Conductor, discriminant, and the Noether formula of arithmetic surfaces. (English) Zbl 0657.14017

From the author’s introduction: In the classical theory of ramification, there is a fundamental equality conductor \(= discri\min ant\). The purpose of the present paper is to establish this equality for relative curves.
The conductor is an integer defined for arbitrary schemes of finite type over a discrete valuation ring with perfect residue field by using \(\ell\)-adic étale cohomology. On the other hand, the discriminant was defined only for finite extensions of discrete valuation rings. Recently, Deligne defined a canonical isomorphism between the determinant invertible sheaves of higher direct images of the sheaves of differentials of proper smooth curves, and called it the discriminant. An integer discriminant is defined using this isomorphism. Roughly speaking, the discriminant is the intersection number with the infinite divisor of the moduli space of stable curves. Using this definition of discriminant, the aforementioned equality is proved.
Reviewer: A.Kustin


14H25 Arithmetic ground fields for curves
14A05 Relevant commutative algebra
14H10 Families, moduli of curves (algebraic)
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