if $N \in {{}_{G}\mathcal M}$, $N|_H$ is the $H$–module obtained by result of the restriction of the $G$–action to an $H$–action. In this situation if the structure of $\Bbbk[G]$–comodule of $N$ is $\chi(n)=\sum n_0
\otimes n_1 \in N \otimes \Bbbk[G]$, the structure of $N|_H$ as a $\Bbbk[H]$–comodule is $({\ensuremath{\mathrm{id}}}\otimes \pi)\chi(n)=\sum n_0 \otimes
\pi(n_1) \in N \otimes \Bbbk[H]$ where $\pi:\Bbbk[G] \to \Bbbk[H]$ is the restriction morphism.

Concerning some algebraic aspects: all algebras will be commutative –unless explicitly stated– and over a base field $\Bbbk$ that is algebraically closed. An algebra is affine if it is commutative, finitely generated and with no non–zero nilpotents.

Morever, the category ${{}_{G}\mathcal M}$ for an affine algebraic group $G$ is abelian, and has enough injectives. This guarantees that the basic machinery of homological algebra is available in the working platform of this survey. In particular, this category has the particularity that $\Bbbk[G] \in {{}_{G}\mathcal M}$ is an injective object and also that if $M \in {{}_{G}\mathcal M}$ is an arbitrary rational $G$–module, then $M \otimes \Bbbk[G]$ is injective. In this manner one has that the coaction map $\chi:M \to M \otimes \Bbbk[G]$ produces an imbedding of $M$ in an injective object and this guarantees that the 