Relatively Noetherian rings, localization and sheaves. II: Structure sheaves
As an application of the results obtained in the first part of the paper [see the preceding review Zbl 0834.16032], the authors now show how the relative version of the strong second layer condition allows us to endow \(K(\lambda)\), the generically closed subset of \(\text{Spec}(R)\) canonically associated to a radical \(\lambda\) in \(R\)-mod, with structure sheaves, which generalize most previously studied sheaf constructions in noncommutative algebra. First they develop some general machinery to construct structure sheaves through localization in a rather generic way. The sheaf construction itself, both in the symmetric and the classical case, depends upon the choice of a suitable topology on \(K(\lambda)\). In the classical case, they provide some extra background on localization at relative Ore sets, generalizing and strengthening the results obtained in the absolute, Noetherian case. Then the results of the first part of the paper allow us to associate to any \(R\)-bimodule \(M\) both a symmetric and a classical sheaf of \(R\)- modules on \(K(\lambda)\), whose global sections reduce to the localization of \(M\) at \(\lambda\). This generalizes most previously studied symmetric and nonsymmetric sheaf constructions for modules over noncommutative rings.
sheaf constructions in noncommutative algebra
scientific article; zbMATH DE number 590952
scientific article