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

 

Torsion freeness of symmetric powers of ideals


Author: Alexandre B. Tchernev
Journal: Trans. Amer. Math. Soc. 359 (2007), 3357-3367
MSC (2000): Primary 13C12, 13D30, 13A30
Published electronically: January 26, 2007
MathSciNet review: 2299459
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Abstract: Let $ I$ be an ideal in a Noetherian commutative ring $ R$ with unit, let $ k\ge 2$ be an integer, and let $ \alpha_k : S_k I\longrightarrow I^k$ be the canonical surjective $ R$-module homomorphism from the $ k$th symmetric power of $ I$ to the $ k$th power of $ I$. When $ \mathrm{pd}_R I\le 1$ or when $ I$ is a perfect Gorenstein ideal of grade $ 3$, we provide a necessary and sufficient condition for $ \alpha_k$ to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of $ I$. When $ I=\mathfrak{m}$ is a maximal ideal of $ R$ we show that $ \alpha_k$ is an isomorphism if and only if $ R_{\mathfrak{m}}$ is a regular local ring. In all three cases for $ I$ our results yield that if $ \alpha_k$ is an isomorphism, then $ \alpha_t$ is also an isomorphism for each $ 1\le t\le k$.


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Additional Information

Alexandre B. Tchernev
Affiliation: Department of Mathematics, University at Albany, SUNY, Albany, New York 12222
Email: tchernev@math.albany.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04135-9
PII: S 0002-9947(07)04135-9
Keywords: Torsion freeness, symmetric algebra, Rees algebra, symmetric powers
Received by editor(s): October 29, 2004
Received by editor(s) in revised form: July 5, 2005
Published electronically: January 26, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.