f-cohomology and motives over number rings

Abstract

This paper is concerned with an interpretation of f-cohomology, a modification of motivic cohomology of motives over number fields, in terms of motives over number rings. Under standard assumptions on mixed motives over finite fields, number fields and number rings, we show that the two extant definitions of f-cohomology of mixed motives M_η over a number field F—one via ramification conditions on ℓ-adic realizations, another one via the K-theory of proper regular models—both agree with motivic cohomology of η_!* M_η[1]. Here η_!* is constructed by a limiting process in terms of intermediate extension functors j_!* defined in analogy to perverse sheaves.