Unramified elements in cycle modules

Let X be an algebraic variety over a field F . We study the functor taking a cycle module M over F to the group of unramified elements M ( F ( X )) nr of M ( F ( X )). We prove that this functor is represented by a cycle module. The existence of pull‐back maps on M ( F ( X )) nr for rational maps (under a mild condition) is established. An application to the R ‐equivalence on classifying varieties of algebraic groups is given.