Meaning of coinvariants of a comodule

Question

Let $M$ be a comodule over the bialgebra $B$, with structure map $\rho:M \to M \otimes B$. The space of coinvariants is defined as $M^{coB}:=\{m \in M~|~\rho(m)=m\otimes 1_B\}$. A book I'm reading calls this "the space of generators" of $M$. I wonder what's the motivation of this naming. How does $M^{coB}$ "generate" $M$?

I know that if $M$ is a free comodule genetared by $X$, that is, $M=X\otimes{B}$, then $M^{coB}=X$, so $M$ is indeed generated by $M^{coB}$. Do we have a similar result for general comodules?

Answer 1

If $H$ is a $\Bbbk$-Hopf algebra ($\Bbbk$ a field), then for every $H$-Hopf module $M$ you have an isomorphism of Hopf modules $M\cong M^{coH}\otimes H$.

In fact, this is an equivalence and it is known as the structure theorem for Hopf modules: if we denote by $\mathfrak{M}$ the category of $\Bbbk$-vector spaces and by $\mathfrak{M}_B^B$ the category of Hopf modules over a bialgebra $B$ then we have an adjunction with left and right adjoint $-\otimes B:\mathfrak{M}\to\mathfrak{M}_B^B$ and $(-)^{coB}:\mathfrak{M}_B^B\to\mathfrak{M}$ respectively. This adjunction is an equivalence if and only if $B$ is a Hopf algebra.

This may justify why your reference calls this "the space of generators". However, it may be useful if you explicitly mention which is the book you are reading.