TAILIEUCHUNG - Swan conductors and torsion in the logarithmic de Rham complex

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. | Turk J Math 34 (2010) , 451 – 464. ¨ ITAK ˙ c TUB doi: Swan conductors and torsion in the logarithmic de Rham complex ¨ Sinan Unver 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]. 1. Introduction Let A be a discrete valuation ring with maximal ideal m, perfect residue field k, and field of fractions K, S = SpecA, with closed point s, and X/S an arithmetic surface over S , . an integral, regular scheme which is proper, flat and of relative dimension one over S . We also assume that the reduced special fiber Xs,red is a strict normal crossings divisor in X, by this we mean that Xs,red is a normal crossings divisor in X, and that the irreducible components of Xs,red are regular schemes. There are two numerical invariants for X/S. One of the invariants is based on etale cohomology: the Swan conductor of the Galois representation on the cohomology of the generic fiber; and the other one on de Rham cohomology: the Euler characteristic of the torsion in the logarithmic de Rham complex. Both invariants measure the bad reduction of the special fiber of the arithmetic surface. Below we will prove that these two numerical invariants are in fact the same. We use [5] as a general reference on logarithmic geometry. Let us endow S with the log structure corresponding to the natural inclusion OS \{0} → OS . Similarly, endow X with the log structure corresponding to the natural map ∗ OX ∩ j∗ OX → OX , K where j : XK → X is the inclusion. Then the structure map from X to S becomes a map of fine log schemes. Let Ω˙X/S,log denote its logarithmic de Rham complex, OX → Ω1X/S,log → Ω2X/S,log . 1991 AMS Mathematics Subject Classification: 14C17, 14C40. 451 ¨ UNVER Taking A-torsion in .

• Toán học
• Swan conductors
• Logarithmic de Rham complex
• Precise logarithmic
• Euler characteristic
• Discrete valuation ring