Actions of central Morita monoidal groupoids, and seven term exact sequences of all types

Abstract

We introduce a strict monoidal groupoid associated with a given commutative ring $K$. It acts on the Clifford extension systems of $K$-central algebras which are graded by a group $G$, as well as on the generalized collective characters, both of type $ϕ\in$ Hom$(G,$ Aut$(K))$. These actions partition the two sets above with respect to graded Morita equivalence. By that we establish a seven term exact sequence of type $ϕ$, which boils down to the celebrated Chase-Harrison-Rosenberg sequence when $ϕ$ is Galois.