Torsion freeness of symmetric powers of ideals

Abstract

Let I I be an ideal in a Noetherian commutative ring R R with unit, let k ≥ 2 k\ge 2 be an integer, and let α k :   S k I ⟶ I k \alpha _k\! :\ S_k I\longrightarrow I^k be the canonical surjective R R -module homomorphism from the k k th symmetric power of I I to the k k th power of I I . When p d R I ≤ 1 \mathrm {pd}_R I\le 1 or when I I is a perfect Gorenstein ideal of grade 3 3 , we provide a necessary and sufficient condition for α k \alpha _k to be an isomorphism in terms of upper bounds for the minimal number of generators of the localisations of  I I . When I = m I=\mathfrak {m} is a maximal ideal of R R we show that α k \alpha _k is an isomorphism if and only if R m R_{\mathfrak {m}} is a regular local ring. In all three cases for I I our results yield that if α k \alpha _k is an isomorphism, then α t \alpha _t is also an isomorphism for each 1 ≤ t ≤ k 1\le t\le k .