zbMATH — the first resource for mathematics

On the arithmetics of two-dimensional schemes. I: Distributions and residues. (Zur Arithmetik zweidimensionaler Schemata. I: Verteilungen und Residuen.) (Russian) Zbl 0358.14012
This is the first part of an interesting series of articles, whose aim is to generalize the notion of repartition of an algebraic curve to higher dimensions, and to use the advantages of this notion to several sort of questions, e.g. cohomology and duality of coherent sheaves, questions related with class field theory, etc. The paper under review deals with a systematic study of the notion of residue of a rational differential form of degree 2 an a geometrically smooth surface \(X\) proper over a perfect field \(k\), at a pair \((x,C)\) consisting of an irreducible curve \(C\subset X\) and a closed point \(x\in X\) which belongs to \(C\). Also one defines and studies in detail the notion of repartition an the surface \(X\), which is, roughly speaking, a function which associates to every such pair \((x,C)\) an element \(f_{x,C}\) of the \((x,C)\)-completion \(K_{x,C}\) of the field of rational functions \(K\) of \(X\), satisfying certain finiteness conditions. For example, if \(x\) is a regular point of the curve \(C\), then \(K_{x,C}\) is nothing but the fraction field of the completion \(\partial_{x,C}\) (with respect to the maximal ideal) of the local ring \(O_{x,C}\). In analogy with the situation of curves [see J.-P. Serre, “Groupes algébriques et corps de classes” (Paris 1959; Zbl 0097.35604); chapter II], one shows that one can interpret the cohomology and the duality of coherent sheaves of \(X\) in this context, getting new proofs of the finiteness of the cohomology vector spaces of a coherent sheaf and the duality theorem of coherent sheaves.

14G99 Arithmetic problems in algebraic geometry; Diophantine geometry
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties