On Kähler's integral differential forms of arithmetic function fields
type: Item
instance of: scholarly article
title: On Kähler's integral differential forms of arithmetic function fields
zbMATH Open document ID: 1045.14009
DOI: 10.1007/BF02941285
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2004-05-18T00:00:00Z
review text: Nearly fifty years ago, in his fundamental treatise ``Geometria Aritmetica'' [Ann. Mat. Pura Appl., IV. Ser. 45 (1958; Zbl 0142.18101)], \textit{E. Kähler} laid the foundations for the algebraic differential calculus on arithmetic varieties. Namely, for arithmetic function fields \(K/\mathbb{Q}\), he introduced the ring \(D(K)\) of integral differential forms as a generalization of the ring of integers \(O_K\), and he studied its structure to a large extent. As another generalization of the ring of algebraic integers, Kähler also defined the module \(D({K\over\vartheta})\) of integral differential forms with respect to the different \(\vartheta\) of the arithmetic function field \(K\), which turned out to be a finitely generated projective module over the ring of algebraic integers in \(K\). The paper under review is devoted to some arithmetic-geometric aspects of Kähler's differential calculus. The authors' main theorem states that if an integral scheme \(X\) of finite type over \(\mathbb{Z}\) is a proper and normal geometric model of the field \(K\) then the Kähler module \(D({K\over\vartheta}\)) is a submodule of the cohomology module \(H^0(X,\omega_{X/\mathbb{Z}})\), where \(\omega_{X/\mathbb{Z}}\) denotes the dualizing sheaf on \(X\) over \(\mathbb{Z}\) [after \textit{E. Kunz} and \textit{R. Waldi}, ``Regular differential forms'', Contemp. Math. 79 (1988; Zbl 0658.13019)]. This cohomological interpretation of the Kähler module is then investigated by means of various concrete examples, in particular for Fermat function fields and elliptic function fields, thereby completing some earlier results by \textit{R. Berndt} (Hamburg) from the 1970s.
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14H05
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 2063952
zbMATH Keywords: arithmetic function fields
zbMATH Keywords: Kähler differentials
zbMATH Keywords: dualizing sheaves
zbMATH Keywords: arithmetic schemes
zbMATH Keywords: elliptic curves
zbMATH Keywords: Fermat curves
zbMATH Keywords: Kähler different
zbMATH Keywords: valuations
author: E. Kunz
author: R. Waldi
reviewed by: Werner Kleinert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2063952