Torsion freeness of symmetric powers of ideals

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 $1\le t\le k$.