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Extended plus closure in complete local rings
Journal of Algebra ( IF 0.8 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.jalgebra.2018.10.006
Raymond C. Heitmann , Linquan Ma

The (full) extended plus closure was developed as a replacement for tight closure in mixed characteristic rings. Here it is shown by adapting Andre's perfectoid algebra techniques that, for complete local rings that have F-finite residue fields, this closure has the colon-capturing property. In fact, more generally, if $R$ is a (possibly ramified) complete regular local ring of mixed characteristic that have F-finite residue fields, $I$ and $J$ are ideals of $R$, and the local domain $S$ is a finite $R$-module, then $(IS:J)\subseteq (I:J)S^{epf}$. A consequence is that all ideals in regular local rings are closed, a fact which implies the validity of the direct summand conjecture and the Briancon-Skoda theorem in mixed characteristic.

中文翻译:

完整局部环中的加长闭合

(全)扩展加封闭是作为混合特征环中紧密封闭的替代品而开发的。这里通过改编 Andre 的完美代数技术表明,对于具有 F 有限余数场的完整局部环,该闭包具有冒号捕获特性。事实上,更一般地,如果 $R$ 是一个(可能是分支的)混合特征的完全正则局部环,具有 F 有限残差域,则 $I$ 和 $J$ 是 $R$ 的理想,而局部域 $ S$ 是有限的 $R$-模,则 $(IS:J)\subseteq (I:J)S^{epf}$。结果是规则局部环中的所有理想都是封闭的,这一事实意味着直接被加数猜想和混合特征的 Briancon-Skoda 定理的有效性。
更新日期:2021-04-01
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