The essential cover and the absolute cover of a schematic space

Abstract

Dedicated to B. V. M.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.2000 Mathematics Subject Classification: Primary 54A05, 54G05, 06D05, 06F20.

Real spectra and ℓ-spectra of algebras and vector lattices over countable fields

“…A map, between generalized spectral spaces, is spectral if the inverse image of any compact open subset is compact open. Borrowing notation from Rump and Yang [27], let us denote by DL 0 the category of all distributive 0-lattices with 0-lattice homomorphisms with cofinal 1 range, and by GSp the category of all generalized spectral spaces with spectral maps. Stone duality, originating in Stone [32], states an equivalence of categories between DL 0 and the opposite category of GSp.…”

The C-topology on lattice-ordered groups

“…The theory of lattice-ordered real vector spaces, i.e., Riesz spaces, has had quite some impacts on functional analysis. [2,3] characterized many interesting topological structures of lattice-ordered groups. Gusić [4] showed that a lattice-ordered group can be equipped with C-topology with respect to an admissible subset C of positive elements, and presented many interesting ideas and results on the topological group properties of lattice-ordered groups.…”

The Absolute of a Topological Space and its Application to Abelian l-groups

“…So the quasicomponents of the absolute of X parametrize the refined irreducible components of X, i.e. its irreducible components together with certain embedded ones, both with possible multiplicities [34]. Our lifting theorem then shows that the passage from an abelian l-group to its strongly projectable hull can be regarded as a process of desingularization.…”

“…This yields a characterization of an extremally disconnected space in terms of its space of quasicomponents, which is an extremally disconnected Tychonoff space (Corollary 5 of Theorem 3). If X is the underlying space of a scheme, the fibers of p X are generic, which implies that p X is a retraction [34].…”

“…Furthermore, the irreducible components of an extremally disconnected space coincide with its quasicomponents (see Section 3). The quasicomponents of the absolute of a space X are mapped bijectively onto closed irreducible subspaces of X which can be regarded as refined irreducible components [34]. So the quasicomponents of the absolute of X parametrize the refined irreducible components of X, i.e.…”