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

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.

1 Introduction

A (balanced) big Cohen–Macaulay (BCM) algebra over a Noetherian local ring $(R,\mathfrak {m})$ is an R-algebra B such that every system of parameters is a regular sequence on B. Its existence implies many fundamental homological conjectures including the direct summand conjecture (now a theorem). Hochster and Huneke [14], [15] proved the existence of a BCM algebra in equal characteristic, and André [1] settled the mixed characteristic case. Recently, using BCM algebras, Ma and Schwede [18], [19] introduced the notion of BCM test ideals as an analog of test ideals in tight closure theory.

The test ideal $\tau (R)$ of a Noetherian local ring R of positive characteristic was originally defined as the annihilator ideal of all tight closure relations of R. Since it turned out that $\tau (R)$ was related to multiplier ideals via reduction to characteristic p, the definition of $\tau (R)$ was generalized in [11], [29] to involve effective $\mathbb {Q}$ -Weil divisors $\Delta $ on $\operatorname {Spec} R$ and ideals $\mathfrak {a}\subseteq R$ with real exponent $t>0$ . In these papers, it was shown that multiplier ideals coincide, after reduction to characteristic $p \gg 0$ , with such generalized test ideals $\tau (R,\Delta ,\mathfrak {a}^t)$ . In positive characteristic, Ma-Schwede’s BCM test ideals are the same as the generalized test ideals. In this paper, we study BCM test ideals in equal characteristic zero.

Using ultraproducts, Schoutens [24] gave a characterization of log-terminal singularities, an important class of singularities in the minimal model program. He also gave an explicit construction of a BCM algebra $\mathcal {B}(R)$ in equal characteristic zero: $\mathcal {B}(R)$ is described as the ultraproduct of the absolute integral closures of Noetherian local domains of positive characteristic. He defined a closure operation associated with $\mathcal {B}(R)$ to introduce the notions of $\mathcal {B}$ -rationality and $\mathcal {B}$ -regularity, which are closely related to BCM rationality and BCM regularity defined in [19], and proved that $\mathcal {B}$ -rationality is equivalent to being rational singularities. The aim of this paper is to give a geometric characterization of BCM test ideals associated with $\mathcal {B}(R)$ . Our main result is stated as follows:

Theorem 1.1 (Theorem 6.4).

Let R be a normal local domain essentially of finite type over $\mathbb {C}$ . Let $\Delta $ be an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, where $K_R$ is a canonical divisor on $\operatorname {Spec} R$ . Suppose that $\widehat {R}$ and $\widehat {\mathcal {B}(R)}$ are the $\mathfrak {m}$ -adic completions of R and $\mathcal {B}(R)$ , and $\widehat {\Delta }$ is the flat pullback of $\Delta $ by the canonical morphism $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ . Then we have

$$ \begin{align*} \tau_{\widehat{\mathcal{B}(R)}}(\widehat{R},\widehat{\Delta})=\mathcal{J}(\widehat{R},\widehat{\Delta}), \end{align*} $$

where $\tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ is the BCM test ideal of $(\widehat {R},\widehat {\Delta })$ with respect to $\widehat {\mathcal {B}(R)}$ and $\mathcal {J}(\widehat {R},\widehat {\Delta })$ is the multiplier ideal of $(\widehat {R},\widehat {\Delta })$ .

The inclusion $\mathcal {J}(\widehat {R},\widehat {\Delta }) \subseteq \tau _{\widehat {\mathcal {B}(R)}}(\widehat {R},\widehat {\Delta })$ is obtained by comparing reductions of the multiplier ideal modulo $p\gg 0$ to its approximations. We prove the opposite inclusion by combining an argument similar to that in [25] with the description of multiplier ideals as the kernel of a map between local cohomology modules in [29]. As an application of Theorem 1.1, we show the next result about a behavior of multiplier ideals under pure ring extensions, which is a generalization of [31, Cor. 5.30].

Theorem 1.2 (Corollary 7.11).

Let $R\hookrightarrow S$ be a pure local homomorphism of normal local domains essentially of finite type over $\mathbb {C}$ . Suppose that R is $\mathbb {Q}$ -Gorenstein. Let $\Delta _S$ be an effective $\mathbb {Q}$ -Weil divisor such that $K_S+\Delta _S$ is $\mathbb {Q}$ -Cartier, where $K_S$ is a canonical divisor on $\operatorname {Spec} S$ . Let $\mathfrak {a}\subseteq R$ be a nonzero ideal, and let $t>0$ be a positive rational number. Then we have

$$ \begin{align*} \mathcal{J}(S,\Delta_S,(\mathfrak{a} S)^t)\cap R\subseteq \mathcal{J}(R,\mathfrak{a}^t). \end{align*} $$

In [31], we defined ultra-test ideals, a variant of test ideals in equal characteristic zero, to generalize the notion of ultra-F-regularity introduced by Schoutens [24]. Theorem 1.2 was proved by using ultra-test ideals under the assumption that $\mathfrak {a}$ is a principal ideal. The description of multiplier ideals as BCM test ideals associated with $\mathcal {B}(R)$ (Theorem 1.1) and a generalization of module closures in [20] enables us to show Theorem 1.2 without any assumptions.

As another application of Theorem 1.1, we give an affirmative answer to one of the conjectures proposed by Schoutens [24, Rem. 3.10], which says that $\mathcal {B}$ -regularity is equivalent to being log-terminal singularities (see Theorem 8.2).

This paper is organized as follows: in the preliminary section, we give definitions of multiplier ideals, test ideals, and BCM test ideals. In §3, we quickly review the theory of ultraproducts in commutative algebra including non-standard and relative hulls. In §4, we prove some fundamental results on BCM algebras constructed via ultraproducts following [23]. In §5, we review the relationship between approximations and reductions modulo $p\gg 0$ and consider approximations of multiplier ideals. In §6, we show Theorem 1.1, the main theorem of this paper. In §7, using a generalized module closure, we show Theorem 1.2 as an application of Theorem 1.1. In §8, we show that $\mathcal {B}$ -regularity is equivalent to log-terminal singularities. Finally in §9, we discuss a question, a variant of [7, Quest. 2.7], to handle BCM algebras that cannot be constructed via ultraproducts, and consider the equivalence of BCM-rationality and being rational singularities.

2 Preliminaries

Throughout this paper, all rings will be commutative with unity.

2.1 Multiplier ideals

Here, we briefly review the definition of multiplier ideals and refer the reader to [16], [21] for more details. Throughout this subsection, we assume that X is a normal integral scheme essentially of finite type over a field of characteristic zero or $X=\operatorname {Spec} \widehat {R}$ , where $(R,\mathfrak {m})$ is a normal local domain essentially of finite type over a field of characteristic zero and $\widehat {R}$ is its $\mathfrak {m}$ -adic completion.

Definition 2.1. A proper birational morphism $f:Y\to X$ between integral schemes is said to be a resolution of singularities of X if Y is regular. When $\Delta $ is a $\mathbb {Q}$ -Weil divisor on X and $\mathfrak {a} \subseteq \mathcal {O}_X$ is a nonzero coherent ideal sheaf, a resolution $f:Y \to X$ is said to be a log resolution of $(X,\Delta ,\mathfrak {a})$ if $\mathfrak {a}\mathcal {O}_Y=\mathcal {O}_Y(-F)$ is invertible and if the union of the exceptional locus $\operatorname {Exc} (f)$ of f and the support F and the strict transform $f_*^{-1}\Delta $ of $\Delta $ is a simple normal crossing divisor.

If $f:Y\to X$ is a proper birational morphism with Y a normal integral scheme and $\Delta $ is a $\mathbb {Q}$ -Weil divisor, then we can choose $K_Y$ such that $f^*(K_X+\Delta )-K_Y$ is a divisor supported on the exceptional locus of f. With this convention:

Definition 2.2. Let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor on X such that $K_X+\Delta $ is $\mathbb {Q}$ -Cartier, let $\mathfrak {a}\subseteq \mathcal {O}_X$ be a nonzero coherent ideal sheaf, and let $t>0$ be a positive real number. Then the multiplier ideal sheaf $\mathcal {J}(X,\Delta ,\mathfrak {a}^t)$ associated with $(X,\Delta ,\mathfrak {a}^t)$ is defined by

$$ \begin{align*} \mathcal{J} (X,\Delta,\mathfrak{a}^t)=f_*\mathcal{O}_Y(K_Y-\lfloor f^*(K_X+\Delta)+tF\rfloor). \end{align*} $$

where $f:Y\to X$ is a log resolution of $(X,\Delta ,\mathfrak {a})$ . Note that this definition is independent of the choice of log resolution.

Definition 2.3. Let X be a normal integral scheme essentially of finite type over a field of characteristic zero. We say that X has rational singularities if X is Cohen–Macaulay at x and if for any projective birational morphism $f:Y\to \operatorname {Spec} \mathcal {O}_{X,x}$ with Y a normal integral scheme, the natural morphism $f_*\omega _Y\to \omega _{X,x}$ is an isomorphism.

2.2 Tight closure and test ideals

In this subsection, we quickly review the basic notion of tight closure and test ideals. We refer the reader to [4], [11], [13], [29].

Definition 2.4. Let R be a normal domain of characteristic $p>0$ , let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor, let $\mathfrak {a}\subseteq R$ be a nonzero ideal, and let $t>0$ be a real number. Let $E=\bigoplus E(R/\mathfrak {m})$ be the direct sum, taken over all maximal ideals $\mathfrak {m}$ of R, of the injective hulls $E_R(R/\mathfrak {m})$ of the residue fields $R/\mathfrak {m}$ .

1. (1) Let I be an ideal of R. The $(\Delta ,\mathfrak {a}^t)$ -tight closure $I^{*\Delta ,\mathfrak {a}^t}$ of I is defined as follows: $x\in I^{*\Delta ,\mathfrak {a}^t}$ if and only if there exists a nonzero element $c\in R^{\circ }$ such that

   $$ \begin{align*} c\mathfrak{a}^{\lceil t(q-1)\rceil}x^q\subseteq I^{[q]}R(\lceil (q-1)\Delta\rceil) \end{align*} $$
   for all large $q=p^e$ , where $I^{[q]}=\{f^q|f\in I\}$ and $R^{\circ }=R\setminus \{0\}$ .

2. (2) If M is an R-module, then the $(\Delta , \mathfrak {a}^t)$ -tight closure $0_M^{*\Delta ,\mathfrak {a}^t}$ is defined as follows: $z\in 0_M^{*\Delta , \mathfrak {a}^t}$ if and only if there exists a nonzero element $c\in R^{\circ }$ such that

   $$ \begin{align*} (c\mathfrak{a}^{\lceil t(q-1)\rceil})^{1/q}\otimes z=0 \quad \text{in} \quad R(\lceil (q-1)\Delta\rceil)^{1/q}\otimes_R M \end{align*} $$
   for all large $q=p^e$ .

3. (3) The (big) test ideal $\tau (R,\Delta ,\mathfrak {a}^t)$ associated with $(R,\Delta ,\mathfrak {a}^t)$ is defined by

   $$ \begin{align*} \tau (R,\Delta,\mathfrak{a}^t)=\operatorname{Ann}_R(0_E^{*\Delta,\mathfrak{a}^t}). \end{align*} $$

   When $\mathfrak {a}=R$ , then we simply denote the ideal $\tau (R,\Delta )$ . We call the triple $(R,\Delta ,\mathfrak {a}^t)$ is strongly F-regular if $\tau (R,\Delta ,\mathfrak {a}^t)=R$ .

Definition 2.5 [8].

Let R be an F-finite Noetherian local domain of characteristic $p>0$ of dimension d. We say that R is F-rational if any ideal $I=(x_1,\dots ,x_d)$ generated by a system of parameters satisfies $I=I^*$ .

2.3 Big Cohen–Macaulay algebras

In this subsection, we will briefly review the theory of BCM algebras. Throughout this subsection, we assume that local rings $(R,\mathfrak {m})$ are Noetherian.

Definition 2.6. Let $(R,\mathfrak {m})$ be a local ring, and let $\mathbf {x} = x_1,\dots , x_n$ be a system of parameters. R-algebra B is said to be BCM with respect to $\mathbf {x}$ if $\mathbf {x}$ is a regular sequence on B. B is called a (balanced) BCM algebra if it is BCM with respect to $\mathbf {x}$ for every system of parameters $\mathbf {x}$ .

Remark 2.7 [5, Cor. 8.5.3].

If B is BCM with respect to $\mathbf {x}$ , then the $\mathfrak {m}$ -adic completion $\widehat {B}$ is (balanced) BCM.

About the existence of BCM algebras of residue characteristic $p>0$ , the following are proved in [3], [14].

Theorem 2.8. If $(R,\mathfrak {m})$ is an excellent local domain of residue characteristic $p>0$ , then the p-adic completion of absolute integral closure $R^+$ is a (balanced) BCM R-algebra.

Using BCM algebras, we can define a class of singularities.

