The extended plus ($\mathsf{epf}$) closure and the rank 1 ($\mathsf{r1f}$) closure are two closure operations introduced by Raymond C. Heitmann for rings of mixed characteristic. Recently, he and Linquan Ma proved that the $\mathsf{epf}$ closure satisfies the usual colon-capturing property under mild conditions. In this paper, we extend their result and prove that the $\mathsf{epf}$ closure satisfies what we call the p-colon-capturing property. Based on that, we define a new closure notion called “weak $\mathsf{epf}$ closure,” and prove that it satisfies the generalized colon-capturing property and some other colon-capturing properties. This gives a new proof of the existence of big Cohen-Macaulay algebras in the mixed characteristic case. We also show that any module-finite extension of a complete local domain is $\mathsf{epf}$-phantom, which generalizes a result of Mel Hochster and Craig Huneke about “phantom extensions.” Finally, we prove some related results in characteristic p.

扩展加号（$\mathsf{epf}$）关闭和等级1（$\mathsf{r1f}$）闭合是Raymond C.Heitmann针对混合特性的环引入的两个闭合操作。最近，他和马林泉证明了$\mathsf{epf}$封闭在温和条件下满足通常的结肠捕获性能。在本文中，我们扩展了他们的结果，并证明了$\mathsf{epf}$闭包满足我们所谓的p-冒号捕获属性。基于此，我们定义了一个新的闭包概念，称为“弱$\mathsf{epf}$”，并证明它满足广义的结肠捕获特性和其他一些结肠捕获特性。这为混合特征情况下大Cohen-Macaulay代数的存在提供了新的证据。我们还表明，完整本地域的任何模块有限扩展都是$\mathsf{epf}$-phantom，它概括了Mel Hochster和Craig Huneke关于“幻像扩展”的结果。最后，我们证明了特征p中的一些相关结果。