# 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 $$V \square_H H ~~~ \simeq V?$$

• Transferred from Stack Exchange due to no answer . Jan 6, 2017 at 11:28
• Always, and you just need $H$ to be a coalgebra with a counit $\varepsilon$: use the maps $\mathrm{id}\otimes\varepsilon:V\otimes H\to V$ and $\mathrm{id}\otimes\varepsilon:V\otimes H\otimes H\to V\otimes H$ to split the equalizer. Jan 6, 2017 at 13:55
• Can you explain "split the equalizer" please? Jan 6, 2017 at 14:28
• It is a standard abstract nonsense trick, see e. g. in nLab Jan 6, 2017 at 14:33

For any $$H$$-comodule $$V$$ you have its structure map $$\rho:V\to V\otimes H$$. The image of $$\rho$$ is the equalizer defining cotensor product.