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.

- K.-H. Ulbrich,
*On modules indued or coinduced from Hopf subalgebras*, Mathematica Scandinavica**67**:2 (1990) 177-182 - S. Caenepeel, S. Raianu, F. van Oystaeyen,
*Induction and coinduction for Hopf algebras: applications*, J. Algebra**165**:1 (1994) 204-222, doi - T. Brzeziński,
*The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois properties*, Algebras Represent. Theory**5**(2002) 389–410 - Johan Kustermans,
*Induced corepresentations of locally compact quantum groups*, J. Funct. Analysis**194**:2 (2002) 410-459 doi - Stefaan Vaes,
*A new approach to induction and imprimitivity results*, J. Funct. Analysis**229**:2 (2005) 317-374 doi

Last revised on November 22, 2019 at 10:21:42. See the history of this page for a list of all contributions to it.