Swan conductors and torsion in the logarithmic de rham complex

Abstract

We prove, for an arithmetic scheme X/S over a discrete valuation ring whose special fiber is a strict normal crossings divisor in X, that the Swan conductor of X/S is equal to the Euler characteristic of the torsion in the logarithmic de Rham complex of X/S This is a precise logarithmic analog of a theorem by Bloch [1].