Definition 2.9. If R is an excellent local ring of dimension d, and let B be a BCM R-algebra. We say that R is BCM-rational with respect to B (or simply $\text {BCM}_B$ -rational) if R is Cohen–Macaulay and if $H_{\mathfrak {m}}^d(R)\to H_{\mathfrak {m}}^d(B)$ is injective. We say that R is BCM-rational if R is $\text {BCM}_B$ -rational for any BCM algebra B.

We explain BCM test ideals introduced in [19].

Setting 2.10. Let $(R,\mathfrak {m})$ be a normal local domain of dimension d.

1. (i) $\Delta \geqslant 0$ is a $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier.

2. (ii) Fixing $\Delta $ , we also fix an embedding $R\subseteq \omega _R \subseteq \operatorname {Frac} R$ , where $\omega _R$ is the canonical module.

3. (iii) Since $K_R + \Delta $ is effective and $\mathbb {Q}$ -Cartier, there exist an integer $n>0$ and $f\in R$ such that $n(K_R+\Delta )=\operatorname {div} (f)$ .

Definition 2.11. With notation as in Setting 2.10, if B is a BCM $R[f^{1/n}]$ -algebra, then we define $0_{H_{\mathfrak {m}}^d(\omega _R)}^{B,K_R+\Delta }$ to be $\operatorname {Ker} \psi $ , where $\psi $ is the homomorphism determined by the below commutative diagram:

If R is $\mathfrak {m}$ -adically complete, then we define

$$ \begin{align*}\tau_B(R, \Delta)=\operatorname{Ann}_{R} 0_{H_{\mathfrak{m}}^d(\omega_R)}^{B,K_R+\Delta}. \end{align*} $$

We call $\tau _B(R,\Delta )$ the BCM test ideal of $(R,\Delta )$ with respect to B. We say that $(R,\Delta )$ is BCM regular with respect to B (or simply $\text {BCM}_{B}$ regular) if $\tau _B(R,\Delta )=R$ .

Proposition 2.12 [19].

Let $(R,\mathfrak {m})$ be a complete normal local domain of characteristic $p>0$ , let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R,$ and let B be a BCM $R^+$ -algebra. Fix an effective canonical divisor $K_R\geqslant 0$ . Suppose that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Then

$$ \begin{align*} \tau_B(R,\Delta)=\tau(R,\Delta). \end{align*} $$

3 Ultraproducts

3.1 Basic notions

In this subsection, we quickly review basic notions from the theory of ultraproduct. The reader is referred to [22], [26] for details. We fix an infinite set W. We use $\mathcal {P}(W)$ to denote the power set of W.

Definition 3.1. A nonempty subset $\mathcal {F} \subseteq \mathcal {P}(W)$ is called a filter if the following two conditions hold.

1. (i) If $A, B \in \mathcal {F}$ , then $A \cap B\in \mathcal {F}$ .

2. (ii) If $A \in \mathcal {F}$ and $A \subseteq B \subseteq W$ , then $B \in \mathcal {F}$ .

Definition 3.2. Let $\mathcal {F}$ be a filter on W.

1. (1) $\mathcal {F}$ is called an ultrafilter if for all $A \in \mathcal {P}(W)$ , we have $A \in \mathcal {F}$ or $A^c \in \mathcal {F}$ , where $A^c$ is the complement of A.

2. (2) $\mathcal {F}$ is called principal if there exists a finite subset $A\subseteq W$ such that $A \in \mathcal {F}$ .

Remark 3.3. By Zorn’s lemma, non-principal ultrafilters always exist.

Remark 3.4. Ultrafilters are an equivalent notion to two-valued finitely additive measures. If we have an ultrafilter $\mathcal {F}$ on W, then

$$ \begin{align*} m(A):= \left\{\!\!\! \begin{array}{ll} 1 & (A\in \mathcal{F}) \\ 0 & (A \notin \mathcal{F}) \end{array} \right. \end{align*} $$

is a two-valued finitely additive measure. Conversely, if $m:\mathcal {P}(W)\to \{0,1\}$ is a nonzero finitely additive measure, then $\mathcal {F}:=\{A\subseteq W|m(A)=1\}$ is an ultrafilter. Here, $\mathcal {F}$ is principal if and only if there exists an element $w_0$ of W such that $m(\{w_0\})=1$ . Hence, $\mathcal {F}$ is not principal if and only if $m(A)=0$ for any finite subset A of W.

Definition 3.5. Let $A_w$ be a family of sets indexed by W and $\mathcal {F}$ be an ultrafilter on W. Suppose that $a_w\in A_w$ for all $w\in W$ and $\varphi $ is a predicate. We say $\varphi (a_w)$ holds for almost all w if $\{w\in W|\varphi (a_w) \text { holds}\}\in \mathcal {F}$ .

Remark 3.6. This is an analog of “almost everywhere” or “almost surely” in analysis. The difference is that m is not countably but finitely additive. We can consider elements in $\mathcal {F}$ as “large” sets and elements in the complement $\mathcal {F}^c$ as “small” sets. If $\mathcal {F}$ is not principal, all finite subsets of W are “small.”

Definition 3.7. Let $A_w$ be a family of sets indexed by W and $\mathcal {F}$ be a non-principal ultrafilter on W. The ultraproduct of $A_w$ is defined by

$$ \begin{align*} \operatorname*{\mathrm{ulim}}_w A_w = A_{\infty} := \prod_w A_w/\sim, \end{align*} $$

where $(a_w)\sim (b_w)$ if and only if $\{w\in W|a_w=b_w\}\in \mathcal {F}$ . We denote the equivalence class of $(a_w)$ by $\operatorname *{\mathrm {ulim}}_w a_w$ .

Remark 3.8 [17, Sec. 3].

If $A_w$ are local rings, then the ultraproduct is equivalent to the localization of $\prod A_w$ at a maximal ideal.

Example 3.9. We use $^*\mathbb {N}$ and $^*\mathbb {R}$ to denote the ultraproduct of $|W|$ copies of $\mathbb {N}$ and $\mathbb {R,}$ respectively. $^*\mathbb {N}$ is a semiring and $^*\mathbb {R}$ is a field (see Definition-Proposition 3.10 and Theorem 3.20). $^*\mathbb {N}$ is a non-standard model of Peano arithmetic. $^*\mathbb {R}$ is a system of hyperreal numbers used in non-standard analysis.

Definition-Proposition 3.10. Let $A_{1w},\dots , A_{nw}$ , $B_w$ be families of sets indexed by W and $\mathcal {F}$ be a non-principal ultrafilter. Suppose that $f_w:A_{1w}\times \dots \times A_{nw}\to B_w$ is a family of maps. Then we define the ultraproduct $f_\infty = \operatorname *{\mathrm {ulim}}_w f_w : A_{1\infty }\times \dots \times A_{n\infty }\to B_\infty $ of $f_w$ by

$$ \begin{align*} f_\infty(\operatorname*{\mathrm{ulim}}_w a_{1w},\dots, \operatorname*{\mathrm{ulim}}_w a_{nw}):=\operatorname*{\mathrm{ulim}}_w f_w(a_{1w},\dots,a_{nw}). \end{align*} $$

This is well-defined.

Corollary 3.11. Let $A_w$ be a family of rings. Suppose that $B_w$ is an $A_w$ -algebra and $M_w$ is an $A_w$ -module for almost all w. Then the following hold:

1. (1) $A_\infty $ is a ring.

2. (2) $B_\infty $ is an $A_\infty $ -algebra.

3. (3) $M_\infty $ is an $A_\infty $ -module.

Proof. Let $0:=\operatorname *{\mathrm {ulim}}_w 0$ , $1:=\operatorname *{\mathrm {ulim}}_w 1$ in $A_\infty $ , $B_\infty $ and $0:=\operatorname *{\mathrm {ulim}}_w 0$ in $M_\infty $ . By the above Definition–Proposition, $A_\infty $ , $B_\infty $ have natural additions, subtractions, and multiplications and we have a natural ring homomorphism $A_\infty \to B_\infty $ . Similarly, $M_\infty $ has a natural addition and a scalar multiplication between elements of $M_\infty $ and $A_\infty $ .

Proposition 3.12. Suppose that, for almost all w, we have an exact sequence

$$ \begin{align*} 0\to L_w\to M_w \to N_w\to 0 \end{align*} $$

of abelian groups. Then

$$ \begin{align*} 0\to \operatorname*{\mathrm{ulim}}_w L_w \to \operatorname*{\mathrm{ulim}}_w M_w \to \operatorname*{\mathrm{ulim}}_w N_w \to 0 \end{align*} $$

is an exact sequence of abelian groups. In particular, $\operatorname *{\mathrm {ulim}}_w:\prod _w\operatorname {Ab}\to \operatorname {Ab}$ is an exact functor.

Proof. Let $f_w:L_w\to M_w$ and $g_w:M_w\to N_w$ be the morphisms in the given exact sequence. Here, we only prove the injectivity of $\operatorname *{\mathrm {ulim}}_w f_w$ and the surjectivity of $\operatorname *{\mathrm {ulim}}_w g_w$ . Suppose that $\operatorname *{\mathrm {ulim}}_w f_w(a_w)=0$ for $\operatorname *{\mathrm {ulim}}_w a_w \in \operatorname *{\mathrm {ulim}}_w L_w$ . Then $f_w(a_w)=0$ for almost all w. Since $f_w$ is injective for almost all w, we have $a_w=0$ for almost all w. Therefore, $\operatorname *{\mathrm {ulim}}_w a_w=0$ in $\operatorname *{\mathrm {ulim}}_w L_w$ . Hence, $\operatorname *{\mathrm {ulim}}_w f_w$ is injective. Next, let $\operatorname *{\mathrm {ulim}}_w c_w$ be any element in $\operatorname *{\mathrm {ulim}}_w N_w$ . Since $g_w$ is surjective for almost all w, there exists $b_w\in M_w$ such that $g_w(b_w)=c_w$ for almost all w. Let $b=\operatorname *{\mathrm {ulim}}_w b_w$ . Then we have $(\operatorname *{\mathrm {ulim}}_w g_w)(b)=\operatorname *{\mathrm {ulim}}_w g_w(b_w)=\operatorname *{\mathrm {ulim}}_w c_w$ . Hence, $\operatorname *{\mathrm {ulim}}_w g_w$ is surjective. The rest of the proof is similar.

Łoś’s theorem is a fundamental theorem in the theory of ultraproducts. We will prepare some notions needed to state the theorem.

Definition 3.13. The language $\mathcal {L}$ of rings is the set defined by

$$ \begin{align*} \mathcal{L}:=\{0,1,+,-,\cdot \}. \end{align*} $$

Definition 3.14. Terms of $\mathcal {L}$ are defined as follows:

1. (i) $0$ , $1$ are terms.

2. (ii) Variables are terms.

3. (iii) If s, t are terms, then ${-(s)}, (s)+(t), (s)\cdot (t)$ are terms.

4. (iv) A string of symbols is a term only if it can be shown to be a term by finitely many applications of the above three rules.

We omit parentheses and “ $\cdot $ ” if there is no ambiguity.

Example 3.15. $1+1$ , $x_1(x_2+1)$ , $-(-x)$ are terms.

Definition 3.16. Formulas of $\mathcal {L}$ are defined as follows:

1. (i) If $s $ , t are terms, then $(s=t)$ is a formula.

2. (ii) If $\varphi , \psi $ are formulas, then $(\varphi \land \psi ), (\varphi \lor \psi ), (\varphi \to \psi ),(\lnot \varphi )$ are formulas.

3. (iii) If $\varphi $ is a formula and x is a variable, then $\forall x \varphi , \exists x \varphi $ are formulas.

4. (iv) A string of symbols is a formula only if it can be shown to be a formula by finitely many applications of the above three rules.

We omit parentheses if there is no ambiguity and use $\neq $ , $\nexists $ in the usual way.

Remark 3.17. $\varphi \land \psi $ means “ $\varphi $ and $\psi $ ,” $\varphi \lor \psi $ means “ $\varphi $ or $\psi $ ,” $\varphi \to \psi $ means “ $\varphi $ implies $\psi $ ,” and $\lnot \varphi $ means “ $\varphi $ does not hold.”

Example 3.18. 0=1, $x=0 \land y\neq 1$ , $\forall x \forall y (xy=yx)$ are formulas.

Remark 3.19. Variables in a formula $\varphi $ which is not bounded by $\forall $ or $\exists $ are called free variables of $\varphi $ . If $x_1,\dots ,x_n$ are free variables of $\varphi $ , we denote $\varphi (x_1,\dots ,x_n)$ and we can substitute elements of a ring for $x_1,\dots ,x_n$ .

Theorem 3.20 (Łoś’s theorem in the case of rings).

