Auslander's -invariants of Gorenstein local rings

Author:
Songqing Ding

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

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Abstract: Let (*R*, , *k*) be a Gorenstein local ring with associated graded ring . It is conjectured that for any integer , Auslander's -invariant of equals 1 if and only if is contained in a parameter ideal of *R*. In an earlier paper we showed that the conjecture holds if is Cohen-Macaulay. In this paper we prove that the conjecture has an affirmative answer if depth and *R* is gradable. We also prove that if *R* is not regular and depth , then if and only if *R* has minimal multiplicity.

**[1]**M. Auslander,*Minimal Cohen-Macaulay approximations*(in preparation).**[2]**M. Auslander and R. O. Buchweitz,*The homological theory of maximal Cohen-Macaulay approximations*, Mém. Soc. Math. France (N.S.), no. 38, Soc. Math. France, Paris, 1989, pp. 5-37. MR**1044344 (91h:13010)****[3]**S. Ding,*A note on the index of Cohen-Macaulay local rings*, Comm. Algebra**21**(1993), 53-71. MR**1194550 (94b:13014)****[4]**-,*The associated graded ring and the index of a Gorenstein local ring*, Proc. Amer. Math. Soc. (to appear) MR**1181160 (94f:13014)****[5]**J. Herzog,*On the index of a homogeneous Gorenstein ring*, preprint, 1992. MR**1266181 (95b:13031)****[6]**J. Sally,*Tangent cones at Gorenstein singularities*, Comput. Math. Appl., vol. 40, Academic Press, New York, 1980, pp. 169-175. MR**563540 (81e:14004)****[7]**K. Watenabe,*Some examples of one dimensional Gorenstein domains*, Nagoya Math. J.**49**(1973), 101-109. MR**0318140 (47:6689)**

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DOI:
https://doi.org/10.1090/S0002-9939-1994-1203983-2

Article copyright:
© Copyright 1994
American Mathematical Society