Strong equivalence of graded algebras

Abstract

We introduce the notion of a strong equivalence between graded algebras and prove that any partially-strongly-graded algebra by a group G is strongly-graded-equivalent to the skew group algebra by a product partial action of G. As to a more general idempotent graded algebra B, we point out that the Cohen-Montgomery duality holds for B, and B is graded-equivalent to a global skew group algebra. We show that strongly-graded-equivalence preserves strong gradings and is nicely related to Morita equivalence of product partial actions. Furthermore, we prove that any product partial group action α is globalizable up to Morita equivalence; if such a globalization β is minimal, then the skew group algebras by α and β are graded-equivalent; moreover, β is unique up to Morita equivalence. Finally, we show that strongly-graded-equivalent partially-strongly-graded algebras are stably isomorphic as graded algebra