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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Torsion freeness of symmetric powers of ideals
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by Alexandre B. Tchernev PDF
Trans. Amer. Math. Soc. 359 (2007), 3357-3367 Request permission

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
  • MR Author ID: 610821
  • Email: tchernev@math.albany.edu
  • Received by editor(s): October 29, 2004
  • Received by editor(s) in revised form: July 5, 2005
  • Published electronically: January 26, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 359 (2007), 3357-3367
  • MSC (2000): Primary 13C12, 13D30, 13A30
  • DOI: https://doi.org/10.1090/S0002-9947-07-04135-9
  • MathSciNet review: 2299459