{}^{H''}A = {}^HA$. This is a crucial observation that reduces some problems in invariant theory to the case of observable subgroups.

Let $H \subseteq G$ be a closed inclusion we say that “the pair $(H,G)$ satisfies the codimension two condition” if there exists a finite dimensional rational $G$–module $V$ and an element $v \in V$ such that: (i) $H=\{x \in V: x\cdot v =v\}$ and $G/H \cong G\cdot v$; (ii) for each irreducible component $C$ of $\overline {(G \cdot v)} \setminus (G \cdot v)$, we have that $\operatorname{codim}_{\overline {G \cdot v}} C \geq 2$.

In that context, the following theorem is proved in [@kn:gross14]:

[@kn:gross Theorem 4.3]\[theo:descent\] For the situation above if $H$ is observable in $G$, the conditions:

1.  The $\Bbbk$–algebra $H' \subseteq \Bbbk[G]$ is finitely generated;

2.  The pair $(H,G)$ satisfies the codimension two condition;

3.  For any finitely generated rational $G$–module; algebra A, $A^H$ is a finitely generated $\Bbbk$–algebra,

are related as follows. Conditions (1) and (2) are equivalent and condition (3) implies both of them. In the case that the action of $G$ on $V$ is separable and $G$ is reductive, the three conditions are equivalent.

For a proof of this theorem we refer the reader 