induced comodule

Given a commutative unital ring kk and a morphism DCD\to C of kk-coalgebras, one can consider the dualized notion of induced module, using the cotensor product instead of tensor product.

If DD is flat as a kk-module (e.g. kk is a field), NN is a left DD- right CC-bicomodule and MM is a left CC-comodule, then the cotensor product N CMN \Box_C M is a DD-subcomodule of N kMN \otimes_k M. In particular, under the flatness assumption, if π:DC\pi : D \rightarrow C is a surjection of coalgebras then DD is a left DD- right CC-bicomodule via Δ D\Delta_D and (idπ)Δ D(\id \otimes \pi) \circ \Delta_D respectively, hence Ind C D:=D C\mathrm{Ind}^D_C := D \Box^C - is a functor from left CC- to left DD-comodules called the induction functor for left comodules from CC to DD.

One can consider this construction more generally for corings.

Revised on September 13, 2016 10:26:20 by heiderich? (