A variant of Wangʼs theorem

Abstract

In this paper, we give a new formula of J:J¯ for any parameter ideal J in a Gorenstein local ring R of positive characteristic in terms of test ideals: J:J¯=J+τ(Jd−1), where τ(Jd−1) denotes the Jd−1-test ideal of R.As an application, we give a variant of Wangʼs theorem. Namely, we prove that if J is a parameter ideal in a Cohen–Macaulay local ring (R,m) of dimension d⩾2 with J⊆ms, then J:m(d−1)(s−1) (resp. J:m(d−1)(s−1)+1) is integral over J (resp. if R is not regular).Moreover, we prove that, after reduction to characteristic p≫0, a similar assertion holds true for Cohen–Macaulay Q-Gorenstein normal local domain essentially of finite type over a field of characteristic zero under some extra assumption.