On Kähler's integral differential forms of arithmetic function fields
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.
scientific article; zbMATH DE number 2063952
scientific article