The essential cover and the absolute cover of a schematic space
Tom 114 / 2009
Colloquium Mathematicum 114 (2009), 53-75 MSC: Primary 54A05, 54G05, 06D05, 06F20. DOI: 10.4064/cm114-1-6
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.