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：

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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。