Big Cohen-Macaulay and seed algebras in equal characteristic zero via ultraproducts

Dietz, Geoffrey D.; Rebecca, R G

doi:10.1216/jca-2019-11-4-511

Cited by 6 publications

(6 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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…”

Section: §8 B-regularitymentioning

confidence: 93%

See 1 more Smart Citation

Big Cohen–macaulay Test Ideals in Equal Characteristic Zero via Ultraproducts

Yamaguchi

2022

Nagoya Math. J.

View full text Add to dashboard Cite

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.

show abstract

“…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…”

Section: §8 B-regularitymentioning

confidence: 93%

“…In §8, we show that -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.…”

Section: Introductionmentioning

confidence: 99%

Big Cohen–macaulay Test Ideals in Equal Characteristic Zero via Ultraproducts

Yamaguchi

2022

Nagoya Math. J.

View full text Add to dashboard Cite

show abstract

“…For our purposes, we will be dealing with rings of equal characteristic 0 that are ultraproducts of rings of characteristic p > 0, as in [DR17].…”

Section: Characteristic-free Test Ideals 779mentioning

confidence: 99%

Characteristic-free test ideals

Pérez¹,

Rebecca

2021

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.

show abstract

“…Among them, we want to remark that André established the following theorem in [3], using the results of this paper. 7 Refer to the paper [11] for a recent progress made on the weakly functorial big Cohen-Macaulay algebras in the equal characteristic zero case, using ultra-products. In [14],…”

Section: Comments and Open Problemsmentioning

confidence: 99%

Integral perfectoid big Cohen–Macaulay algebras via André’s theorem

Shimomoto

2018

Math. Ann.

View full text Add to dashboard Cite

The main result of this article is to prove that any Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay algebra. The proof of this result is based on the construction of almost Cohen-Macaulay algebras in mixed characteristic due to Yves André. Moreover, we prove that the absolute integral closure of a complete Noetherian local domain of mixed characteristic maps to an integral perfectoid big Cohen-Macaulay algebra.Macaulay R-module with respect to x. The notion of big Cohen-Macaulay modules was introduced by Hochster in 1970's for the purpose of studying the homological conjectures (see [16] for a brief survey and [18] in the equal characteristic case). Hochster proved that any local ring (R, m) of equal characteristic admits a big Cohen-Macaulay module. The issue whether such a module exists in mixed characteristic has remained unclear for a long time. Quite recently, André proved that big Cohen-Macaulay algebras exist in the mixed 1 2 K.SHIMOMOTO characteristic case in [1] and [2], which also implies that the Direct Summand Conjecture is fully settled (see [7] for the derived variant of this conjecture).Theorem 1.1 (Y. André). Every complete Noetherian local domain of mixed characteristic admits a big Cohen-Macaulay algebra.This theorem can be deduced from Theorem 1.2, which we explain below. The primary aim of this article is to show abundance of big Cohen-Macaulay algebras with certain distinguished properties based on Theorem 1.2 stated below. First, we state the main theorem (see Theorem 6.3).Main Theorem 1. Let (R, m) be a Noetherian local domain of mixed characteristic. Then there exists an R-algebra T such that T is an integral perfectoid big Cohen-Macaulay R-algebra. Moreover, we have the following assertions:(1) Assume that R is a complete Noetherian local domain of mixed characteristic with perfect residue field. Then there exists an integral perfectoid big Cohen-Macaulay R-algebra T with the property that R → T factors as R → S → T , such that S is an integral pre-perfectoid normal domain that is integral over R and(2) Assume that R is a complete Noetherian local domain of mixed characteristic. Let B be an integral almost perfectoid, almost Cohen-Macaulay R-algebra such that R → B factors as R → S → B and S is an integral perfectoid algebra containing compatible systems of elements: {p 1 p n } n≥0 , {x 1 p n 2 } n≥0 , . . . , {x 1 p nd } n≥0 for a system of parameters p, x 2 , . . . , x d of R. Then there is a ring homomorphism S → T such that T is an integral perfectoid big Cohen-Macaulay R-algebra.One can indeed prove the following stronger result (see Corollary 6.5 and Remark 6.6).Main Theorem 2. Let (R, m) be a Noetherian local domain of mixed characteristic and let R + be its absolute integral closure. Then R + maps to an integral perfectoid big Cohen-Macaulay R-algebra.For basic notion and notations, we refer the reader to § 2. An integral perfectoid algebra is a p-torsion free, p-adically complete algebras on which the Frobenius endomorphism is su...

show abstract

Big Cohen-Macaulay and seed algebras in equal characteristic zero via ultraproducts

Cited by 6 publications

References 13 publications

Big Cohen–macaulay Test Ideals in Equal Characteristic Zero via Ultraproducts

Big Cohen–macaulay Test Ideals in Equal Characteristic Zero via Ultraproducts

Characteristic-free test ideals

Integral perfectoid big Cohen–Macaulay algebras via André’s theorem

Contact Info

Product

Resources

About