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

Abstract

We introduce the characteristic class of an ℓ-adicétale sheaf using a cohomological pairing due to Verdier (SGA5). As a consequence of the Lefschetz-Verdier trace formula, its trace computes the Euler-Poincaré characteristic of the sheaf. We compare the characteristic class to two other invariants arising from ramification theory. One is the Swan class of and the other is the 0-cycle class defined by Kato for rank 1 sheaves in [15].

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

“…We call this equality the localized Abbes-Saito formula. The Abbes-Saito formula mentioned above follows from the compatibility of the characteristic class with the pull-back in [1,Corollary 2.1.11] and an explicit computation of the characteristic class in the tamely ramified case in [1,Corollary 2.2.5]. The localized Abbes-Saito formula is proved in a similar manner to the proof of them.…”

“…See [14,16,18]. This formula is generalized to an arbitrary dimension by Abbes, Kato and Saito in [1] and [11]. To generalize this formula, Kato and Saito define the Swan class of an -adic sheaf on a variety of an arbitrary dimension by using alteration and logarithmic blow-up in [11,Sect.…”

“…Let F denote an étale sheaf of -modules on X . Abbes and Saito define the characteristic class C(F) ∈ H 0 (X, K X ) by using the Verdier pairing introduced in [6] and study its properties in [1]. By wherek denotes a separable closure of k. Let j : U → X be an open immersion with dense image.…”

“…Let cl : C H 0 (X \U ) Q −→ H 0 (X, K X ) be the cycle class map. In [1,Theorem 3.3.1.2], they prove that the characteristic class C( j ! F ) − rk(F) · C( j ! )…”

“…F ) ∈ H 0 X \U (X, K X ) with support on the boundary and show that it is a refinement of the difference C( j ! F ) − rk(F) · C( ) in [1,Sect. 5].…”