Pierce sheaves and commutative idempotent generated algebras

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 LSp(R) of locally Specker R-algebras and bundles Y -> 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 IG(R) of commutative idempotent generated R-algebras. We show that LSp(R) is the least epicoreflective subcategory of 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 LSp(R) homologically in IG(R) by examining the algebras in IG(R) that when considered as an R-module are free, projective or flat. If R is an indecomposable ring, then for algebras in 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.