Hopf module

Summary

A Hopf module over a Hopf algebra $H$ is a vector space that is both an $H$-module and an $H$-comodule. The coaction must be compatible with the action. The fundamental theorem of Hopf modules, due to Larson and Sweedler, says that every Hopf module is free. It has the form $V \otimes H$, where $V$ is its space of coinvariants. This theorem is the main tool behind the uniqueness of integrals and the Frobenius property of finite-dimensional Hopf algebras.