On the higher delta invariants of a Gorenstein local ring
HTML articles powered by AMS MathViewer
- by Yuji Yoshino
- Proc. Amer. Math. Soc. 124 (1996), 2641-2647
- DOI: https://doi.org/10.1090/S0002-9939-96-03376-X
- PDF | Request permission
Abstract:
Let $(R , \mathfrak {m} )$ be a Gorenstein complete local ring. Auslander’s higher delta invariants are denoted by $\delta _R ^n(M)$ for each module $M$ and for each integer $n$. We propose a conjecture asking if $\delta _R ^n (R/\mathfrak {m} ^{\ell }) = 0$ for any positive integers $n$ and $\ell$. We prove that this is true provided the associated graded ring of $R$ has depth not less than $\operatorname {dim} R -1$. Furthermore we show that there are only finitely many possibilities for a pair of positive integers $(n, \ell )$ for which $\delta _R ^n (R/ \mathfrak {m} ^{\ell }) >0$.References
- Maurice Auslander and Ragnar-Olaf Buchweitz, The homological theory of maximal Cohen-Macaulay approximations, Mém. Soc. Math. France (N.S.) 38 (1989), 5–37 (English, with French summary). Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR 1044344
- Maurice Auslander, Songqing Ding, and Øyvind Solberg, Liftings and weak liftings of modules, J. Algebra 156 (1993), no. 2, 273–317. MR 1216471, DOI 10.1006/jabr.1993.1076
- Songqing Ding, Cohen-Macaulay approximation and multiplicity, J. Algebra 153 (1992), no. 2, 271–288. MR 1198202, DOI 10.1016/0021-8693(92)90156-G
- Songqing Ding, A note on the index of Cohen-Macaulay local rings, Comm. Algebra 21 (1993), no. 1, 53–71. MR 1194550, DOI 10.1080/00927879208824550
- Songqing Ding, The associated graded ring and the index of a Gorenstein local ring, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1029–1033. MR 1181160, DOI 10.1090/S0002-9939-1994-1181160-1
- Jürgen Herzog, On the index of a homogeneous Gorenstein ring, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 95–102. MR 1266181, DOI 10.1090/conm/159/01506
- A. Shida, On indices of local rings along local homomorphisms, Preprint, Nagoya Univ. (1994).
Bibliographic Information
- Yuji Yoshino
- Affiliation: Institute of Mathematics, Faculty of Integrated Human Studies, Kyoto University, Yoshida-Nihonmatsu, Sakyo-ku, Kyoto 606-01, Japan
- Email: yoshino@math.h.kyoto-u.ac.jp
- Received by editor(s): January 20, 1995
- Received by editor(s) in revised form: March 9, 1995
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2641-2647
- MSC (1991): Primary 13C14, 13D02, 13H10, 16G50
- DOI: https://doi.org/10.1090/S0002-9939-96-03376-X
- MathSciNet review: 1327054
Dedicated: To the memory of Professor Maurice Auslander