# Coinvariant Subalgebras of Hopf Comodules and Quotients

For $H$ a Hopf algebra, let $V$ be a right $H$-comodule with coaction $\Delta_R$. Moreover, let $W$ be a subspace of $V$ such that $\Delta_R(W) \subseteq W \otimes H$, and note that this implies that $\Delta_R$ restricts to a coaction $V/W \to V/W \otimes H$. If we denote, $$V^H := \lbrace v \in V ~ | ~ \Delta_R(v) = v \otimes 1 \rbrace,$$ and analagously $$(V/W)^H := \lbrace [v] \in V/W) ~ | ~ \Delta_R([v]) = [v] \otimes 1 \rbrace,$$ where $[v]$ denotes the coset of $v$, and $\pi:V \to V/W$ be the canonical projection, then when do we have $$\pi(V^H) = (V/W)^H?$$

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Notice that $F(V)=V^H$ for all $V\in\Com$.

If $$\tag{\star}0\to W\to V\to U\to0$$ is a short exact sequence in $\Com$, then we have a long exact sequence for the derived functors $R^pF$, which starts with $$0\to W^H\to V^H\to U^H\to\Ext^1_{\Com}(k,W)\to\cdots$$

We can conclude, then, as usual, that the map $V^H\to U^H$ is surjective if, for example $\Ext^1_{\Com}(k,W)=0$. This can happen for various reasons: one obvious one is that $W$ be an injective comodule. A draconiant version of this is the condition that $H$ be cosemisimple.

To say something more intelligent, one would probably need to know more details about your concrete situation, though.


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Thanks for your answer. I guess it's what I was looking for, since my real question was: Is it obvious that this is always true? You've certainly shown that it isn't. –  Ago Szekeres Apr 24 '12 at 21:29