nLab
coinvariant

Let $G$ be a discrete group, $k$ a commutative unital ring, $k[G]$ the group ring of $G$ and $M$ a left $k[G]$-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 ${\Delta }_C(\chi )=\chi \otimes \chi$, and $\rho :V\to V\otimes C$ a right $C$-coaction. Any element $v\in V$ such that $\rho (v)=v\otimes \chi$ is called a $(\rho ,\chi )$-coinvariant element in the $C$-comodule $(V,\rho )$. 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 $(\rho ,1)$-coinvariants simply $\rho$-coinvariants. The subset of $\rho$-coinvariants in $A$ is a subalgebra, called the subalgebra of coinvariants.