## Pierce sheaves and commutative idempotent generated algebras

### Volume 240 / 2018

#### Abstract

Let $R$ be a commutative ring. Pierce duality between the category of commutative rings and the category of ringed Stone spaces with commutative indecomposable stalks can be adapted to the category of commutative $R$-algebras. Examination of morphisms under this duality leads in a natural way to a class of faithfully flat commutative idempotent generated $R$-algebras that we term locally Specker $R$-algebras. We study locally Specker $R$-algebras in detail. We show that every such $R$-algebra $A$ is uniquely determined by a Boolean algebra homomorphism from the Boolean algebra of idempotents of $R$ into that of $A$; this in turn leads to a dual equivalence between the category ${\sf LSp}_R$ of locally Specker $R$-algebras and bundles $Y \rightarrow X$, where $Y$ is a Stone space and $X$ is the Pierce spectrum of $R$. We also show that the concept of a locally Specker $R$-algebra generalizes functorial constructions of Bergman and Rota.

The algebraic and categorical properties of locally Specker $R$-algebras are useful for illuminating the category ${\sf IG}_R$ of commutative idempotent generated $R$-algebras. We show that ${\sf LSp}_R$ is the least epicoreflective subcategory of ${\sf IG}_R$, and hence every commutative idempotent generated $R$-algebra can be presented as the image of a locally Specker $R$-algebra in a canonical way. We also situate the category ${\sf LSp}_R$ homologically in ${\sf IG}_R$ by examining the algebras in ${\sf IG}_R$ that when considered as an $R$-module are free, projective or flat. If $R$ is an indecomposable ring, then for algebras in ${\sf IG}_R$ all three notions coincide with that of being locally Specker. If $R$ is not indecomposable, then in general the four notions diverge. However, we identify the classes of rings $R$ for which any two coincide.