## The essential cover and the absolute cover of a schematic space

### Volume 114 / 2009

Colloquium Mathematicum 114 (2009), 53-75
MSC: Primary 54A05, 54G05, 06D05, 06F20.
DOI: 10.4064/cm114-1-6

#### Abstract

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.