Suppose that $\varphi (x_1,\dots ,x_n$ ) is a formula of $\mathcal {L}$ and $A_w$ is a family of rings indexed by a set W endowed with a non-principal ultrafilter. Let $a_{iw}\in A_w$ . Then $\varphi (\operatorname *{\mathrm {ulim}}_w a_{1w}, \dots , \operatorname *{\mathrm {ulim}}_w a_{nw})$ holds in $A_\infty $ if and only if $\varphi (a_{1w},\dots ,a_{nw})$ holds in $A_w$ for almost all w.

Remark 3.21. Even if $A_w$ are not rings, replacing $\mathcal {L}$ properly, we can get the same theorem as above. We use one in the case of modules.

Example 3.22. Let A be a ring. If a property of rings is written by some formula, we can apply Łoś’s theorem.

1. (1) A is a field if and only if $\forall x (x=0 \lor \exists y(xy=1))$ holds.

2. (2) A is a domain if and only if $\forall x \forall y(xy=0\to (x=0\lor y=0))$ holds.

3. (3) A is a local ring if and only if

   $$ \begin{align*} \forall x \forall y (\nexists z (xz=1)\land \nexists w(yw=1)\to \nexists u((x+y)u=1)) \end{align*} $$
   holds.

4. (4) The condition that A is an algebraically closed field is written by countably many formulas, that is, the formula in (1) and for all $n\in \mathbb {N}$ ,

   $$ \begin{align*}\forall a_0 \dots a_{n-1} \exists x (x^n+a_{n-1}x^{n-1}+\dots +a_0=0).\end{align*} $$

5. (5) The condition that A is Noetherian cannot be written by formulas. Indeed, if $W=\mathbb {N}$ with some non-principal ultrafilter and , then $\operatorname *{\mathrm {ulim}}_n x^n\neq 0$ is in $\cap _n \mathfrak {m}_\infty ^n$ , where $\mathfrak {m}_\infty $ is the maximal ideal of $A_\infty $ . Hence, $A_\infty $ is not Noetherian.

Proposition 3.23 ([22, 2.8.2]; see Example 3.22).

If almost all $K_w$ are algebraically closed field, then $K_\infty $ is an algebraically closed field.

Theorem 3.24 (Lefschetz principle [22, Th. 2.4]).

Let W be the set of prime numbers endowed with some non-principal ultrafilter. Then

$$ \begin{align*} \operatorname*{\mathrm{ulim}}_{p\in\mathcal{W}} \overline{\mathbb{F}_p}\cong \mathbb{C}. \end{align*} $$

Proof. Let $C=\operatorname *{\mathrm {ulim}}_p\overline {\mathbb {F}_p}$ . By the above theorem, C is an algebraically closed field. For any prime number q, we have $q\neq 0$ in $\overline {\mathbb {F}_p}$ for almost all p. Hence, $q\neq 0$ in C, that is, C is of characteristic zero. We can check that C has the same cardinality as $\mathbb {C}$ . If two algebraically closed uncountable field of characteristic zero have the equal cardinality, then they are isomorphic. Hence, $C\cong \mathbb {C}$ . (Note that this isomorphism is not canonical.)

3.2 Non-standard hulls

In this subsection, we will introduce the notion of non-standard hulls along [22], [26]. Throughout this subsection, let $\mathcal {P}$ be the set of prime numbers and we fix a non-principal ultrafilter on $\mathcal {P}$ and an isomorphism $\operatorname *{\mathrm {ulim}}_p \overline {\mathbb {F}_p}\cong \mathbb {C}$ .

Let $\mathbb {C}[X_1,\dots ,X_n]_\infty :=\operatorname *{\mathrm {ulim}}_p\overline {\mathbb {F}_p}[X_1,\dots ,X_n]$ . Then we have the following proposition.

Proposition 3.25 [22, Th. 2.6].

We have a natural map $\mathbb {C}[X_1,\dots ,X_n]\to \mathbb {C}[X_1,\dots ,X_n]_\infty $ , which is faithfully flat.

Definition 3.26. The ring $\mathbb {C}[X_1,\dots ,X_n]_\infty $ is said to be the non-standard hull of $\mathbb {C}[X_1,\dots ,X_n]$ .

Remark 3.27. If $n\geqslant 1$ , then $\mathbb {C}[X_1,\dots ,X_n]_\infty $ is not Noetherian. Let $y=\operatorname *{\mathrm {ulim}}_p X_1^p$ . Then, for any integer $l\geqslant 1$ , $X_1^p\in (X_1,\dots ,X_n)^l$ for almost all p. Hence, $y\in (X_1,\dots ,X_n)^l$ for any l by Łoś’s theorem. Therefore, $\cap _l (X_1,\dots ,X_n)^l\neq 0$ . By Krull’s intersection theorem, $\mathbb {C}[X_1,\dots ,X_n]_\infty $ is not Noetherian.

Definition 3.28. Suppose that R is a finitely generated $\mathbb {C}$ -algebra. Let

$$ \begin{align*} R\cong \mathbb{C}[X_1,\dots,X_n]/I \end{align*} $$

be a presentation of R. The non-standard hull $R_\infty $ of R is defined by

$$ \begin{align*} R_\infty:=\mathbb{C}[X_1,\dots,X_n]_\infty/I\mathbb{C}[X_1,\dots,X_n]_\infty. \end{align*} $$

Remark 3.29. The non-standard hull is independent of a representation of R. If $R\cong \mathbb {C}[X_1,\dots ,X_n]/I\cong \mathbb {C}[Y_1,\dots ,Y_m]/J$ , then $\overline {\mathbb {F}_p}[X_1,\dots ,X_n]/I_p \cong \overline {\mathbb {F}_p}[Y_1,\dots ,Y_m]/J_p$ for almost all p (see Definitions 3.33 and 3.35).

Remark 3.30. The natural map $R\to R_\infty $ is faithfully flat since this is a base change of the homomorphism $\mathbb {C}[X_1,\dots ,X_n]\to \mathbb {C}[X_1,\dots ,X_n]_\infty $ . By faithfully flatness, we have $IR_\infty \cap R=R$ for any ideal $I\subseteq R$ .

Definition 3.31. Let $a\in \mathbb {C}$ . Since $\operatorname *{\mathrm {ulim}}_p \overline {\mathbb {F}_p}\cong \mathbb {C}$ , we have a family $(a_p)_p$ of elements of $\overline {\mathbb {F}_p}$ such that $\operatorname *{\mathrm {ulim}} a_p=a$ . Then we call $(a_p)_p$ an approximation of a.

Proposition 3.32. Let $I=(f_1,\dots ,f_s)$ be an ideal of $\mathbb {C}[X_1,\dots , X_n]$ and $f_i=\sum a_{i\nu }X^\nu $ . Let $I_p=(f_{1p},\dots ,f_{sp})\overline {\mathbb {F}_p}[X_1,\dots ,X_n]$ , where $f_{ip}=\sum a_{i\nu p}X^\nu $ and each $(a_{i\nu p})_p$ is an approximation of $a_{i\nu }$ . Then we have

$$ \begin{align*} I\mathbb{C}[X_1,\dots,X_n]_\infty =\operatorname*{\mathrm{ulim}}_p I_p \end{align*} $$

and

$$ \begin{align*} R_\infty\cong \operatorname*{\mathrm{ulim}}_p (\overline{\mathbb{F}_p}[X_1,\dots,X_n]/I_p). \end{align*} $$

Definition 3.33. Let R be a finitely generated $\mathbb {C}$ -algebra.

1. (1) In the setting of Proposition 3.32, a family $R_p$ is said to be an approximation of R if $R_p$ is an $\overline {\mathbb {F}_p}$ -algebra and $R_p \cong \overline {\mathbb {F}_p}[X_1,\dots ,X_n]/I_p$ for almost all p. Then we have $R_\infty \cong \operatorname *{\mathrm {ulim}}_p R_p$ .

2. (2) For an element $f\in R$ , a family $f_p$ is said to be an approximation of f if $f_p\in R_p$ for almost all p and $f=\operatorname *{\mathrm {ulim}}_p f_p$ in $R_\infty $ . For $f\in R_\infty $ , we define an approximation of f in the same way.

3. (3) For an ideal $I=(f_1,\dots ,f_s) \subseteq R$ , a family $I_p$ is said to be an approximation of I if $I_p$ is an ideal of $R_p$ and $I_p=(f_{1p},\dots ,f_{sp})$ for almost all p. For finitely generated ideal $I\subseteq R_\infty $ , we define an approximation of I in the same way.

Remark 3.34. This is an abuse of notation since approximations should be denoted by $(R_p)_p$ , $(f_p)_p$ , $(I_p)_p$ , and so forth.

Definition 3.35. Let $\varphi :R\to S$ be a $\mathbb {C}$ -algebra homomorphism between finitely generated $\mathbb {C}$ -algebras. Suppose that $R\cong \mathbb {C}[X_1,\dots ,X_n]/I$ and $S\cong \mathbb {C}[Y_1,\dots ,Y_m]/J$ . Let $f_i\in \mathbb {C}[Y_1,\dots ,Y_m]$ be a lifting of the image of $X_i\ \mod I$ under $\varphi $ . Then we define an approximation $\varphi _p:R_p\to S_p$ of $\varphi $ as the morphism induced by $X_i\longmapsto f_{ip}$ . Let $\varphi _\infty :=\operatorname *{\mathrm {ulim}}_p\varphi _p$ , then the following diagram commutes.

Proposition 3.36 [22, Cor. 4.2], [26, Th. 4.3.4].

Let R be a finitely generated $\mathbb {C}$ -algebra. An ideal $I\subseteq R$ is prime if and only if $I_p$ is prime for almost all p if and only if $IR_\infty $ is prime.

Definition 3.37. Let R be a local ring essentially of finite type over $\mathbb {C}$ . Suppose that $R\cong S_{\mathfrak {p}}$ , where S is a finitely generated $\mathbb {C}$ -algebra and $\mathfrak {p}$ is a prime ideal of S. Then we define the non-standard hull $R_\infty $ of R by

$$ \begin{align*} R_\infty:=(S_\infty)_{\mathfrak{p} S_\infty}. \end{align*} $$

Remark 3.38. Since $S\to S_\infty $ is faithfully flat, $R\to R_\infty $ is faithfully flat.

Definition 3.39. Let S be a finitely generated $\mathbb {C}$ -algebra, let $\mathfrak {p}$ be a prime ideal of S, and let $R\cong S_{\mathfrak {p}}$ .

1. (1) A family $R_p$ is said to be an approximation of R if $R_p$ is an $\overline {\mathbb {F}_p}$ -algebra and $R_p \cong (S_p)_{\mathfrak {p}_p}$ for almost all p. Then we have $R_\infty \cong \operatorname *{\mathrm {ulim}}_p R_p$ .

2. (2) For an element $f\in R$ , a family $f_p$ is said to be an approximation of f if $f_p\in R_p$ for almost all p and $f=\operatorname *{\mathrm {ulim}}_p f_p$ in $R_\infty $ . For $f\in R_\infty $ , we define an approximation of f in the same way.

3. (3) For an ideal $I=(f_1,\dots ,f_s) \subseteq R$ , a family $I_p$ is said to be an approximation of I if $I_p$ is an ideal of $R_p$ and $I_p=(f_{1p},\dots ,f_{sp})$ for almost all p. For finitely generated ideal $I\subseteq R_\infty $ , we define an approximation of I in the same way.

Definition 3.40. Let $S_1,S_2$ be finitely generated $\mathbb {C}$ -algebras, and let $\mathfrak {p}_1,\mathfrak {p}_2$ be prime ideals of $S_1,S_2,$ respectively. Suppose that $R_i\cong (S_i)_{\mathfrak {p}_i}$ and $\varphi :R_1\to R_2$ is a local $\mathbb {C}$ -algebra homomorphism. Let $S_1\cong \mathbb {C}[X_1,\dots ,X_n]/I$ and $f_j/g_j$ be the image of $X_j$ under $\varphi $ , where $f_j\in S_2$ , $g_j\in S_2\setminus \mathfrak {p}_2$ . Then we say that a homomorphism $R_{1p}\to R_{2p}$ induced by $X_j\longmapsto f_{jp}/g_{jp}$ is an approximation of $\varphi $ . Let $\varphi _\infty :=\operatorname *{\mathrm {ulim}}_p \varphi _p$ . Then the following commutative diagram commutes:

Definition 3.41. Let R be a finitely generated $\mathbb {C}$ -algebra or a local ring essentially of finite type over $\mathbb {C}$ , and let M be a finitely generated R-module. Write M as the cokernel of a matrix A, that is, given by an exact sequence

$$ \begin{align*} R^m\xrightarrow{A} R^n\to M \to 0, \end{align*} $$

where $m,n$ are positive integers. Let $A_p$ be an approximation of A defined by entrywise approximations. Then the cokernel $M_p$ of the matrix $A_p$ is called an approximation of M and the ultraproduct $M_\infty :=\operatorname *{\mathrm {ulim}}_p M_p$ is called the non-standard hull of M. $M_\infty $ is a finitely generated $R_\infty $ -module and independent of the choice of matrix A.

