GRADED MORITA THEORY FOR GROUP CORING AND GRADED MORITA-TAKEUCHI THEORY

A Graded Morita context is constructed for any comodule of a group coring. For any right G-C-comodule M with dual graded ring R, we define a graded ring T = HOMG,C(M,M) = circle plus g is an element of G HOMG,C(M, M)(g), and a G-graded R-T bimodule Q = circle plus(g is an element of G) Q(9) where Qg is a family of right A-linear maps q(alpha)(g); M alpha -> R-g alpha in M-A. We construct a graded Morita context M = (T, R, circle plus(alpha is an element of G) M alpha, Q, tau, mu) with connecting homomorphisms tau : T(circle plus)(alpha is an element of G) M-alpha) circle times(R) Q(T) -> T, m circle times q -> mq(-), mu : (R)Q circle times(T) (circle plus(alpha is an element of G) M alpha)(R) -> R, q circle times m -> q(m), which generalized the Morita context in [3, 5-7, 10, 13]. Meanwhile, we prove the graded Morita-Takeuchi theory as a generalization of Morita-Takeuchi theory which characterize the equivalence of comodule over field.