Closure operations that induce big Cohen-Macaulay modules and classification of singularities

JOURNAL OF ALGEBRA

卷:467

Article

Rebecca, R. G.¹ 

[1] Univ Michigan, Dept Math, 2074 East Hall,530 Church St, Ann Arbor, MI 48109 USA

关键词: Commutative algebra;    Closure operations;    Big Cohen-Macaulay modules;    Homological conjectures;   

DOI  :  10.1016/j.jalgebra.2016.08.009

来源: Elsevier

【 摘 要 】

Geoffrey Dietz introduced a set of axioms for a closure operation on a complete local domain R so that the existence of such a closure operation is equivalent to the existence of a big Cohen-Macaulay module. These closure operations are called Dietz closures. In complete rings of characteristic p > 0, tight closure and plus closure satisfy the axioms. We define module closures and discuss their properties. For many of these properties, there is a smallest closure operation satisfying the property. In particular, we discuss properties of big Cohen-Macaulay module closures, and prove that every Dietz closure is contained in a big Cohen-Macaulay module closure. Using this result, we show that under mild conditions, a ring R is regular if and only if all Dietz closures on R are trivial. Finally, we show that solid closure in equal characteristic 0, integral closure, and regular closure are not Dietz closures, and that all Dietz closures are contained in liftable integral closure. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free