A formalism of $F$-modules for rings with complete local finite $F$-representation type

We develop a formalism of unit $F$-modules in the style of Lyubeznik and Emerton-Kisin for rings which 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 $H_{I_1}^{n_1}\circ \cdots \circ H_{I_s}^{n_s}(R)$ have finitely many associated primes, and that all local cohomology modules $H_I^n(R\mathrm{/}gR)$ have closed support when $g$ is a nonzerodivisor on $R$.