induced comodule

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

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

One can consider this construction more generally for corings.

Last revised on September 13, 2016 at 10:26:20. See the history of this page for a list of all contributions to it.