Anodality

Abstract

Anodality (also termed u-closedness) is a notion coming from K-theory and is usefull when studying Pic A[X, X-1]. An injective ring morphism A → B is called anodal if an element x in B belongs to A whenever Then a ring A with total quotient ring K is said to be anodal if A → K is anodal. We show that a weak Baer ring A is anodal if and only if for each pair (a, b) ∈ A2 such that there is some z ∈ A such that . Anodality of weak Baer rings is descended by faithfully flat or anodal morphisms. We define u-integral morphisms similar to R. Swan's subintegral morphisms and show that for any injective morphism A → B there is a u-closure such that is u-integral. A u-integral morphism is locally an epimorphism and is a direct limit of unramified morphisms. This result has many consequences. An injective integral morphism which is injective on the spectrum is locally anodal whence anodal. If A is a weak Baer ring, then A is locally anodal if A is locally unibranched; the converse is true if, in addition, A is one-dimensional and Noetherian. We introduce new closures for injective ring morphisms A → B in order to evaluate . In some measure, seminormality and anodality are dual notions : among other results, we get that Pic f is injective for a u-integral ring morphism f, while Pic f is surjective if f is subintegral.