Functorial test modules

Abstract

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:X→Y of F-finite schemes one has a natural isomorphism f!∘τ≅τ∘f!. If f is quasi-finite and of finite type we construct a natural transformation τ∘f⁎→f⁎∘τ.