# Cotensoring by a Hopf Algebra

For $H$ a Hopf algebra, with bijective antipode. For a right, and a left, $H$-comodule $(V,\alpha_R)$, and $(W,\alpha_L)$ respectively, the cotensor product of $V$ and $W$ is $$V \square_H W := \ker(\alpha_R \otimes \text{id} - \text{id} \otimes \alpha_L:V \otimes W \to V \otimes H \otimes W).$$

When does it hold that $$V \square_H H ~~~ \simeq V?$$

One case works right off the bat, group algebras $H = kG$. An $H$-comodule is a $G$-graded vector space, so we may write $M = \oplus_{g\in G} M_g$, where $m \in M_g$ if and only if $\rho(m) = m\otimes g$. In particular $M\Box_H H$ consists of $\oplus_{g\in G} M_g \Box g$.