In an abstract category with suitable notions of subobject, closure and point, we discuss the separation axioms and . Each of the arising subcategories is reflective. We give an iterative construction of the reflectors and present characteristic examples.
Dans cet article on étudie les -modules dont le support singulier est un croisement normal dans , par l’intermédiaire de la catégorie équivalente de faisceaux pervers. On montre qu’ils sont caractérisés, à isomorphisme près, par la donnée suivante : un hypercube constitué par des espaces vectoriels de dimension finie indexés par les parties de , et des applications linéaires soumises à certaines conditions de commutativité et d’inversibilité. Ce résultat est exprimé sous forme d’une équivalence...
It is known that a ring is left Noetherian if and only if every left -module has an injective (pre)cover. We show that if is a right -coherent ring, then every right -module has an -injective (pre)cover; if is a ring such that every -injective right -module is -pure extending, and if every right -module has an -injective cover, then is right -coherent. As applications of these results, we give some characterizations of -rings, von Neumann regular rings and semisimple rings....
Given an axiomatic account of the category of locales the closed subgroup theorem is proved. The theorem is seen as a consequence of a categorical account of the Hofmann-Mislove theorem. The categorical account has an order dual providing a new result for locale theory: every compact subgroup is necessarily fitted.