Ramification of local fields with imperfect residue fields

Abstract

We define two decreasing filtrations by ramification groups on the absolute Galois group of a complete discrete valuation field whose residue field may not be perfect. In the classical case where the residue field is perfect, we recover the classical upper numbering filtration. The definition uses rigid geometry and log-structures. We also establish some of their properties.

The characteristic class and ramification of an ℓ-adic étale sheaf

“…More quantitatively, the Swan conductor measures the necessary blow-up to kill the ramification. This fits nicely with the ramification theory in [1]. The authors expect that this argument should work with arbitrary rank (cf.…”

Relative singular support and the semi-continuity of characteristic cycles for étale sheaves

“…It is an essential step to obtain characteristic cycles of ℓ-adic sheaves. For an ℓ-adic sheaf on a smooth variety, one can associate a divisor called the total dimension divisor using Abbes and Saito's non-logarithmic ramification filtration of Galois group of local fields [AS02,Sa13]. In [HY], the authors proved the lower semi-continuity for total dimension divisors of ℓ-adic sheaves on a smooth fibration.…”

On differential modules associated to de Rham representations in the imperfect residue field case

“…The Galois group of a complete valuation field F is canonically endowed with non-log and log ramification filtrations in the sense of Abbes-Saito ( [AS02]). By using the ramification filtrations, one can define Artin and Swan conductors of Galois representations, which are important arithmetic invariants.…”