Remark 3.42. Tensoring the above exact sequence with $R_\infty $ , we have an exact sequence

$$ \begin{align*} R_\infty^m \xrightarrow{A} R_\infty^n \to M\otimes_R R_\infty \to 0. \end{align*} $$

Taking the ultraproduct of exact sequences

$$ \begin{align*} R_p^m\xrightarrow{A_p} R_p^n \to M_p \to 0, \end{align*} $$

we have an exact sequence

$$ \begin{align*} R_\infty^m\xrightarrow{A} R_\infty^n \to M_\infty\to 0. \end{align*} $$

Therefore, $M_\infty \cong M\otimes _R R_\infty $ . Note that if $m,n$ are not integers but infinite cardinals, then the naive definition of an approximation of A does not work and the ultraproduct of $R_p^{\oplus n}$ is not necessarily equal to $R_\infty ^{\oplus n}$ .

Here, we state basic properties about non-standard hulls and approximations.

Proposition 3.43 [22, 2.9.5, 2.9.7, Ths. 4.5 and 4.6], [26, §4.3]; cf. [2, 5.1].

Let R be a local ring essentially of finite type over $\mathbb {C}$ , then the following hold:

1. (1) R has dimension d if and only if $R_p$ has dimension d for almost all p.

2. (2) $\mathbf {x}=x_1,\dots ,x_i$ is an R-regular sequence if and only if $\mathbf {x}_p=x_{1p},\dots , x_{ip}$ is an $R_p$ -regular sequence for almost all p if and only if $\mathbf {x}$ is an $R_\infty $ -regular sequence.

3. (3) $\mathbf {x}=x_1,\dots ,x_d$ is a system of parameters of R if and only if  $\mathbf {x}_p$ is a system of parameters of $R_p$ for almost all p.

4. (4) R is regular if and only if $R_p$ is regular for almost all p.

5. (5) R is Gorenstein if and only if $R_p$ is Gorenstein for almost all p.

6. (6) R is Cohen–Macaulay if and only if $R_p$ is Cohen–Macaulay for almost all p.

Proposition 3.44 [31, Prop. 3.9].

Let R be a local ring essentially of finite type over $\mathbb {C}$ . The following conditions are equivalent to each other.

1. (1) R is normal.

2. (2) $R_p$ is normal for almost all p.

3. (3) $R_\infty $ is normal.

Definition 3.45. Let R be a normal local domain essentially of finite type over $\mathbb {C}$ , and let $\Delta =\sum _i a_i\Delta _i$ be a $\mathbb {Q}$ -Weil divisor. Assume that $\Delta _i$ are prime divisors and $\mathfrak {p}_i$ is a prime ideal associated with $\Delta _i$ for each i. Suppose that $\mathfrak {p}_{ip}$ is an approximation of $\mathfrak {p}_{i}$ and $\Delta _{ip}$ is a divisor associated with $\mathfrak {p}_{ip}$ . We say $\Delta _p:=\sum _i a_i\Delta _{ip}$ is an approximation of $\Delta $ .

Remark 3.46. If $\Delta $ is an effective integral divisor, then this definition is compatible with Definition 3.33 by [22, Th. 4.4]. Hence, if $\Delta $ is $\mathbb {Q}$ -Cartier, then $\Delta _p$ is $\mathbb {Q}$ -Cartier for almost all p.

Lastly, we review some singularities introduced by Schoutens via ultraproducts.

Definition 3.47 [22, Def. 5.2], [25, Def. 3.1].

Suppose that R is a finitely generated $\mathbb {C}$ -algebra or a local domain essentially of finite type over $\mathbb {C}$ . Let $I\subseteq R$ be an ideal. The generic tight closure $I^{*\operatorname {gen}}$ of I is defined by

$$ \begin{align*} I^{*\operatorname{gen}}=(\operatorname*{\mathrm{ulim}}_p I_p)^*\cap R. \end{align*} $$

Remark 3.48. The generic tight closure $I^{*\operatorname {gen}}$ of I does not depend on the choice of approximation of I since any two approximations are almost equal.

Definition 3.49 [25, Def. 4.1 and Rem. 4.7], [23, Def. 4.3].

Suppose that R is a finitely generated $\mathbb {C}$ -algebra or a local ring essentially of finite type over $\mathbb {C}$ .

1. (1) R is said to be weakly generically F-regular if $I^{*\operatorname {gen}}=I$ for any ideal $I\subseteq R$ .

2. (2) R is said to be generically F-regular if $R_{\mathfrak {p}}$ is weakly generically F-regular for any prime ideal $\mathfrak {p}\in \operatorname {Spec} R$ .

3. (3) Let R be a local ring essentially of finite type over $\mathbb {C}$ . R is said to be generically F-rational if $I^{*\operatorname {gen}}=I$ for some ideal I generated by a system of parameters.

Proposition 3.50 [25, Th. 4.3].

If R is generically F-rational, then $I^{*\operatorname {gen}}=I$ for any ideal I generated by part of a system of parameters.

Proposition 3.51 [25, Th. 6.2], [23, Prop. 4.5 and Th. 4.12].

If R is generically F-rational if and only if $R_p$ is F-rational for almost all p if and only if R has rational singularities.

Definition 3.52 [24, 3.2].

Let R be a local ring essentially of finite type over $\mathbb {C}$ and $R_p$ be an approximation. Let $\varepsilon :=\operatorname *{\mathrm {ulim}}_p e_p \in {^*\mathbb {N}}$ . Then an ultra-Frobenius $F^\varepsilon :R\to R_\infty $ associated with $\varepsilon $ is defined by $x\longmapsto \operatorname *{\mathrm {ulim}}_p (F_p^{e_p}(x_p))$ , where $F_p$ is a Frobenius morphism in characteristic p.

Definition 3.53 [24, Def. 3.3].

Let R be a local domain essentially of finite type over $\mathbb {C}$ . R is said to be ultra-F-regular if, for each $c\in R^{\circ }$ , there exists $\varepsilon \in {^*\mathbb {N}}$ such that

$$ \begin{align*} R \xrightarrow{cF^\varepsilon} R_\infty \end{align*} $$

is pure.

Proposition 3.54 [24, Th. A].

Let R be a $\mathbb {Q}$ -Gorenstein normal local domain essentially of finite type over $\mathbb {C}$ . Then R is ultra-F-regular if and only if R has log-terminal singularities.

3.3 Relative hulls

In this subsection, we introduce the concept of relative hulls and approximations of schemes, cohomologies, and so forth. We refer the reader to [22], [24], [25].

Definition 3.55 (Cf. [25]).

Let R be a local ring essentially of finite type over $\mathbb {C}$ . Suppose that X is a finite tuple of indeterminates and $f\in R[X]$ is a polynomial such that $f=\sum _{\nu } a_\nu X^\nu $ , where $\nu $ is a multi-index. If $a_{\nu p}$ is an approximation of $a_{\nu }$ for each $\nu $ , then the sequence of polynomials $f_p:=\sum _\nu a_{\nu p} X^\nu $ is said to be an R-approximation of f. If $I:=(f_1,\dots ,f_s)$ is an ideal in $R[X]$ , then we call $I_{p}:=(f_{1p},\dots ,f_{sp})R_p[X]$ an R-approximation of I, and if $S=R[X]/I$ , then we call $S_p:=R_p[X]/I_p$ an R-approximation of S.

Remark 3.56. Any two R-approximations of a polynomial f are almost equal. Similarly, any two R-approximations of an ideal I are almost equal.

Definition 3.57 (Cf. [25]).

Let S be a finitely generated R-algebra, and let $S_p$ be an R-approximation of S, then we call $S_\infty =\operatorname *{\mathrm {ulim}} _p S_p$ the (relative) R-hull of S.

Definition 3.58 (Cf. [24]).

If X is an affine scheme $\operatorname {Spec} S$ of finite type over $\operatorname {Spec} R$ , then we call $X_p:=\operatorname {Spec} S_p$ is an R-approximation of X.

Definition 3.59 (Cf. [24]).

Suppose that $f:Y\to X$ is a morphism of affine schemes of finite type over $\operatorname {Spec} R$ . If $X=\operatorname {Spec} S, Y=\operatorname {Spec} T$ and $\varphi :S\to T$ is the morphism corresponding to f, then we call $f_p:Y_p\to X_p$ is an R-approximation of f, where $f_p$ is a morphism of $R_p$ -schemes induced by an R-approximation $\varphi _p:S_p\to T_p$ .

Definition 3.60 (Cf. [24]).

Let S be a finitely generated R-algebra, and let M be a finitely generated S-module. Write M as the cokernel of a matrix A, that is, given by an exact sequence

$$ \begin{align*} S^m\xrightarrow{A} S^n\to M\to 0, \end{align*} $$

where $m,n$ are positive integers. Let $A_p$ be an R-approximation of A defined by entrywise R-approximations. Then the cokernel $M_p$ of the matrix $A_p$ is called an R-approximation of M and the ultraproduct $M_\infty :=\operatorname *{\mathrm {ulim}}_p M_p$ is called the R-hull of M. $M_\infty $ is independent of the choice of the matrix A and $M_\infty \cong M\otimes _S S_\infty $ .

Remark 3.61. If M is not finitely generated, then we cannot define an R-approximation of M in this way. It is crucial that any two R-approximations of A is equal for almost all p.

Definition 3.62 [24].

Let X be a scheme of finite type over $\operatorname {Spec} R$ . Let $\mathfrak {U}=\{U_i\}$ is a finite affine open covering of X and $U_{ip}$ be an R-approximation of $U_i$ . Gluing $\{U_{ip}\}$ together, we obtain a scheme $X_p$ of finite type over $\operatorname {Spec} R_p$ . We call $X_p$ an R-approximation of X.

Remark 3.63. Suppose that $\{U_{ijk}\}_k$ is a finite affine open covering of $U_i\cap U_j$ and $\varphi _{ijk}:\mathcal {O}_{U_i}|_{U_k}\cong \mathcal {O}_{U_j}|_{U_k}$ are isomorphisms. Then R-approximations $\varphi _p:\mathcal {O}_{U_{ip}}|_{U_{kp}}\to \mathcal {O}_{U_{jp}}|_{U_{kp}}$ are isomorphisms for almost all p (note that indices $ijk$ are finitely many). Hence, we can glue these together. For any other choice of finite affine open covering $\mathfrak {U}'$ of X, the resulting R-approximation $X^{\prime }_p$ is isomorphic to $X_p$ for almost all p.

Definition 3.64 (Cf. [24]).

Suppose that $f:Y\to X$ is a morphism between schemes of finite type over $\operatorname {Spec} R$ . Let $\mathfrak {U}$ , $\mathfrak {V}$ be finite affine open coverings of X and Y, respectively, such that for any $V\in \mathfrak {V}$ , there exists some $U\in \mathfrak {U}$ such that $f(V)\subseteq U$ . Let $\mathfrak {U}_p$ , $\mathfrak {V}_p$ be R-approximations of $\mathfrak {U}$ , $\mathfrak {V}$ and $(f|_V)_p$ an R-approximation of $f|_V$ . We define an R-approximation $f_p$ of f by the morphism determined by $(f|V)_p$ .

Remark 3.65. In the same way as the above Remark 3.63, $(f|_V)_p$ and $(f|_{V'})_p$ agree on $V\cap V'$ for any two opens $V,V'\in \mathfrak {V}$ for almost all p.

Definition 3.66 (Cf. [24]).

Let X be a scheme of finite type over $\operatorname {Spec} R$ , and let $\mathcal {F}$ be a coherent $\mathcal {O}_X$ -module. Let $\mathfrak {U}$ be a finite affine open covering of X. For any $U\in \mathfrak {U}$ , we have an R-approximation $M_{Up}$ of $M_{U}$ such that $M_{U}$ is a finitely generated $\mathcal {O}_U$ -module and $\widetilde {M_U}\cong \mathcal {F}|_U$ . We define an R-approximation $\mathcal {F}_p$ of $\mathcal {F}$ by the coherent $\mathcal {O}_{X_p}$ -module determined by $\widetilde {M_{Up}}$ .

Definition 3.67 (Cf. [24]).

