coinvariant

Let $V$ be a comodule with a right coaction $t:V\to V\otimes A$ of a coalgebra $A$.An element $v\in V$ is right coinvariant if

$$t(v)=v\otimes {1\mathrm{I}}_A.$$(1)

The set of coinvariants of $A$ is a sub-comodule with the trivial coaction of $A$.The sub-comodule of right (or left) coinvariants of $V$ is sometimes denoted by $V^{\mathrm{co}A}$ (or ${}^{\mathrm{co}A}V$).