Ring epimorphisms and C(X)

Michael Barr ; W.D. Burgess ; R. Raphael

Theory and applications of categories, Tome 11 (2003), pp. 283-308.

Résumé

This paper studies the homomorphism of rings of continuous functions $\rho : C(X)\to C(Y)$, $Y$ a subspace of a Tychonoff space $X$, induced by restriction. We ask when $\rho$ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When $X$ is first countable and perfectly normal (e.g., a metric space), $\rho$ is a CR -epimorphism if and only if it is a R/N-epimorphism if and only if $Y$ is locally closed in $X$. It is also shown that the restriction of $\rho$ to $C^*(X)\to C^*(Y)$, when $X$ is normal, is a CR-epimorphism if and only if it is a surjection. In general spaces the picture is more complicated, as is shown by various examples. Information about $Spec \rho$ and $Spec \rho$ restricted to the proconstructible set of prime z-ideals is given.

Détail

Classification : 18A20, 54C45, 54B30
Keywords: epimorphism, ring of continuous functions, category of rings