The essential cover and the absolute cover of a schematic space

Wolfgang Rump¹ ; Yi Chuan Yang²

¹ Institute for Algebra and Number Theory University of Stuttgart Pfaffenwaldring 57 D-70550 Stuttgart, Germany
² Department of Mathematics & LMIB Beihang University Beijing 100083, P.R. China

Colloquium Mathematicum, Tome 114 (2009) no. 1, pp. 53-75.

Résumé

A theorem of Gleason states that every compact space admits a projective cover. More generally, in the category of topological spaces with continuous maps, covers exist with respect to the full subcategory of extremally disconnected spaces. Such a cover of a space is called its absolute. We prove that the absolute exists within the category of schematic spaces, i.e. the spaces underlying a scheme. For a schematic space, we use the absolute to generalize Bourbaki's concept of irreducible component, so that embedded and multiple components may arise. We introduce the essential cover of a schematic space, and show that it parametrizes the generalized components.

Détail

DOI : 10.4064/cm114-1-6

Keywords: theorem gleason states every compact space admits projective cover generally category topological spaces continuous maps covers exist respect full subcategory extremally disconnected spaces cover space called its absolute prove absolute exists within category schematic spaces spaces underlying scheme schematic space absolute generalize bourbakis concept irreducible component embedded multiple components may arise introduce essential cover schematic space parametrizes generalized components