Let X be a separated scheme of finite type over $\operatorname {Spec} R$ , and let $\mathcal {F}$ be a coherent $\mathcal {O}_X$ -module. Then the ultra-cohomology of $\mathcal {F}$ is defined by

$$ \begin{align*} H_\infty^i(X,\mathcal{F}):=\operatorname*{\mathrm{ulim}}_p H^i(X_p,\mathcal{F}_p). \end{align*} $$

Remark 3.68. In the above setting, let $\mathfrak {U}=\{U_i\}_{i=1,\dots ,n}$ be a finite affine open covering of X, let

$$ \begin{align*} C^j(\mathfrak{U}, \mathcal{F}):=\prod_{i_0<\dots<i_j}\mathcal{F}(U_{i_0\dots i_j}), \end{align*} $$

where $U_{i_0\dots i_j}:=U_{i_0}\cap \dots \cap U_{i_j}$ , and let

$$ \begin{align*} (C^j(\mathfrak{U},\mathcal{F}))_p:=\prod_{i_0\dots i_j}(\mathcal{F}(U_{i_0\dots i_j}))_p, \end{align*} $$

where $\mathcal {F}(U_{i_0\dots i_j})_p$ is an R-approximation considered as $\mathcal {O}(U_{i_0\dots i_j})$ -module. Then

$$ \begin{align*} (C^j(\mathfrak{U},\mathcal{F}))p \end{align*} $$

coincides with the jth term of the Čech complex of $X_p$ , $\mathfrak {U}_p,$ and $\mathcal {F}_p$ . We have a commutative diagram

Since $\operatorname *{\mathrm {ulim}}_p (\text{-})$ is an exact functor, we have

$$ \begin{align*} \check{H}^j(\mathfrak{U},\mathcal{F})\to \operatorname*{\mathrm{ulim}}_p \check{H}^j(\mathfrak{U}_p,\mathcal{F}_p). \end{align*} $$

If X is separated, then $X_p$ is separated for almost all p. This can be checked by taking a finite affine open covering and observing that if the diagonal morphism $\Delta _{X/\operatorname {Spec} R}$ is a closed immersion, then $\Delta _{X_p/\operatorname {Spec} R_p}$ is also a closed immersion for almost all p. Hence, we have the map

$$ \begin{align*} H^j(\mathfrak{U},\mathcal{F})\to \operatorname*{\mathrm{ulim}}_p H^j(\mathfrak{U}_p,\mathcal{F}_p). \end{align*} $$

Note that we do not know whether this map is injective or not.

Proposition 3.69. Let R be a local ring essentially of finite type over $\mathbb {C}$ of dimension d, $\mathbf {x}=x_1,\dots ,x_d$ a system of parameters and M a finitely generated R-module. Then we have a natural homomorphism $H_{\mathfrak {m}}^d(M)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(M_p)$ .

Proof. Since $M_{x_1\cdots \hat {x_i}\cdots x_d}$ is a finitely generated $R_{x_1\cdots \hat {x_i}\cdots x_d}$ -module and $M_{x_1\cdots x_d}$ is a finitely generated $R_{x_1\cdots x_d}$ -module, we have an R-approximation $(M_{x_1\cdots \hat {x_1}\cdots x_d})_p\cong (M_p)_{x_{1p}\cdots \hat {x_{ip}}\cdots x_{dp}}$ and $(M_{x_1\cdots x_d})_p\cong (M_p)_{x_{1p}\cdots x_{dp}}$ for almost all p. We have a commutative diagram

Taking the cokernel of rows, we have the desired map.

Remark 3.70. We do not know whether $H_{\mathfrak {m}}^d(M)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(M_p)$ is injective or not.

Proposition 3.71. Let R be a local ring essentially of finite type over $\mathbb {C}$ of dimension d, $\mathbf {x}=x_,\dots ,x_d$ be a system of parameters and $M_p$ be an $R_p$ -module for almost all p. Then we have a natural homomorphism $H_{\mathfrak {m}}^d(\operatorname *{\mathrm {ulim}}_p M_p)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}}^d(M_p)$ .

Proof. We have a commutative diagram

Taking the cokernel of rows, we have the desired map.

4 Big Cohen–Macaulay algebras constructed via ultraproducts

In [23], Schoutens constructed the canonical BCM algebra in characteristic zero. Following the idea of [23], we will deal with BCM algebras constructed via ultraproducts in slightly general settings. In this section, suppose that $(R,\mathfrak {m})$ is a local domain essentially of finite type over $\mathbb {C}$ and $R_p$ is an approximation of R.

Definition 4.1 [23, §2].

Suppose that R is a local domain essentially of finite type over $\mathbb {C}$ . Then we define the canonical BCM algebra $\mathcal {B}(R)$ of R by

$$ \begin{align*}\mathcal{B}(R):=\operatorname*{\mathrm{ulim}}_p R_p^{+}. \end{align*} $$

Setting 4.2. Let R be a local domain essentially of finite type over $\mathbb {C}$ of dimension d, and let $B_p$ be a BCM ${R_p}^+$ -algebra for almost all p. We use B to denote $\operatorname *{\mathrm {ulim}}_p B_p$ .

Remark 4.3. By Theorem 2.8, we can set $B_p=R_p^+$ and $B=\mathcal {B}(R)$ in Setting 4.2.

Proposition 4.4. $\mathcal {B}(R)$ is a domain over $R^+$ -algebra.

Proof. By Łoś’s theorem, $\mathcal {B}(R)$ is a domain over $R_\infty =\operatorname *{\mathrm {ulim}}_p R_p$ . Hence, $\mathcal {B}(R)$ is an R-algebra. Let $f=\sum a_{n} x^n\in \mathcal {B}(R)[x]$ be a monic polynomial in one variable over $\mathcal {B}(R)$ and let $f_p=\sum a_{np}x^n$ be an approximation of f. Since $f_p$ is a monic polynomial for almost all p and $R_p^+$ is absolutely integrally closed, $f_p$ has a root $c_p$ in $R_p^+$ for almost all p. Hence, $c:=\operatorname *{\mathrm {ulim}}_p c_p\in \mathcal {B}(R)$ is a root of f by Łoś’s theorem. Hence, $\mathcal {B}(R)$ is absolutely integrally closed. In particular, $\mathcal {B}(R)$ contains an absolute integral closure $R^+$ of R.

Corollary 4.5. In Setting 4.2, B is an $R^+$ -algebra.

Proof. Since $B_p$ is an $R_p^+$ -algebra for almost all p, B is an $R^+$ -algebra by the above proposition.

Proposition 4.6. In Setting 4.2, B is a BCM R-algebra.

Proof. Assume that B is not a BCM R-algebra. Since $B_p\neq \mathfrak {m}_p B_p$ for almost all p, we have $B\neq \mathfrak {m} B$ . Hence, there exists part of system of parameters $x_1,\dots ,x_i$ of R such that $(x_1,\dots ,x_{i-1})B\subsetneq (x_1,\dots ,x_{i-1})B:_B x_i$ . Then there exists $y\in B$ such that $x_i y\in (x_1,\dots ,x_{i-1})B$ and $y\notin (x_1,\dots ,x_{i-1})B$ . Taking approximations, we have $x_{ip}y_p\in (x_{1p},\dots ,x_{(i-1)p})B_p$ and $y_p\notin (x_{1p}\dots ,x_{(i-1)p})B_p$ for almost all p. Since $x_{1p},\dots ,x_{ip}$ is part of a system of parameters of $R_p$ and $B_p$ is a BCM $R_p$ -algebra for almost all p, $x_{1p},\dots ,x_{ip}$ is a regular sequence for almost all p. This is a contradiction. Therefore, B is a BCM R-algebra.

Lemma 4.7. In Setting 4.2, the natural homomorphism $H_{\mathfrak {m}}^d(B)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(B_p)$ is injective.

Proof. Let $x=x_1\cdots x_d$ be the product of a system of parameters and $\lbrack \frac {z}{x^t}\rbrack $ be an element of $H_{\mathfrak {m}}^d(B)$ such that the image in $\operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(B_p)$ is zero. Then there exists $s_p\in \mathbb {N}$ such that $x^{s_p}z\in (x_{1p}^{s_p+t},\dots ,x_{dp}^{s_p+t})B_p$ for almost all p. Since $B_p$ is a BCM $R_p$ -algebra for almost all p, $z\in (x_{1p}^t,\dots ,x_{dp}^t)B_p$ for almost all p. Hence, $z\in (x_1^t,\dots ,x_d^t)B$ and $\lbrack \frac {z}{x^t} \rbrack =0$ in $H_{\mathfrak {m}}^d(B)$ .

We generalize [23, Th. 4.2] to the cases other than the canonical BCM algebra.

Proposition 4.8 (Cf. [23, Th. 4.2], [19, Prop. 3.7]).

In Setting 4.2, R is $\text {BCM}_B$ -rational if and only if R has rational singularities. In particular, R has rational singularities if R is BCM-rational.

Proof. Let $x:=x_1\cdots x_d$ is the product of a system of parameters. Suppose that R has rational singularities. By [23, Prop. 4.11] and [9], $R_p$ is F-rational for almost all p. Let $\eta :=\lbrack \frac {z}{x^t} \rbrack $ be an element of $H_{\mathfrak {m}}^d(R)$ such that $\eta =0$ in $H_{\mathfrak {m}}^d(B)$ . Then we have a commutative diagram

By [19, Prop. 3.5], $H_{\mathfrak {m}_p}^d(R_p)\to H_{\mathfrak {m}_p}^d(B_p)$ is injective for almost all p. Hence, $\operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(R_p)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(B_p)$ is injective. Therefore, $\lbrack \frac {z_p}{x_p^t}\rbrack =0$ in $H_{\mathfrak {m}_p}^d(R_p)$ for almost all p. Since $R_p$ is Cohen–Macaulay for almost all p, we have $z_p\in (x_{1p}^t,\dots ,x_{dp}^t)$ for almost all p. Hence, $z\in (x_1^t,\dots ,x_d^t)$ by Łoś’s theorem. Therefore, $H_{\mathfrak {m}}^d(R)\to H_{\mathfrak {m}}^d(B)$ is injective. Conversely, suppose that R is $\text {BCM}_B$ -rational. Let $I=(x_1,\dots ,x_d)$ be an ideal generated by the system of parameters. Let $z\in I^{*\operatorname {gen}}$ . Since $I_p^*\subseteq I_pB_p\cap R_p$ by [27, Th. 5.1] for almost all p, we have $\lbrack \frac {z_p}{x_p}\rbrack =0$ in $H_{\mathfrak {m}_p}^d(B_p)$ for almost all p. Since $H_{\mathfrak {m}}^d(B)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(B_p)$ and $H_{\mathfrak {m}}^d(R)\to H_{\mathfrak {m}}^d(B)$ are injective, we have $\lbrack \frac {z}{x}\rbrack =0$ in $H_{\mathfrak {m}}^d(R)$ . Since R is Cohen–Macaulay, $z\in I$ . Therefore, R is generically F-rational. By Proposition 3.51 (see [25, Th. 6.2]), R has rational singularities.

5 Approximations of multiplier ideals

In this section, we will explain the relationship between approximations and reductions modulo $p\gg 0$ . Note that an isomorphism $\operatorname *{\mathrm {ulim}}_p \overline {\mathbb {F}_p}\cong \mathbb {C}$ is fixed.

Definition 5.1. Let R be a finitely generated $\mathbb {C}$ -algebra. A pair $(A,R_A)$ is called a model of R if the following two conditions hold:

1. (i) $A \subseteq \mathbb {C}$ is a finitely generated $\mathbb {Z}$ -subalgebra.

2. (ii) $R_A$ is a finitely generated A-algebra such that $R_A\otimes _A\mathbb {C}\cong R$ .

Proposition 5.2 [23, Lem. 4.10].

Let A be a finitely generated $\mathbb {Z}$ -subalgebra of $\mathbb {C}$ . There exists a family $(\gamma _p)_p$ which satisfies the following two conditions:

1. (i) $\gamma _p:A\to \overline {\mathbb {F}_p}$ is a ring homomorphism for almost all p.

2. (ii) For any $x\in A$ , $x=\operatorname *{\mathrm {ulim}}_p\gamma _p(x)$ .

Proposition 5.3 (Cf. [23, Cor. 4.10]).

Let R be a finitely generated $\mathbb {C}$ -algebra, and let $\mathbf {a}=a_1,\dots , a_l$ be finitely many elements of R. Let $R_p$ be an approximation of R. Then there exists a model $(A,R_A)$ which satisfies the following conditions:

1. (i) There exists a family $(\gamma _p)$ as in Proposition 5.2.

2. (ii) $\mathbf {a}\subseteq R_A$ .

3. (iii) $R_A\otimes _A \overline {\mathbb {F}_p}\cong R_p$ for almost all p.

4. (iv) For any $x\in R_A$ , the ultraproduct of the image of x under $\operatorname {id}_{R_A}\otimes _A \gamma _p$ is x.

Proof. Let $X=X_1,\dots ,X_n$ and $R\cong \mathbb {C}[X]/I$ for some ideal $I\subseteq \mathbb {C}[X]$ . Take any model $(A,R_A)$ which contains $\mathbf {a}$ . Enlarging this model, we may assume that there exits an ideal $I_A\subseteq A[X]$ such that $R_A\cong A[X]/I_A$ and $I_A\otimes _A \mathbb {C}=I$ in $\mathbb {C}[X]$ . Take $(\gamma _p)$ as in Proposition 5.2. Let $I=(f_1,\dots , f_m)$ . For $f=\sum _\nu c_\nu X^\nu \in A[X]\subseteq \mathbb {C}[X]$ , by the definition of approximations, $f_p:=\sum _\nu \gamma _p(c_\nu )X^\nu \in \overline {\mathbb {F}_p}[X]$ is an approximation of f. Hence, by the definition of approximations of finitely generated $\mathbb {C}$ -algebras, $R_A\otimes _A\overline {\mathbb {F}_p}\cong \overline {\mathbb {F}_p}[X]/(f_{1p},\dots ,f_{mp})\overline {\mathbb {F}_p}[X]$ is an approximation of R. Since two approximations are isomorphic for almost all p, $R_A\otimes _A\overline {\mathbb {F}_p}\cong R_p$ for almost all p. The condition (iv) is clear by the above argument.

