f-cohomology and motives over number rings

TL;DR

This paper interprets f-cohomology of motives over number fields in terms of motives over number rings, showing the equivalence of different definitions under standard assumptions, and introduces a motivic interpretation via intermediate extensions.

Contribution

It provides a motivic interpretation of f-cohomology over number rings, unifying ramification-based and K-theory-based definitions under standard assumptions.

Findings

Both definitions of f-cohomology agree with motivic cohomology of $ ext{eta}_{!*} M_ ext{eta}[1]$

The paper constructs $ ext{eta}_{!*}$ via a limiting process similar to perverse sheaves

Under standard assumptions, the two approaches to f-cohomology are shown to coincide.

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 $M_\eta$ over F--one via ramification conditions on $\ell$-adic realizations, another one via the K-theory of proper regular models--both agree with motivic cohomology of $\eta_{!*} M_\eta[1]$. Here $\eta_{!*}$ is constructed by a limiting process in terms of intermediate extension functors $j_{!*}$ defined in analogy to perverse sheaves.