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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Auslander's $ \delta$-invariants of Gorenstein local rings


Author: Songqing Ding
Journal: Proc. Amer. Math. Soc. 122 (1994), 649-656
MSC: Primary 13H10; Secondary 13A15, 13C14
MathSciNet review: 1203983
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1994-1203983-2
PII: S 0002-9939(1994)1203983-2
Article copyright: © Copyright 1994 American Mathematical Society