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 [Reference Hochster and Huneke14], [Reference Hochster and Huneke15] proved the existence of a BCM algebra in equal characteristic, and André [Reference André1] settled the mixed characteristic case. Recently, using BCM algebras, Ma and Schwede [Reference Ma and Schwede18], [Reference Ma and Schwede19] 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 [Reference Hara and Yoshida11], [Reference Takagi29] 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 [Reference Schoutens24] 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 [Reference Ma and Schwede19], 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 [Reference Schoutens25] with the description of multiplier ideals as the kernel of a map between local cohomology modules in [Reference Takagi29]. 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 [Reference Yamaguchi31, 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 [Reference Yamaguchi31], 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 [Reference Schoutens24]. 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 [Reference Pérez and Rebecca20] 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 [Reference Schoutens24, 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 [Reference Schoutens23]. 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 [Reference Dietz and Rebecca7, 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 [Reference Lazarsfeld16], [Reference Sato and Takagi21] 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 [Reference Blickle, Schwede, Takagi and Zhang4], [Reference Hara and Yoshida11], [Reference Hochster and Huneke13], [Reference Takagi29].
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) 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 for all large
$$ \begin{align*} c\mathfrak{a}^{\lceil t(q-1)\rceil}x^q\subseteq I^{[q]}R(\lceil (q-1)\Delta\rceil) \end{align*} $$
$q=p^e$
, where
$I^{[q]}=\{f^q|f\in I\}$
and
$R^{\circ }=R\setminus \{0\}$
.
-
(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 for all large
$$ \begin{align*} (c\mathfrak{a}^{\lceil t(q-1)\rceil})^{1/q}\otimes z=0 \quad \text{in} \quad R(\lceil (q-1)\Delta\rceil)^{1/q}\otimes_R M \end{align*} $$
$q=p^e$
.
-
(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 [Reference Fedder and Watanabe8].
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 [Reference Bruns and Herzog5, 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 [Reference Bhatt3], [Reference Hochster and Huneke14].
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 [Reference Ma and Schwede19].
Setting 2.10. Let
$(R,\mathfrak {m})$
be a normal local domain of dimension d.
-
(i)
$\Delta \geqslant 0$
is a
$\mathbb {Q}$
-Weil divisor on
$\operatorname {Spec} R$
such that
$K_R+\Delta $
is
$\mathbb {Q}$
-Cartier. -
(ii) Fixing
$\Delta $
, we also fix an embedding
$R\subseteq \omega _R \subseteq \operatorname {Frac} R$
, where
$\omega _R$
is the canonical module. -
(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 [Reference Ma and Schwede19].
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 [Reference Schoutens22], [Reference Schoutens26] 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.
-
(i) If
$A, B \in \mathcal {F}$
, then
$A \cap B\in \mathcal {F}$
. -
(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)
$\mathcal {F}$
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)
$\mathcal {F}$
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 [Reference Lyu17, 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)
$A_\infty $
is a ring. -
(2)
$B_\infty $
is an
$A_\infty $
-algebra. -
(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:
-
(i)
$0$
,
$1$
are terms. -
(ii) Variables are terms.
-
(iii) If s, t are terms, then
${-(s)}, (s)+(t), (s)\cdot (t)$
are terms. -
(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:
-
(i) If
$s $
, t are terms, then
$(s=t)$
is a formula. -
(ii) If
$\varphi , \psi $
are formulas, then
$(\varphi \land \psi ), (\varphi \lor \psi ), (\varphi \to \psi ),(\lnot \varphi )$
are formulas. -
(iii) If
$\varphi $
is a formula and x is a variable, then
$\forall x \varphi , \exists x \varphi $
are formulas. -
(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) A is a field if and only if
$\forall x (x=0 \lor \exists y(xy=1))$
holds. -
(2) A is a domain if and only if
$\forall x \forall y(xy=0\to (x=0\lor y=0))$
holds. -
(3) A is a local ring if and only if
holds.
$$ \begin{align*} \forall x \forall y (\nexists z (xz=1)\land \nexists w(yw=1)\to \nexists u((x+y)u=1)) \end{align*} $$
-
(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) 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 ([Reference Schoutens22, 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 [Reference Schoutens22, 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 [Reference Schoutens22], [Reference Schoutens26]. 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 [Reference Schoutens22, 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) 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) 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) 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 [Reference Schoutens22, Cor. 4.2], [Reference Schoutens26, 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) 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) 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) 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 [Reference Schoutens22, 2.9.5, 2.9.7, Ths. 4.5 and 4.6], [Reference Schoutens26, §4.3]; cf. [Reference Aschenbrenner and Schoutens2, 5.1].
