Pierce's representation and separable algebras
Andy R. Magid · Illinois Journal of Mathematics · 1971
Pierce [4] gives a representation of an arbitrary commutative ring R as the ring of global sections of a sheaf of connected rings over a compact, totally disconnected, Hausdorff space.Here we apply this representation to the study of central separable R-algebras.Pierce's sheaf has the interesting property that modules and algebras over the stalks may be extended to mod- ules and algebras over R. We carry out these constructions in Section 1 be- low and use the results to compute the Brauer group of R in terms of the Brauer groups of the stalks.In Section 2 we establish that properties ana- logous to the Skolem-Noether Theorem, the existence of Galois splitting rings, and the generation of separable algebras by units holds for R if they are true at each stalk.Our results apply in particular to commutative Von Neumann (regular) rings, which are characterized [4, p. 41, 10.3] by the property that each stalk of the associated sheaf is a field.We will assume all rings and algebras have identities and all modules are unitary.R always denotes the fixed commutative base ring and unsub- scripted tensor means over R.