Self-intersection 0-cycles and coherent sheaves on arithmetic schemes. (English) Zbl 0687.14004

Let S be the spectrum of a discrete valuation ring with perfect residue field and X be a regular flat proper S-scheme purely of relative dimension 1 with smooth general fiber. Bloch has defined a number \((\Delta_ X,\Delta_ X)_ S\) which is an equivalent of the usual intersection number for the equal characteristic case. The coherent sheaves \(\Omega^ p_{X/S,tors}\) are concentrated in the closed fiber of X. This makes a definition possible for the Euler characteristic of the sheaf. The author proves that \[ \deg (\Delta_ X,\Delta_ X)_ S=\sum_{p}(-1)^ p\chi (X,\Omega^ p_{X/S,tors}). \] This gives an equivalence of two expressions for the conductor of X over S conjectured by Bloch and Kato.
Reviewer: A.N.Parshin


14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
14F45 Topological properties in algebraic geometry
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