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Torsion freeness of symmetric powers of ideals

Authors:	Alexandre B Tchernev

Institution:	Department of Mathematics, University at Albany, SUNY, Albany, New York 12222

Abstract:	Let be an ideal in a Noetherian commutative ring with unit, let $k\ge 2$ be an integer, and let $\alpha_k : S_k I\longrightarrow I^k$ be the canonical surjective -module homomorphism from the th symmetric power of to the th power of . When $\mathrm{pd}_R I\le 1$ or when is a perfect Gorenstein ideal of grade , 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 . When $I=\mathfrak{m}$ is a maximal ideal of 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 our results yield that if $\alpha_k$ is an isomorphism, then $\alpha_t$ is also an isomorphism for each $1\le t\le k$ .

Keywords:	Torsion freeness symmetric algebra Rees algebra symmetric powers

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