Remark 5.4. Let $\mathfrak {p} =(x_1,\dots ,x_n) \subseteq R$ be a prime ideal. Enlarging the model $(A,R_A)$ , we may assume that $x_1,\dots ,x_n\in R_A$ . Let $\mu _p$ be the kernel of $\gamma _p:A\to \overline {\mathbb {F}_p}$ . Then this is a maximal ideal of A and $A/\mu _p$ is a finite field. $\mathfrak {p}_{\mu _p}=(x_1,\dots ,x_n) R_A/\mu _p R_A $ is prime for almost all p since this is a reduction to $p\gg 0$ . On the other hand, $\mathfrak {p}_p:=(x_1,\dots ,x_n)R_A\otimes _A\overline {\mathbb {F}_p}\subseteq R_p$ is an approximation of $\mathfrak {p}$ . Hence, $\mathfrak {p}_p$ is prime for almost all p. Here, $(R_p)_{\mathfrak {p}_p}$ is an approximation of $R_{\mathfrak {p}}$ . Thus we have a flat local homomorphism $(R_A/\mu _p R_A)_{\mathfrak {p}_{\mu _p}}\to R_p$ with $\mathfrak {p}_{\mu _p}R_p=\mathfrak {p}_p$ . Moreover, if $\mathfrak {p}$ is maximal, then $\mathfrak {p}_{\mu _p},\mathfrak {p}_p$ are maximal for almost all p. Then, the map $R_A/\mathfrak {p}_{\mu _p}\to R_p/\mathfrak {p}_p\cong \overline {\mathbb {F}_p}$ is a separable field extension since $R_A/\mathfrak {p}_{\mu _p}$ is a finite field.

The next result is a generalization of [31, Th. 4.6] from ideal pairs to triples.

Proposition 5.5. Let R be a normal local domain essentially of finite type over $\mathbb {C}$ , let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier, let $\mathfrak {a}$ be a nonzero ideal, and let $t>0$ be a real number. Suppose that $R_p$ , $\Delta _p$ , $\mathfrak {a}_p$ are approximations. Then $\tau (R_p,\Delta _p,\mathfrak {a}_p^t)$ is an approximation of $\mathcal {J}(\operatorname {Spec} R,\Delta ,\mathfrak {a}^t)$ .

Proof. Let $R=S_{\mathfrak {p}}$ , where S is a normal domain of finite type over $\mathbb {C}$ and $\mathfrak {p}$ is a prime ideal. Let $\mathfrak {m}$ be a maximal ideal contains $\mathfrak {p}$ . Then there exists a model $(A,S_A)$ of S such that the properties in Proposition 5.3 hold and $S_A$ containing a system of generators of $\mathcal {J}(\operatorname {Spec} R,\Delta ,\mathfrak {a}^t)$ and $\Delta _A$ , $\mathfrak {a}_A$ can be defined properly. Let $\mu _p$ be maximal ideals of $S_A$ as in Remark 5.4, and let $\mathfrak {m}_{\mu _p}, \mathfrak {p}_{\mu _p}$ be reductions to $p\gg 0$ . Since, for almost all p, $(S_A/\mu _p)_{\mathfrak {m}_{\mu _p}}\to (S_{\mathfrak {m}})_p$ is a flat local homomorphism such that $S_A/\mathfrak {m}_{\mu _p}\to (S/\mathfrak {m})_p\cong \overline {\mathbb {F}_p}$ is a separable field extension, we have

$$ \begin{align*} \tau((S_A/\mu_p)_{\mathfrak{m}_{\mu_p}}, \Delta_{(S_A/\mu_p)_{\mathfrak{m}_{\mu_p}}},\mathfrak{a}^t_{(S_A/\mu_p)_{\mathfrak{m}_{\mu_p}}})(S_{\mathfrak{m}})_p=\tau((S_{\mathfrak{m}})_p,\Delta_{\mathfrak{m}_p},\mathfrak{a}_{\mathfrak{m}_p}^t), \end{align*} $$

by a generalization of [28, Lem. 1.5]. Since the localization commutes with test ideals [10, Prop. 3.1], we have

$$ \begin{align*} \tau((S_A/\mu_p)_{\mathfrak{p}_{\mu_p}}, \Delta_{(S_A/\mu_p)_{\mathfrak{p}_{\mu_p}}},\mathfrak{a}^t_{(S_A/\mu_p)_{\mathfrak{p}_{\mu_p}}})R_p=\tau(R_p,\Delta_p,\mathfrak{a}_p^t) \end{align*} $$

for almost all p. Since the reduction of multiplier ideals modulo $p\gg 0$ is the test ideal [29, Th. 3.2], $\tau ((S_A/\mu _p)_{\mathfrak {p}_{\mu _p}}, \Delta _{(S_A/\mu _p)_{\mathfrak {p}_{\mu _p}}},\mathfrak {a}^t_{(S_A/\mu _p)_{\mathfrak {p}_{\mu _p}}})$ is a reduction of

$$ \begin{align*} \mathcal{J}(\operatorname{Spec} R,\Delta,\mathfrak{a}^t) \end{align*} $$

to characteristic $p\gg 0$ . Hence, $\tau (R_p,\Delta _p,\mathfrak {a}^t_p)$ is an approximation of $\mathcal {J}(\operatorname {Spec} R,\Delta ,\mathfrak {a}^t)$ .

6 BCM test ideal with respect to a big Cohen–Macaulay algebra constructed via ultraproducts

Throughout this section, we assume that $(R,\mathfrak {m})$ is a normal local domain essentially of finite type over $\mathbb {C}$ . Fix a canonical divisor $K_R$ such that $R\subseteq \omega _R:=R(K_R)\subseteq \operatorname {Frac}(R)$ . Let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Suppose that $\operatorname {div} f=n(K_R+\Delta )$ for $f\in R^{\circ }$ , $n\in \mathbb {N}$ . Let $B_p$ be a BCM $R_p^+$ -algebra for almost all p and $B:=\operatorname *{\mathrm {ulim}}_p B_p$ . We use $\widehat {R}$ to denote the completion of R with respect to $\mathfrak {m}$ and $\widehat {\Delta }$ to denote the flat pullback of $\Delta $ by $\operatorname {Spec} \widehat {R}\to \operatorname {Spec} R$ .

Proposition 6.1. In the setting as above, we have

$$ \begin{align*} \mathcal{J}(\widehat{R},\widehat{\Delta})\subseteq \tau_{\widehat{B}}(\widehat{R},\widehat{\Delta}). \end{align*} $$

Proof. Consider the following commutative diagram:

By Proposition 2.12, we have

$$ \begin{align*} 0_{H_{\mathfrak{m}_p}^d(\omega_{R_p})}^{B_p,K_{R_p}+\Delta_p}=0_{H_{\mathfrak{m}_p}^d(\omega_{R_p})}^{*\Delta_p} \end{align*} $$

for almost all p. Let $x_1,\dots ,x_d$ be a system of parameters, and let $x=x_1\cdots x_d$ be the product of them. Take $a\in \mathcal {J}(R,\Delta )=\operatorname *{\mathrm {ulim}}_p \tau (R_p,\Delta _p)\cap R$ and $\lbrack \frac {z}{x^t}\rbrack \in 0_{H_{\mathfrak {m}}^d(\omega _{R})}^{B,K_R+\Delta }$ . Let J be a divisorial ideal which is isomorphic to $\omega _R$ and $g\in R^\circ $ an element such that $\omega _R \xrightarrow {\cdot g} J$ is an isomorphism. As in Proof of [29, Th. 2.8], we have $g_p z_p x_p^t\in ((x_{1p}^{2t},\dots ,x_{dp}^{2t})J_p)^{*\Delta _p}$ for almost all p. Hence, $a_p g_p z_p x_p^t\in (x_{1p}^{2t},\dots ,x_{dp}^{2t})J_p$ for almost all p. Therefore, $agzx^t\in (x_1^{2t},\dots ,x_d^{2t})J$ and $\lbrack \frac {az}{x^t} \rbrack =0$ in $H_{\mathfrak {m}}^d(\omega _R)$ . Hence, we have $a\in \operatorname {Ann}_R 0_{H_{\mathfrak {m}}^d(\omega _{R})}^{B,K_R+\Delta }$ . In conclusion, we have $\mathcal {J} (R,\Delta )\widehat {R}\subseteq \tau _{\widehat {B}}(\widehat {R},\widehat {\Delta })$ .

Lemma 6.2 [29, Th. 2.13].

Let $(R,\mathfrak {m})$ be an F-finite normal local domain of characteristic $p>0$ and $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor on $X:=\operatorname {Spec} R$ such that $K_X+\Delta $ is $\mathbb {Q}$ -Cartier. Let $f:Y\to X$ be a proper birational morphism with X normal. Suppose that $Z:=f^{-1}(\mathfrak {m})$ and $\delta :H_{\mathfrak {m}}^d(R(K_X))\to H_Z^d(Y,\mathcal {O}_Y(\lfloor f^{*}(K_X+\Delta )\rfloor )$ is the Matlis dual of the natural inclusion map $H^0(Y,\mathcal {O}_Y(\lceil K_Y-f^*(K_X+\Delta ) \rceil ))\hookrightarrow R$ . Then $\operatorname {Ker} \delta \subseteq 0_E^{*\Delta }$ , where E is the injective hull of the residue field $R/\mathfrak {m}$ of R.

Proof. By [29, Th. 2.13], we have $\tau (R,\Delta )\subseteq H^0(Y,\mathcal {O}_Y(\lceil K_Y-f^*(K_X+\Delta ) \rceil ))$ . Hence,

$$ \begin{align*} \operatorname{Ker} \delta&=\operatorname{Ann}_E H^0(Y,\mathcal{O}_Y(\lceil K_Y-f^*(K_X+\Delta) \rceil))\\ &\subseteq \operatorname{Ann}_E \tau(R,\Delta) \\ &=\operatorname{Ann}_E \tau(R,\Delta)\widehat{R} \\ &=\operatorname{Ann}_E \tau(\widehat{R},\widehat{\Delta}) \\ &=\operatorname{Ann}_E \operatorname{Ann}_{\widehat{R}} 0_E^{*\Delta} \\ &= 0_E^{*\Delta}.\\[-36pt] \end{align*} $$

Remark 6.3. Moreover, we have $\operatorname {Ker} \delta =0_E^{*\Delta }$ if f is a reduction of a log resolution in characteristic zero modulo $p\gg 0$ by [29, Th. 3.2].

Theorem 6.4. Let R be a normal local domain essentially of finite type over $\mathbb {C}$ . Fix an effective canonical divisor $K_R\geqslant 0$ on $\operatorname {Spec} R$ . Let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier and $B_p$ is a BCM $R_p^+$ -algebra for almost all p. Suppose that $n(K_R+\Delta )=\operatorname {div} (f)$ for $f\in R^\circ , n\in \mathbb {N}$ . Then we have

$$ \begin{align*} \tau_{\widehat{B}}(\widehat{R},\widehat{\Delta})=\mathcal{J}(\widehat{R},\widehat{\Delta}). \end{align*} $$

Proof. Thanks to Proposition 6.1, it suffices to prove $\tau _{\widehat {B}}(\widehat {R},\widehat {\Delta })\subseteq \mathcal {J}(\widehat {R},\widehat {\Delta })$ . Let $\mu :Y\to X:=\operatorname {Spec} R$ be a log resolution of $(X,\Delta )$ , and let $Z:=\mu ^{-1}(\mathfrak {m})$ . Considering approximations, we have a corresponding morphisms $\mu _p:Y_p\to X_p:=\operatorname {Spec} R_p$ , $Z_p=\mu _p^{-1}(\mathfrak {m}_p)$ for almost all p. Then we have a commutative diagram

where $\mathcal {L}:=\mathcal {O}_Y(\lfloor \mu ^*(K_X+\Delta ) \rfloor )$ and the middle row is exact. Similarly, we have the following commutative diagram for almost all p:

where the middle row is exact. Assume that $\eta \in \operatorname {Ker} \delta $ . Then $u^{d-1}(\gamma (\eta ))\in \operatorname {Im} \rho _\infty ^{d-1}$ . Therefore, $\gamma _p(\eta _p) \in \operatorname {Im}\rho _p^{d-1}$ for almost all p. Hence, $\eta _p\in \operatorname {Ker} \delta _p$ for almost all p. By Lemma 6.2, $\eta _p\in 0_{H_{\mathfrak {m}_p}^d(\omega _{R_p})}^{*\Delta _p}$ for almost all p. Hence, by Propositon 2.12, we have $\eta _p\in 0_{H_{m_p}^d(\omega _{R_p})}^{B_p,K_{R_p}+\Delta _p}$ for almost all p. We have a commutative diagram

where $\psi $ , $\psi _p$ are the morphisms as in Definition 2.11. Since $\psi _\infty (\operatorname *{\mathrm {ulim}}_p \eta _p)=0$ and $H_{\mathfrak {m}}^d(B) \to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(B_p)$ is injective by Lemma 4.7, we have $\psi (\eta )=0$ in $H_{\mathfrak {m}}^d(B)$ . Hence, $\eta \in 0_{H_{\mathfrak {m}}^d(\omega _R)}^{B,K_R+\Delta }$ . Therefore, we have

$$ \begin{align*} \tau_{\widehat{B}}(\widehat{R},\widehat{\Delta})&\subseteq \operatorname{Ann}_{\widehat{R}}(\operatorname{Ker} \delta)\\ &= \operatorname{Ann}_{\widehat{R}}\operatorname{Ann}_{H_{\mathfrak{m}}^d(\omega_R)}\mathcal{J}(R,\Delta)\\ &= \mathcal{J}(\widehat{R},\widehat{\Delta}).\\[-36pt] \end{align*} $$

Remark 6.5. We can generalize the notion of ultra-test ideals in [31, Def. 5.5] to the pair $(R,\Delta )$ . Using Lemma 6.2 instead of [11, Th. 6.9], we can show that generalized ultra-test ideals are equal to multiplier ideals.

