The Briançon-Skoda theorem and coefficient ideals for non-𝔪-primary ideals

Abstract

We generalize a Briançon-Skoda type theorem first studied by Aberbach and Huneke. With some conditions on a regular local ring ( R , m ) (R,\mathfrak {m}) containing a field, and an ideal I I of R R with analytic spread ℓ \ell and a minimal reduction J J , we prove that for all w ≥ − 1 w \geq -1 , I ℓ + w ¯ ⊆ J w + 1 a ( I , J ) , \overline {I^{\ell +w}} \subseteq J^{w+1} \mathfrak {a} (I,J), where a ( I , J ) \mathfrak {a}(I,J) is the coefficient ideal of I I relative to J J , i.e. the largest ideal b \mathfrak {b} such that I b = J b I\mathfrak {b}=J\mathfrak {b} . Previously, this result was known only for m \mathfrak {m} -primary ideals.