Functorial Test Modules

Abstract

In this article we introduce a slight modification of the definition of test modules which is an additive functor $\tau$ 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 $f:X\to Y$ of $F$-finite schemes one has a natural isomorphism $f^{!}\circ \tau \cong \tau \circ f^{!}$. If $f$ is quasi-finite and of finite type we construct a natural transformation $\tau \circ f_{\ast }\to f_{\ast }\circ \tau$.

34 pages, minor corrections