Let R be a local ring essentially of finite type over
$\mathbb {C}$
, then the following hold:
-
(1) R has dimension d if and only if
$R_p$
has dimension d for almost all p. -
(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)
$\mathbf {x}=x_1,\dots ,x_d$
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) R is regular if and only if
$R_p$
is regular for almost all p. -
(5) R is Gorenstein if and only if
$R_p$
is Gorenstein for almost all p. -
(6) R is Cohen–Macaulay if and only if
$R_p$
is Cohen–Macaulay for almost all p.
Proposition 3.44 [Reference Yamaguchi31, 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) R is normal.
-
(2)
$R_p$
is normal for almost all p. -
(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 [Reference Schoutens22, 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 [Reference Schoutens22, Def. 5.2], [Reference Schoutens25, 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 [Reference Schoutens25, Def. 4.1 and Rem. 4.7], [Reference Schoutens23, 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) R is said to be weakly generically F-regular if
$I^{*\operatorname {gen}}=I$
for any ideal
$I\subseteq R$
. -
(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) 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 [Reference Schoutens25, 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 [Reference Schoutens25, Th. 6.2], [Reference Schoutens23, 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 [Reference Schoutens24, 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 [Reference Schoutens24, 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 [Reference Schoutens24, 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 [Reference Schoutens22], [Reference Schoutens24], [Reference Schoutens25].
Definition 3.55 (Cf. [Reference Schoutens25]).
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. [Reference Schoutens25]).
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. [Reference Schoutens24]).
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. [Reference Schoutens24]).
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. [Reference Schoutens24]).
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 [Reference Schoutens24].
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. [Reference Schoutens24]).
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. [Reference Schoutens24]).
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. [Reference Schoutens24]).
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 [Reference Schoutens23], Schoutens constructed the canonical BCM algebra in characteristic zero. Following the idea of [Reference Schoutens23], 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 [Reference Schoutens23, §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
$\operatorname *{\mathrm {ulim}}_p B_p$
.
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 [Reference Schoutens23, Th. 4.2] to the cases other than the canonical BCM algebra.
Proposition 4.8 (Cf. [Reference Schoutens23, Th. 4.2], [Reference Ma and Schwede19, 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 [Reference Schoutens23, Prop. 4.11] and [Reference Hara9],
$R_p$
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 [Reference Ma and Schwede19, 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 [Reference Smith27, 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 [Reference Schoutens25, 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:
-
(i)
$A \subseteq \mathbb {C}$
is a finitely generated
$\mathbb {Z}$
-subalgebra. -
(ii)
$R_A$
is a finitely generated A-algebra such that
$R_A\otimes _A\mathbb {C}\cong R$
.
Proposition 5.2 [Reference Schoutens23, 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:
-
(i)
$\gamma _p:A\to \overline {\mathbb {F}_p}$
is a ring homomorphism for almost all p. -
(ii) For any
$x\in A$
,
$x=\operatorname *{\mathrm {ulim}}_p\gamma _p(x)$
.
Proposition 5.3 (Cf. [Reference Schoutens23, 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:
-
(i) There exists a family
$(\gamma _p)$
as in Proposition 5.2. -
(ii)
$\mathbf {a}\subseteq R_A$
. -
(iii)
$R_A\otimes _A \overline {\mathbb {F}_p}\cong R_p$
for almost all p. -
(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 [Reference Yamaguchi31, 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 [Reference Srinivas and Takagi28, Lem. 1.5]. Since the localization commutes with test ideals [Reference Hara and Takagi10, 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 [Reference Takagi29, 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
