BIG COHEN–MACAULAY TEST IDEALS IN EQUAL CHARACTERISTIC ZERO VIA ULTRAPRODUCTS

Abstract

Abstract Utilizing ultraproducts, Schoutens constructed a big Cohen–Macaulay (BCM) algebra $\mathcal {B}(R)$ over a local domain R essentially of finite type over $\mathbb {C}$ . We show that if R is normal and $\Delta $ is an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, then the BCM test ideal $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ coincides with the multiplier ideal $\mathcal {J}(\widehat {R},\widehat {\Delta })$ of $(\widehat {R},\widehat {\Delta })$ , where $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$ -adic completions of R and $\mathcal {B}(R)$ , respectively, and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ . As an application, we obtain a result on the behavior of multiplier ideals under pure ring extensions.