Strong equivalence of graded algebras (Q6612188)

The aim of this paper is to study the Cohen-Montgomery duality and graded Morita equivalences for more general \(G\)-graded algebras, where \(G\) is a group. The first main result says that the Cohen-Montgomery duality holds for any idempotent algebra \(G\)-graded \(B\), where \(M_n(B)\) replaced by the algebra \(\mathrm{FMat}_G(B)\) of matrices over with finitely many non-zero entries.\N\NThe authors define graded-equivalence of idempotent graded algebras using graded Morita contexts and show that if \(B\) is an idempotent \(G\)-graded algebra, then \(B\) is graded-equivalent to the skew group algebra \((B\#G) \rtimes_{\beta^B} G\), where \(\beta^B\) is the usual global action of \(G\) on the smash product \(B\#G\). Then strong-graded-equivalence is introduced and studied for partially-strongly-graded algebras \(A\) Another main result says that the skew group algebra \(A \rtimes_\alpha G\) by a product partial action \(\alpha\) is graded-equivalent to the skew group algebra \(B\rtimes_\beta G\), where \(\beta\) is a minimal globalization of \(\alpha\).\N\NGiven a partially-strongly-graded algebra \(B\), the dual action \(\beta^B\) of \(G\) on \(B\#G\) can be restricted to a certain ideal \(I^B\) of \(B\# G\), called the partial smash product, resulting in a product partial action \(\gamma^B\), called the canonical partial action associated to \(B\), such that \(B\) and the skew group algebra \(I^B \rtimes_{\gamma^B} G\) are strongly-graded-equivalent. The authors prove that the global skew group algebra \((B\# G) \rtimes_{\beta^B} G\) and the partial one \(I^B \rtimes_{\gamma^B} G\) are graded-equivalent.\N\NAnother main result says that if \(B\) is a partially-strongly-graded algebra, then \(B\) is strongly graded-equivalent to \(I^B \rtimes_{\gamma^B} G\). There are several consequences: If \(B\) is a strongly-graded algebra, then \(B\) is strongly-graded-equivalent to the (global) skew group algebra \((B\#G) \rtimes_{\beta^B} G\); the crossed product by any twisted partial group action is strongly-graded-equivalent to the skew group algebra of a product partial action; moreover, the canonical partial actions \(\gamma^A\) and \(\gamma^B\) associated to strongly-graded-equivalent partially strongly-\(G\)-graded algebras \(A\) and \(B\) are Morita equivalent.\N\NIt turns out that two skew group algebras by product partial actions \(\alpha\) and \(\alpha'\) are strongly-equivalent if and only if \(\alpha\) and \(\alpha'\) are Morita equivalent. Then, the authors prove that any product partial action \(\alpha\) of a group \(G\) has a minimal globalization of a product partial action \(\beta\) which is Morita equivalent to \(\alpha\); moreover, \(\beta\) is unique up to Morita equivalence, and the skew group algebras \(A\rtimes_\alpha G\) and \(B\rtimes_\beta G\) are graded-equivalent.\N\NFinally, given strongly-graded-equivalent partially-strongly-\(G\)-graded algebras \(A\) and \(B\) with orthogonal local units, there exists a graded isomorphism of algebras \(\mathrm{FMat}_\mathcal{X} (A) \simeq \mathrm{FMat}_\mathcal{X} (B)\), where \(\mathcal{X}\) is a sufficiently large cardinal.