标题： Profinite 直和及其在 $Φ_R$ 型 Profinite 群中的应用 显示英文标题

标题： Profinite Direct Sums with Applications to Profinite Groups of Type $Φ_R$

摘要： 我们证明了“拟有限直和”是拟有限模的良好无穷直和概念，具有与抽象模的直和类似的性质。 例如，拟有限直和下的投射模仍然是投射模，并且存在描述拟有限模的Mackey公式，这些公式使用了这些直和。 作为应用，我们证明了类型为 $\Phi_R$ 的拟有限群类在子群下是封闭的。 显示英文摘要

摘要： We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective, and there is a Mackey's Formula for profinite modules described using these sums. As an application, we prove that the class of profinite groups of type $\Phi_R$ is closed under subgroups.