A characterization of multiplier ideals via ultraproducts

Abstract

In this paper, using ultra-Frobenii, we introduce a variant of Schoutens’ non-standard tight closure (Schoutens in Manuscr Math 111:379–412, 2003), ultra-tight closure, on ideals of a local domain R essentially of finite type over $$\mathbb {C}$$ . We prove that the ultra-test ideal $$\tau _{\textrm{u}}(R,\mathfrak {a}^t)$$ , the annihilator ideal of all ultra-tight closure relations of R, coincides with the multiplier ideal $$\mathcal {J}({\text {Spec}}R,\mathfrak {a}^t)$$ if R is normal $$\mathbb {Q}$$ -Gorenstein. As an application, we study a behavior of multiplier ideals under pure ring extensions.