7 Generalized module closures and applications

We introduce the notion of generalized module closures inspired by [20]. Using the generalized module closures, we will generalize [31, Cor. 5.30]. We also use [19, §6.1] as reference in the following arguments.

Setting 7.1. Suppose that R is a normal local domain essentially of finite type over $\mathbb {C}$ of dimension d, $K_R\geqslant 0$ is a fixed effective canonical divisor and $\Delta \geqslant 0$ is an effective $\mathbb {Q}$ -Weil divisor such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Moreover, we assume that $B_p$ is a BCM $R_p^+$ -algebra for almost all p, $B:=\operatorname *{\mathrm {ulim}}_p B_p$ and $r(K_R+\Delta )=\operatorname {div} f$ for $f\in R$ , $r\in \mathbb {N}$ . Let $R'\subseteq R^+$ be an integrally closed finite extension of R such that $f^{1/r}\in R'$ and $\pi ^*\Delta $ is Weil divisor, where $\pi :\operatorname {Spec} R'\to \operatorname {Spec} R$ .

Definition 7.2. Assume Setting 7.1 and let $g\in R^\circ $ and $t>0$ be a positive rational number. We use $\widehat {B_\Delta }$ to denote

$$ \begin{align*} B\otimes_{R'}R'(\pi^*\Delta)\otimes_R \widehat{R}. \end{align*} $$

For any $\widehat {R}$ -modules $N\subseteq M$ , we define $N_M^{\operatorname {cl}_{\widehat {B_\Delta },g^t}}$ as follows: $x\in N_M^{\operatorname {cl}_{\widehat {B_\Delta },g^t}}$ if and only if $g^t\otimes x\in \operatorname {Im}(\widehat {B_\Delta }\otimes _{\widehat {R}}N\to \widehat {B_\Delta }\otimes _{\widehat {R}}M)$ . We use $\tau _{\operatorname {cl}_{\widehat {B_\Delta },g^t}}(\widehat {R})$ to denote

$$ \begin{align*} \bigcap_{N\subseteq M}(N:_{\widehat{R}}N_M^{\operatorname{cl}_{\widehat{B_\Delta}},g^t}), \end{align*} $$

where M runs through all $\widehat {R}$ -modules and N runs through all $\widehat {R}$ -submodules of M.

Proposition 7.3. In Setting 7.1, if $g\in R^\circ $ and $t>0$ is a positive rational number, then we have

$$ \begin{align*} \tau_{\operatorname{cl}_{\widehat{B_\Delta},g^t}}(\widehat{R})=\bigcap_{M}\operatorname{Ann}_{\widehat{R}}0_M^{\operatorname{cl}_{\widehat{B_\Delta},g^t}}=\operatorname{Ann}_{\widehat{R}}0_E^{\operatorname{cl}_{\widehat{B_\Delta},g^t}}, \end{align*} $$

where M runs through all $\widehat {R}$ -modules and E is the injective hull of the residue field of R.

Proof. We can prove this by arguments similar to [20, Lem. 3.3 and Prop. 3.9].

Proposition 7.4. In Setting 7.1, if $g\in R^\circ $ and $t>0$ is a positive rational number, then we have

$$ \begin{align*} 0_E^{B,K_R+\Delta+t\operatorname{div} g}=0_E^{\operatorname{cl}_{\widehat{B_\Delta},g^t}}. \end{align*} $$

Proof. Since the reflexive hull $(R'(\pi ^*\Delta )\otimes _R \omega _R)^{**}$ is equal to $R'(\operatorname {div} (f^{\frac {1}{r}}))$ , we have $H_{\mathfrak {m}}^d(R'(\pi ^*\Delta )\otimes _R \omega _R)\cong H_{\mathfrak {m}}^d(R'(\operatorname {div} (f^{\frac {1}{r}})))$ . Hence, we have

$$ \begin{align*} \widehat{B_\Delta}\otimes_{\widehat{R}} E &\cong B\otimes_{R'}H_{\mathfrak{m}}^d(R'(\pi^*\Delta)\otimes_R\omega_R) \\ &\cong B\otimes_{R'}H_{\mathfrak{m}}^d(R'(\operatorname{div} (f^{\frac{1}{r}}))). \end{align*} $$

Then there exists a commutative diagram

where $\psi $ is the second map of

$$ \begin{align*} \cdot f^{\frac{1}{r}}g^t : H_{\mathfrak{m}}^d(B) \to H_{\mathfrak{m}}^d(B\otimes_R\omega_R) \to H_{\mathfrak{m}}^d(B). \end{align*} $$

The result follows by the above commutative diagram.

Definition 7.5. Let $R\hookrightarrow S$ be an injective local homomorphism of normal local domains essentially of finite type over $\mathbb {C}$ . Fix $K_R, K_S\geqslant 0$ effective canonical divisors on $\operatorname {Spec} R$ and on $\operatorname {Spec} S$ , respectively. Let $\Delta _R,\Delta _S\geqslant 0$ be effective $\mathbb {Q}$ -Weil divisors on $\operatorname {Spec} R$ and on $\operatorname {Spec} S$ , respectively, such that $K_R+\Delta _R$ , $K_S+\Delta _S$ are $\mathbb {Q}$ -Cartier. Let $\mathfrak {a}\subseteq R$ be a nonzero ideal and $t>0$ be a positive rational number. Suppose that $\widehat {B_{\Delta _R}}$ and $\widehat {B_{\Delta _S}}$ are defined as in Definition 7.2. Then, for an $\widehat {R}$ -module M and an $\widehat {S}$ -module N, we define $0_{M}^{\operatorname {cl}_{\widehat {B_{\Delta _R}},\mathfrak {a}^t}}$ , $0_{N}^{\operatorname {cl}_{\widehat {B_{\Delta _S}}\mathfrak {a}^t}}$ by

$$ \begin{align*} 0_{M}^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}:=\bigcap_{n\in\mathbb{N}}\bigcap_{g\in\mathfrak{a}^{\lceil nt \rceil}}0_{M}^{\operatorname{cl}_{\widehat{B_{\Delta_R}},g^{\frac{1}{n}}}},\\ 0_{N}^{\operatorname{cl}_{\widehat{B_{\Delta_S}}\mathfrak{a}^t}}:=\bigcap_{n\in\mathbb{N}}\bigcap_{g\in\mathfrak{a}^{\lceil nt \rceil}}0_{N}^{\operatorname{cl}_{\widehat{B_{\Delta_S}}g^{\frac{1}{n}}}}. \end{align*} $$

We use $\tau _{\operatorname {cl}_{\widehat {B_{\Delta _R}},\mathfrak {a}^t}}(\widehat {R})$ , $\tau _{\operatorname {cl}_{\widehat {B_{\Delta _S}},\mathfrak {a}^t}}(\widehat {S})$ to denote

$$ \begin{align*} \bigcap_M \operatorname{Ann}_{\widehat{R}}0_M^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}, \\ \bigcap_N \operatorname{Ann}_{\widehat{S}}0_N^{\operatorname{cl}_{\widehat{B_{\Delta_S}},\mathfrak{a}^t}}, \end{align*} $$

where M runs through all $\widehat {R}$ -modules and N runs through all $\widehat {S}$ -modules.

Proposition 7.6. In the setting of Definition 7.5, we have

$$ \begin{align*} \operatorname{Ann}_{\widehat{R}}0_{E_R}^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}=\bigcap_M \operatorname{Ann}_{\widehat{R}}0_M^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}, \\ \operatorname{Ann}_{\widehat{S}}0_{E_S}^{\operatorname{cl}_{\widehat{B_{\Delta_S}},\mathfrak{a}^t}}=\bigcap_N \operatorname{Ann}_{\widehat{S}}0_N^{\operatorname{cl}_{\widehat{B_{\Delta_S}},\mathfrak{a}^t}}, \end{align*} $$

where $M,N$ run through all $\widehat {R}$ -modules and $\widehat {S}$ -modules, respectively, and $E_R$ , $E_S$ are the injective hulls of the residue fields of R and S, respectively.

Proof. We can show this by arguments similar to Proposition 7.3.

Proposition 7.7. In the setting of Definition 7.5, we have

$$ \begin{align*} \tau_{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}(\widehat{R})=\mathcal{J}(\widehat{R},\widehat{\Delta},(\mathfrak{a}\widehat{R})^t). \end{align*} $$

Proof. Let E be the injective hull of the residue field of R. Then

$$ \begin{align*} 0_E^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}} &= \bigcap_{n\in \mathbb{N}}\bigcap_{g\in\mathfrak{a}^{\lceil nt \rceil}} 0_E^{\operatorname{cl}_{\widehat{B_{\Delta_R}}, g^{\frac{1}{n}}}} \\ &= \bigcap_{n\in\mathbb{N}}\bigcap_{g\in\mathfrak{a}^{\lceil nt \rceil}}\operatorname{Ann}_E \mathcal{J}(\widehat{R},\widehat{\Delta},g^{\frac{1}{n}}) \\ &= \operatorname{Ann}_E \sum_{n\in \mathbb{N}}\sum_{g\in\mathfrak{a}^{\lceil nt \rceil}}\mathcal{J} (\widehat{R},\widehat{\Delta},g^{\frac{1}{n}}) \\ &= \operatorname{Ann}_E \mathcal{J}(\widehat{R},\widehat{\Delta},(\mathfrak{a} \widehat{R})^t), \end{align*} $$

where the second equality follows from Theorem 6.4. Hence, we have

$$ \begin{align*} \operatorname{Ann}_{\widehat{R}}0_E^{\operatorname{cl}_{\widehat{B_{\Delta_R}},\mathfrak{a}^t}}=\mathcal{J}(\widehat{R},\widehat{\Delta},(\mathfrak{a} \widehat{R})^t).\\[-36pt] \end{align*} $$

The next lemma is a generalization of [30, Th. 3.2].

Lemma 7.8. Let R be a normal local domain essentially of finite type over $\mathbb {C}$ , and let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Let $\mathfrak {a}_1,\dots ,\mathfrak {a}_n\subseteq R$ be nonzero ideals, and let $t>0$ be a positive rational number. Then we have

$$ \begin{align*} \mathcal{J}(R,\Delta,(\mathfrak{a}_1+\dots+\mathfrak{a}_n)^t)=\sum_{\lambda_1+\dots+\lambda_n=t}\mathcal{J}(R,\Delta,\mathfrak{a}_1^{\lambda_1}\cdots \mathfrak{a}_n^{\lambda_n}). \end{align*} $$

Lemma 7.9. In the setting of Definition 7.5, we have

$$ \begin{align*} \sum_{n\in\mathbb{N}}\sum_{g\in\mathfrak{a}^{\lceil nt \rceil}}\mathcal{J}(S,\Delta_S,g^{\frac{1}{n}})=\mathcal{J}(S,\Delta_S,(\mathfrak{a} S)^t). \end{align*} $$

Proof. $\sum _{n\in \mathbb {N}}\sum _{g\in \mathfrak {a}^{\lceil nt \rceil }}\mathcal {J}(S,\Delta _S,g^{1/n})\subseteq \mathcal {J}(S,\Delta _S,(\mathfrak {a} S)^t)$ is clear. If $t=q/p$ , $p,q>0$ and $\mathfrak {a}=(g_1,\dots ,g_l)$ , then

$$ \begin{align*} \sum_{n\in\mathbb{N}}\sum_{g\in\mathfrak{a}^{\lceil nt \rceil}}\mathcal{J}(S,\Delta_S,g^{\frac{1}{n}}) & \supseteq \sum_{n\in \mathbb{N}}\sum_{i_1+\dots+i_l=nq}\mathcal{J}(S,\Delta_S,(g_1^{i_1}\cdots g_l^{i_l})^{\frac{1}{np}}) \\ &=\mathcal{J}(S,\Delta_S,(\mathfrak{a} S)^t), \end{align*} $$

by the above lemma.

