### Ramification of local fields with imperfect residue fields. II.

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For a variety over a local field, Bloch proposed a conjectural formula for the alternating sum of Artin conductor of $\ell $-adic cohomology. We prove that the formula is valid modulo 2 if the variety has even dimension.

In [6], S. Bloch conjectures a formula for the Artin conductor of the ℓ-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.

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