nLab coinvariant

Let $G$ be a discrete group, $k$ a commutative unital ring, $k\left[G\right]$ the group ring of $G$ and $M$ a left $k\left[G\right]$-module. Then there is well-defined $k$-module ${M}_{G}=M/⟨m-gm⟩$ called the module of $G$-coinvariants. Here $⟨m-gm⟩$ denotes the smallest sub-$k$-module of $M$ containing all expressions of the form $m-gm$ where $g\in G$ and $m\in M$.

Let $C$ be a $k$-coalgebra, $\chi$ a group-like element, that is, an element such that ${\Delta }_{C}\left(\chi \right)=\chi \otimes \chi$, and $\rho :V\to V\otimes C$ a right $C$-coaction. Any element $v\in V$ such that $\rho \left(v\right)=v\otimes \chi$ is called a $\left(\rho ,\chi \right)$-coinvariant element in the $C$-comodule $\left(V,\rho \right)$. Suppose $H$ is a bialgebra, $A$ an algebra and $\rho :A\to A\otimes H$ a coaction making $A$ into a right $H$-comodule algebra. The unit element ${1}_{H}$ is a group-like element, and we call $\left(\rho ,1\right)$-coinvariants simply $\rho$-coinvariants. The subset of $\rho$-coinvariants in $A$ is a subalgebra, called the subalgebra of coinvariants.

Revised on November 28, 2012 06:45:46 by Anonymous Coward (186.84.46.246)