Theorem 7.10. Let $R\hookrightarrow S$ be a pure local homomorphism of normal local domains essentially of finite type over $\mathbb {C}$ . Fix effective canonical divisors $K_R$ and $K_S$ on $\operatorname {Spec} R$ and $\operatorname {Spec} S$ , respectively. Let $\Delta _R, \Delta _S \geqslant 0$ be effective $\mathbb {Q}$ -Weil divisors on $\operatorname {Spec} R$ , $\operatorname {Spec} S$ such that $K_R+\Delta _R$ , $K_S+\Delta _S$ are $\mathbb {Q}$ -Cartier. Take normal domains $R',S'$ and morphisms $\pi _R,\pi _S$ as in Setting 7.1. Moreover, let $\mathfrak {a}\subseteq R$ be a nonzero ideal, and let $t>0$ be a positive rational number. If $R'(\pi _R^*\Delta _R)\subseteq S'(\pi _S^*\Delta _S)$ , then we have

$$ \begin{align*} \mathcal{J}(S,\Delta_S,(\mathfrak{a} S)^t)\cap R\subseteq \mathcal{J}(R,\Delta_R,\mathfrak{a}^t). \end{align*} $$

Proof. Since $R\hookrightarrow S$ is pure, $\widehat {R} \hookrightarrow \widehat {S}$ is pure (see [6, Cor. 3.2.1]). Since $R\to \widehat {R}$ , $S\to \widehat {S}$ are pure, it is enough to show

$$ \begin{align*} \mathcal{J}(\widehat{S},\widehat{\Delta_S},(\mathfrak{a} \widehat{S})^t)\cap \widehat{R}\subseteq \mathcal{J}(\widehat{R},\widehat{\Delta_R},(\mathfrak{a} \widehat{R})^t). \end{align*} $$

Let $\mathcal {B}(R)$ , $\mathcal {B}(S)$ be the canonical BCM algebras. Let $\widehat {\mathcal {B}_{\Delta _R}}:=\widehat {\mathcal {B}(R)_{\Delta _R}}$ and $\widehat {\mathcal {B}_{\Delta _S}}:=\widehat {\mathcal {B}(S)_{\Delta _S}}$ . Take an $\widehat {R}$ -module M. Then we have a commutative diagram

Tensoring the commutative diagram with M, we have

Hence, we have

$$ \begin{align*} 0_{M}^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_R}},\mathfrak{a}^t}}\subseteq 0_{\widehat{S}\otimes_{\widehat{R}}M}^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},\mathfrak{a}^t}}. \end{align*} $$

Then we have

$$ \begin{align*} \mathcal{J}(\widehat{R},\widehat{\Delta_R},\mathfrak{a}^t)&=\bigcap_{M} \operatorname{Ann}_{\widehat{R}}0_M^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_R}},\mathfrak{a}^t}} \\ &\supseteq \bigcap_{M}\operatorname{Ann}_{\widehat{R}}0_{M\otimes S}^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},\mathfrak{a}^t}} \\ &\supseteq \bigcap_N \operatorname{Ann}_{\widehat{R}}0_N^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},\mathfrak{a}^t}}\\ &= \bigcap_N (\operatorname{Ann}_{\widehat{S}}0_N^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},\mathfrak{a}^t}}\cap \widehat{R}) \\ &= (\operatorname{Ann}_{\widehat{S}}0_{E_S}^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},\mathfrak{a}^t}})\cap \widehat{R} \\ &= (\operatorname{Ann}_{\widehat{S}}\bigcap_{n\in\mathbb{N}}\bigcap_{g\in\mathfrak{a}^{\lceil nt \rceil}} 0_{E_S}^{\operatorname{cl}_{\widehat{\mathcal{B}_{\Delta_S}},g^{\frac{1}{n}}}})\cap \widehat{R} \\ &= (\operatorname{Ann}_{\widehat{S}}\operatorname{Ann}_{E_S}\sum_{n\in\mathbb{N}}\sum_{g\in \mathfrak{a}^{\lceil nt \rceil}}\mathcal{J}(\widehat{S},\widehat{\Delta_S},g^{\frac{1}{n}}))\cap \widehat{R} \\ &= \mathcal{J}(\widehat{S},\widehat{\Delta_S},(\mathfrak{a} \widehat{S})^t))\cap \widehat{R}, \end{align*} $$

where M runs through all $\widehat {R}$ -modules, N runs through all $\widehat {S}$ -modules, and $E_S$ is the injective hull of the residue field of S.

As a corollary, we have a generalization of [31, Cor. 5.30] to the case that $\mathfrak {a}$ is not necessarily a principal ideal.

Corollary 7.11. Let $R\hookrightarrow S$ be a pure local homomorphism of normal local domains essentially of finite type over $\mathbb {C}$ . Suppose that R is $\mathbb {Q}$ -Gorenstein. Fix effective canonical divisors $K_R$ and $K_S$ on $\operatorname {Spec} R$ and $\operatorname {Spec} S$ , respectively. Let $\Delta _S$ be an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} S$ such that $K_S+\Delta _S$ is $\mathbb {Q}$ -Cartier. Let $\mathfrak {a}\subseteq R$ be a nonzero ideal and $t>0$ a positive rational number. Then we have

$$ \begin{align*} \mathcal{J}(S,\Delta_S,(\mathfrak{a} S)^t)\cap R\subseteq \mathcal{J}(R,\mathfrak{a}^t). \end{align*} $$

Proof. Let $R'$ be the integral closure of $R[f^{1/r}]$ in $R^+$ . Then the result follows from Theorem 7.10.

8 $\mathcal {B}$ -regularity

As another application of the main theorem, we will give a partial answer to [24, Rem. 3.10]. For this, we will review the definition of $\mathcal {B}$ -regularity.

Definition 8.1 [23, Def. 4.3].

Let R be a normal $\mathbb {Q}$ -Gorenstein local domain essentially of finite type over $\mathbb {C}$ .

1. (1) R is said to be weakly $\mathcal {B}$ -regular if $R\to \mathcal {B}(R)$ is cyclically pure.

2. (2) R is said to be $\mathcal {B}$ -regular if every localization of R at a prime ideal is weakly $\mathcal {B}$ -regular.

Theorem 8.2. Let R be a normal $\mathbb {Q}$ -Gorenstein local domain. Then the following are equivalent:

1. (1) R has log-terminal singularities.

2. (2) R is ultra-F-regular.

3. (3) R is weakly generically F-regular.

4. (4) R is generically F-regular.

5. (5) R is weakly $\mathcal {B}$ -regular.

6. (6) R is $\mathcal {B}$ -regular.

7. (7) $\widehat {R}$ is $\text {BCM}_{\widehat {\mathcal {B}(R)}}$ -regular.

Proof. The equivalence of (1) and (2) follows from Proposition 3.54 and the equivalence of (1) and (7) follows from Theorem 6.4. Since, if R has log-terminal singularities, then every localization of R at a prime ideal is log-terminal, it is enough to show the equivalence of (1), (3), and (5). (1) is equivalent to (3) by [31, Th. 5.24 and Proof of Th. 5.25]. Lastly, we will show the equivalence of (5) and (7). Let E be the injective hull of the residue field of R. By Proposition 7.4, we have $0_E^{\operatorname {cl}_{\mathcal {B}(R)\otimes _R\widehat {R}}}=0_E^{\mathcal {B}(R),K_R}$ . Hence, $E\to \mathcal {B}(R)\otimes _R E$ is injective if and only if $\widehat {R}$ is $\text {BCM}_{\widehat {\mathcal {B}(R)}}$ -regular. $R\to \mathcal {B}(R)$ is pure if and only if $E\to \mathcal {B}(R)\otimes _R E$ is injective by [15, Lem. 2.1(e)]. $R\to \mathcal {B}(R)$ is pure if and only if $R\to \mathcal {B}(R)$ is cyclically pure by [12, Th. 1.7]. Therefore, (5) is equivalent to (7).

Remark 8.3. For the equivalence of (5) and (7) (see [19, Prop. 6.14]).

9 Further questions and remarks

In this section, we will consider whether R is BCM-rational if R has rational singularities. The next question is a variant of [7, Quest. 2.7].

Question 1. Let R be a local domain essentially of finite type over $\mathbb {C}$ , and let B be a BCM R-algebra. If S is finitely generated R-algebra such that the following diagram commutes:

then does there exist a BCM $R_p$ -algebra for almost all p which fits into the following commutative diagram:

where $S_p$ is an R-approximation of S?

Proposition 9.1 (Cf. [19, Conj. 3.9]).

Let R be a normal local domain essentially of finite type over $\mathbb {C}$ of dimension d. Suppose that R has rational singularities. If Question 1 has an affirmative answer, then R is BCM-rational.

Proof. Let B be a BCM $R^+$ -algebra. Suppose that $\eta \in \operatorname {Ker} (H_m^d(R)\to H_{\mathfrak {m}}^d(B))$ . Then there exists a finitely generated R-subalgebra of B such that the image of $\eta $ in $H_m^d(S)$ is zero. If Question 1 has an affirmative answer, we can take $S_p$ and $B_p$ as in Question 1. Then we have a commutative diagram

By the proof of Proposition 4.8, $\operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(R_p)\to \operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(S_p)$ is injective. Therefore, the image of $\eta $ in $\operatorname *{\mathrm {ulim}}_p H_{\mathfrak {m}_p}^d(R_p)$ is zero. Suppose that $\eta =\lbrack \frac {y}{x^t}\rbrack $ , where $y\in R$ , $t\in \mathbb {N}$ and x is the product of a system of parameters $x_1,\dots ,x_d$ of R. Since $R_p$ is Cohen–Macaulay for almost all p, $y_p\in (x_{1p}^t,\dots , x_{dp}^t)$ for almost all p. Hence, $y\in (x_1^t,\dots ,x_d^t)$ and $\eta =0$ in $H_{\mathfrak {m}}^d(R)$ . Thus, $H_{\mathfrak {m}}^d(R)\to H_{\mathfrak {m}}^d(B)$ is injective.

The next result follows from a similar argument.

Proposition 9.2. Let R be a normal local domain essentially of finite type over $\mathbb {C}$ of dimension d. Fix an effective canonical divisor $K_R$ on $\operatorname {Spec} R$ . Let $\Delta \geqslant 0$ be an effective $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Suppose that C is a BCM $R^+$ -algebra. If Question 1 has an affirmative answer, then we have

$$ \begin{align*} \mathcal{J}(R,\Delta)\subseteq \tau_{\widehat{C}}(\widehat{R},\widehat{\Delta}). \end{align*} $$

Definition 9.3 (Cf. [19, Def. 6.9]).

Let R be a normal local domain essentially of finite type over $\mathbb {C}$ . Fix an effective canonical divisor $K_R$ on $\operatorname {Spec} R$ . Let $\Delta \geqslant 0$ be a $\mathbb {Q}$ -Weil divisor on $\operatorname {Spec} R$ such that $K_R+\Delta $ is $\mathbb {Q}$ -Cartier. Suppose that $n(K_R+\Delta )=\operatorname {div}(f)$ for $f\in R^\circ $ , $n\in \mathbb {N}$ . We define

$$ \begin{align*} 0_{H_{\mathfrak{m}}^d(R)}^{\mathscr{B},K_R+\Delta}:=\{\eta\in H_{\mathfrak{m}}^d(R)| \quad &\exists C \text{ BCM } R^+ \text{-algebra}\\ &{\text{such that }f^{\frac{1}{n}}\eta =0 \text{ in } H_{\mathfrak{m}}^d(C)\}.} \end{align*} $$

We define the BCM test ideal $\tau _{\mathscr {B}}(R,\Delta )$ of $(\widehat {R},\widehat {\Delta })$ by

$$ \begin{align*} \tau_{\mathscr{B}}(\widehat{R},\widehat{\Delta}):=\operatorname{Ann}_{\omega_{\widehat{R}}}0_{H^d_{\mathfrak{m}}(R)}^{\mathscr{B},K_R+\Delta}. \end{align*} $$

Corollary 9.4 (Cf. [19, Th. 6.21]).

In the setting of the above proposition, if Question 1 has an affirmative answer, then we have

$$ \begin{align*} \tau_{\mathscr{B}}(\widehat{R},\widehat{\Delta})=\mathcal{J}(\widehat{R},\widehat{\Delta}). \end{align*} $$