Abstract
Abstract We develop a formalism of unit $F$-modules in the style of Lyubeznik and Emerton-Kisin for rings that have finite $F$-representation type after localization and completion at every prime ideal. As applications, we show that if $R$ is such a ring then the iterated local cohomology modules $H^{n_{1}}_{I_{1}} \circ \cdots \circ H^{n_{s}}_{I_{s}}(R)$ have finitely many associated primes, and that all local cohomology modules $H^{n}_{I}(R / gR)$ have closed support when $g$ is a nonzerodivisor on $R$.
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