Categorical methods in Hopf algebras
DOI:
https://doi.org/10.5556/j.tkjm.32.2001.343Abstract
In this paper, we first study the adjoint relationship of general functors between two Grothendieck categories. Then apply those results to obtain information about the connections of the categories $ A \# H $-mod and $ A^H $-mod, resp. $ {\cal M}^{C \rtimes H} $ and $ {\cal M}^{C^{coH}} $, where $ A $ is an $ H $-module algebra and $ C $ is an $ H $-comodule coalgebra. In particular, the relationships between the (co)generators, injectivity and projectivity of $ A \# H $-modules and the corresponding property of $ A^H $-modules are given, the relationships between the injective dimension $ A_A $ the corresponding dimension of $ A_{A^H}^H $ and $ A \# H_{A \# H} $ are also established.