Auslander’s $\delta$-invariants of Gorenstein local rings
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Abstract:
Let (R, $\mathfrak {m}$, k) be a Gorenstein local ring with associated graded ring $G(R)$. It is conjectured that for any integer $n > 0$, Auslander’s $\delta$-invariant $\delta (R/{\mathfrak {m}^n})$ of $R/{\mathfrak {m}^n}$ equals 1 if and only if ${\mathfrak {m}^n}$ is contained in a parameter ideal of R. In an earlier paper we showed that the conjecture holds if $G(R)$ is Cohen-Macaulay. In this paper we prove that the conjecture has an affirmative answer if depth $G(R) = \dim R - 1$ and R is gradable. We also prove that if R is not regular and depth $G(R) \geq \dim R - 1$, then $\delta (R/{\mathfrak {m}^2}) = 1$ if and only if R has minimal multiplicity.References
-
M. Auslander, Minimal Cohen-Macaulay approximations (in preparation).
- 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
- 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
- Judith D. Sally, Tangent cones at Gorenstein singularities, Compositio Math. 40 (1980), no. 2, 167–175. MR 563540
- Keiichi Watanabe, Some examples of one dimensional Gorenstein domains, Nagoya Math. J. 49 (1973), 101–109. MR 318140, DOI 10.1017/S0027763000015312
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 122 (1994), 649-656
- MSC: Primary 13H10; Secondary 13A15, 13C14
- DOI: https://doi.org/10.1090/S0002-9939-1994-1203983-2
- MathSciNet review: 1203983