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When is $ R \ltimes I$ an almost Gorenstein local ring?


Authors: Shiro Goto and Shinya Kumashiro
Journal: Proc. Amer. Math. Soc. 146 (2018), 1431-1437
MSC (2010): Primary 13H10; Secondary 13H05, 13H15
DOI: https://doi.org/10.1090/proc/13835
Published electronically: November 7, 2017
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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.


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  • [1] Valentina Barucci and Ralf Fröberg, One-dimensional almost Gorenstein rings, J. Algebra 188 (1997), no. 2, 418-442. MR 1435367, https://doi.org/10.1006/jabr.1996.6837
  • [2] Joseph P. Brennan, Jürgen Herzog, and Bernd Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), no. 2, 181-203. MR 947472, https://doi.org/10.7146/math.scand.a-12198
  • [3] 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.
  • [4] 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, https://doi.org/10.1090/S0002-9939-02-06436-5
  • [5] Shiro Goto, Naoyuki Matsuoka, and Tran Thi Phuong, Almost Gorenstein rings, J. Algebra 379 (2013), 355-381. MR 3019262, https://doi.org/10.1016/j.jalgebra.2013.01.025
  • [6] 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, https://doi.org/10.1016/j.jalgebra.2015.12.022
  • [7] 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, https://doi.org/10.1016/j.jpaa.2016.04.007
  • [8] 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
  • [9] 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
  • [10] 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).
  • [11] 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, https://doi.org/10.1016/j.jpaa.2014.09.022
  • [12] 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
  • [13] Idun Reiten, The converse to a theorem of Sharp on Gorenstein modules, Proc. Amer. Math. Soc. 32 (1972), 417-420. MR 0296067, https://doi.org/10.2307/2037829
  • [14] Ryo Takahashi, On $ G$-regular local rings, Comm. Algebra 36 (2008), no. 12, 4472-4491. MR 2473342, https://doi.org/10.1080/00927870802179602
  • [15] N. Taniguchi, On the almost Gorenstein property of determinantal rings, arXiv:1701.06690v1.

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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
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

DOI: https://doi.org/10.1090/proc/13835
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
Article copyright: © Copyright 2017 American Mathematical Society

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