On the injectivity of blowing-up ring morphisms

Let I=(x1,…,xr) be a finitely generated ideal in a commutative ring R and let n⩾2 be an integer. Let αI,n:Sn(I)→In be the canonical morphism from the nth symmetric power of I onto the nth power of I. Recently Tchernev asked for when αI,n being an isomorphism implies that αI,p is an isomorphism for each 2⩽p⩽n. We give an affirmative answer provided that the ideal J=(x1,…,xr−1) verifies that αJ,p:Sp(J)→Jp is an isomorphism for all 2⩽p⩽n. In addition, for every n⩾2, we give an example of an ideal I such that αI,p is an isomorphism for all p⩾n+1 and αI,n is not.