On localizations of the characteristic classes of ℓ-adic sheaves and conductor formula in characteristic p > 0

Abstract

Abbes, Kato and Saito generalize the Grothendieck-Ogg-Shafarevich formula to an arbitrary dimension (Kato and Saito in Ann. Math. 168:33-96, 2008; Abbes and Saito in Invent. Math. 168:567-612, 2007). In this paper, assuming the strong resolution of singularities, we prove a localized version of a formula proved using the characteristic class of an -adic sheaf by Abbes and Saito (Invent Math 168:567-612, 2007). We prove a localized version of the Lefschetz-Verdier trace formula proved in Grothendieck (Formule de Lefschetz, exposé III, SGA 5, Lect. Notes Math., vol 589, pp 372-406, Exp. X, Springer, Berlin, 1977 [Théorème 4.4]). As an application, we prove a conductor formula in an arbitrary dimension in the equal characteristic case.Again by the projection formula, these isomorphisms induce the following isomorphism Rf

Wild Ramification of Schemes and Sheaves

“…For surfaces dim U = 2, we can remove the assumption on the reduced closed fiber X F,red , since the strong resolution of singularity is now obtained by blow-up in dimension 2 [23]. In a geometric case, a formula analogous to (14) is obtained by using a localized refinement of the characteristic class in [43], assuming resolution of singularities.…”

Ramification theory for varieties over a local field

“…In this paper, we will focus on a mixed characteristic case. Another approach in a geometric equal characteristic case is studied in [44].…”

Refined characteristic class and conductor formula

“…This article is devoted to the proof of a conductor formula for ℓ-adic sheaves in a geometric situation (1.3.1) which generalizes the classical Grothendieck-Ogg-Shafarevich formula ( [11] X 7.1) as well as the index formula of Saito ([18] 3.8). It uses the ramification theory developed by Abbes and Saito and it relies on a previous work of Tsushima, who proved a special case ( [23] 5.9). 1.2.…”

“…1.5. The analogous approach for the proof of the conductor formula (1.3.1) was started by Tsushima in [23]. He refined the characteristic class of an ℓ-adic sheaf into a cohomology class with support in the wild locus, called in this article the refined characteristic class.…”

“…He refined the characteristic class of an ℓ-adic sheaf into a cohomology class with support in the wild locus, called in this article the refined characteristic class. He proved a Lefschetz-Verdier formula for this class ( [23] 5.4) which amounts to say that it commutes with proper push-forward. On a smooth curve, the refined characteristic class gives the Swan conductor ( [23] 4.1).…”