, Proceedings of the 48th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Yamanashi, 2016, pp. 152–159 (Japanese, with English summary). MR 3524258
- K. Yoshida, S. Goto, N. Taniguchi, and N. Matsuoka, Almost Gorenstein Rees algebras, Proceedings of the 48th Symposium on Ring Theory and Representation Theory, Symp. Ring Theory Represent. Theory Organ. Comm., Yamanashi, 2016, pp. 152–159 (Japanese, with English summary). MR 3524258
- S. Goto, M. Rahimi, N. Taniguchi, and H. L. Truong, When are the Rees algebras of parameter ideals almost Gorenstein graded rings?, Kyoto J. Math. (to appear).
- Shiro Goto, Ryo Takahashi, and Naoki Taniguchi, Almost Gorenstein rings—towards a theory of higher dimension, J. Pure Appl. Algebra 219 (2015), no. 7, 2666–2712. MR 3313502, DOI 10.1016/j.jpaa.2014.09.022
- Jürgen Herzog and Ernst Kunz (eds.), Der kanonische Modul eines Cohen-Macaulay-Rings, Lecture Notes in Mathematics, Vol. 238, Springer-Verlag, Berlin-New York, 1971. Seminar über die lokale Kohomologietheorie von Grothendieck, Universität Regensburg, Wintersemester 1970/1971. MR 0412177
- Idun Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417–420. MR 296067, DOI 10.1090/S0002-9939-1972-0296067-7
- Ryo Takahashi, On $G$-regular local rings, Comm. Algebra 36 (2008), no. 12, 4472–4491. MR 2473342, DOI 10.1080/00927870802179602
- N. Taniguchi, On the almost Gorenstein property of determinantal rings, arXiv:1701.06690v1.
Additional Information
- Shiro Goto
- Affiliation: Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki 214-8571, Japan
- MR Author ID: 192104
- Email: shirogoto@gmail.com
- Shinya Kumashiro
- Affiliation: Department of Mathematics and Informatics, Graduate School of Science and Technology, Chiba University, Chiba-shi 263, Japan
- Email: polar1412@gmail.com
- Received by editor(s): March 17, 2017
- Received by editor(s) in revised form: May 11, 2017
- Published electronically: November 7, 2017
- Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 25400051. Both authors are partially supported by JSPS Bilateral Programs (Joint Research) and International Research Supporting Program of Meiji University
- Communicated by: Irena Peeva
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1431-1437
- MSC (2010): Primary 13H10; Secondary 13H05, 13H15
- DOI: https://doi.org/10.1090/proc/13835
- MathSciNet review: 3754330