Abstract
It is shown that certain aspects of the theory of tight closure are well behaved under localization. Let $J$ be the parameter test ideal for $R$, a complete local Cohen-Macaulay ring of positive prime characteristic. For any multiplicative system $U \subset R$, it is shown that $J{U^{ - 1}}R$ is the parameter test ideal for ${U^{ - 1}}R$. This is proved by proving more general localization results for the here-introduced classes of "${\text {F}}$-ideals" of $R$ and "${\text {F}}$-submodules of the canonical module" of $R$, which are annihilators of $R$ modules with an action of Frobenius. It also follows that the parameter test ideal cannot be contained in any parameter ideal of $R$.
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