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 .
"Swan conductors and torsion in the logarithmic de Rham complex,"
Turkish Journal of Mathematics: Vol. 34:
4, Article 3.
Available at: https://journals.tubitak.gov.tr/math/vol34/iss4/3