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Functorial Test Modules

In this article we introduce a slight modification of the definition of test modules which is an additive functor $τ$ on the category of coherent Cartier modules. We show that in many situations this modification agrees with the usual definition of test modules. Furthermore, we show that for a smooth morphism $f \colon X \to Y$ of $F$-finite schemes one has a natural isomorphism $f^! \circ τ\cong τ\circ f^!$. If $f$ is quasi-finite and of finite type we construct a natural transformation $τ\circ f_* \to f_* \circ τ$.