On the conductor formula of Bloch

Abstract

In [6], S. Bloch conjectures a formula for the Artin conductor of the -adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The formula, which we call the conductor formula of Bloch, enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. In this paper, we prove the formula in arbitrary dimension under the assumption that the reduced closed fiber has normal crossings.

The logarithmic cotangent complex

“…Secondly, we are hopeful that the development of a theory of log cotangent complexes can be useful in the study of intersection theory on degenerations of varieties. In fact, the beginnings of a theory of log cotangent complexes can be found in the recent work of K. Kato and T. Saito ( [12]) on logarithmic localized intersection product, where it is used to study a proper scheme over a complete discrete valuation ring whose reduced closed fiber is a divisor with normal crossings.…”

Wild Ramification of Schemes and Sheaves

“…For an explicit local description, see Example 2.2 in Section 2.2. For more intrinsic definition in the language of log geometry, we refer to [30,Section 4]. We introduce the log product in order to focus on the wild ramification.…”

“…For the generality on log schemes, we refer to [26] and [30,Section 4]. In the classical case where the residue field is perfect, the two filtrations are the same up to the shift by 1.…”

Miscellany on traces in ℓ-adic cohomology: a survey

“…As X is regular, this group is naturally identified with the Grothendieck group G(Y ) of coherent sheaves of Y , with Kato-Saito's notation in [KS1]. As X /S is proper, we have a degree homomorphism (Euler-Poincaré characteristic)…”