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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the depth of the symmetric algebra


Authors: J. Herzog, M. E. Rossi and G. Valla
Journal: Trans. Amer. Math. Soc. 296 (1986), 577-606
MSC: Primary 13C15; Secondary 13H10
MathSciNet review: 846598
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Abstract: Let $ (R,\mathfrak{m})$ be a local ring. Assume that $ R = A/I$, where $ (A,\mathfrak{n})$ is a regular local ring and $ I \subseteq {\mathfrak{n}^2}$ is an ideal. The depth of the symmetric algebra $ S(\mathfrak{m})$ of $ \mathfrak{m}$ over $ R$ is computed in terms of the depth of the associated graded module $ {\text{gr}_\mathfrak{n}}(I)$ and the so-called "strong socle condition." Explicit results are obtained, for instance, if $ I$ is generated by a super-regular sequence, if $ I$ has a linear resolution or if $ I$ has projective dimension one.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0846598-8
PII: S 0002-9947(1986)0846598-8
Article copyright: © Copyright 1986 American Mathematical Society