Axiomatic closure operations, phantom extensions, and solidity

JOURNAL OF ALGEBRA

卷:502

Article

Dietz, Geoffrey D.¹ 

[1] Gannon Univ, Dept Math, Erie, PA 16541 USA

关键词: Big Cohen-Macaulay modules;    Closure operations;    Tight closure;    Phantom extensions;    Solid modules and algebras;   

DOI  :  10.1016/j.jalgebra.2018.01.023

来源: Elsevier

【 摘 要 】

In this article, we generalize a previously defined set of axioms for a closure operation that induces balanced big Cohen-Macaulay modules. While the original axioms were only defined in terms of finitely generated modules, these new ones will apply to all modules over a local domain. The new axioms will lead to a notion of phantom extensions for general modules, and we will prove that all modules that are phantom extensions can be modified into balanced big Cohen-Macaulay modules and are also solid modules. As a corollary, if R has characteristic p > 0 and is F-finite, then all solid algebras are phantom extensions. If R also has a big test element (e.g., if R is complete), then solid algebras can be modified into balanced big Cohen-Macaulay modules. (Hochster and Huneke have previously demonstrated that there exist solid algebras that cannot be modified into balanced big Cohen-Macaulay algebras.) We also point out that tight closure over local domains in characteristic p generally satisfies the new axioms and that the existence of a big Cohen-Macaulay module induces a closure operation satisfying the new axioms. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free