Separable Groupoid Rings

Abstract

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.

Key Words:

• Graded module
• Graded ring
• Separable functor
• Separable ring extension

Mathematics Subject Classification:

• Primary 16S34, 16H05
• Secondary 16B50, 16D90

Notes

Communicated by H. Schneider.