2 00 6 The characteristic class and ramification of an l - adic

We introduce the characteristic class of an l-adic étale 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 Kato-Saito [16] and the other is the 0-cycle class defined by Kato for rank 1 sheaves in [15]. Let k be a perfect field, l be a prime number invertible in k and Λ be a finite extension of either Fl or Ql. Let X be a separated k-scheme of finite type. Let f : X → Spec k denote the structural morphism and put KX = Rf Λ. For an étale sheaf F of Λ-modules on X, its characteristic class is defined as follows. We put H = RHom(pr2F , Rpr1F) and H∗ = RHom(pr1F , Rpr2F) on X×X and let ∆ = X ⊂ X× X denote the diagonal. Then, we have 1 ∈ End(F) = H ∆(X×X,H) = H ∆(X×X,H). The natural pairing H⊗H∗ → KX×X induces a pairing 〈 , 〉 : H ∆(X×X,H)⊗H ∆(X× X,H∗) → H ∆(X ×X,KX×X) = H(X,KX). We define the characteristic class of F , denoted C(F), to be the pairing 〈1, 1〉 ∈ H(X,KX). If X is smooth of dimension d, we have C(F) ∈ H(X,Λ(d)). By the Lefschetz-Verdier trace formula [11] Théorème 4.4, the trace Tr C(F) gives the Euler-Poincaré characteristic χ(Xk̄,F) if X is proper, where k̄ denotes a separable closure of k. By devissage, computations of the characteristic classes are reduced to a computation of C(j!F) where j : U → X is an open immersion, U is smooth and F is a smooth sheaf on U . In this paper, we compute the characteristic class of C(j!F) or rather the difference C(j!F)−rank F ·C(j!Λ) in terms of the ramification of F along the boundary X \U . More precisely, we prove that C(j!F)− rank F ·C(j!Λ) is equal to the following two invariants, under certain assumptions. One is the Swan class Sw(F) of F defined in [16]. The other is the 0-cycle class cF defined by Kato [15] in rank 1 case. We also define a localization of the difference C(j!F) − rank F · C(j!Λ) in H X\U (X,KX) as a cohomology class with support on the boundary.