Hopf module

Remarks

Clause 3 says that $\rho$ is a module map when $M \otimes H$ carries the action $(m \otimes x) \cdot h = \sum m \cdot h_{(1)} \otimes x h_{(2)}$. Example: for a vector space $V$, the space $M = V \otimes H$ with $(v \otimes x) \cdot h = v \otimes xh$ and $\rho(v \otimes x) = \sum v \otimes x_{(1)} \otimes x_{(2)}$. Clause 3 holds because $\Delta(xh) = \Delta(x)\Delta(h)$. Non-example: $H = KG$ for a group $G \ne 1$, acting on itself by multiplication, with the coaction $\rho(x) = x \otimes 1$. This is a module and a comodule, but for $g \ne 1$ in $G$, $\rho(1 \cdot g) = g \otimes 1$ while the right side of clause 3 is $g \otimes g$. The fundamental theorem of Hopf modules states that the map $M^{\mathrm{co}H} \otimes H \to M$, $m \otimes h \mapsto m \cdot h$, is an isomorphism of Hopf modules.