to [@kn:gross] or to a more recent exposition appearing in [@kn:nosotros2 Section 13.5,13.6].

The so called “codimension two condition” is used in order to apply the following theorem on the extension of regular functions. “Let $X$ be an irreducible normal variety and $f \in \mathcal O_X(U)$ be a function defined in an open subset U such that $\operatorname{codim}_X(X \setminus U) \geq 2$, then $f$ can be extended to a function defined in X”. See [@kn:gross14 Lemma 1] or [@kn:nosotros2 Theorem 2.6.14] for (similar) proofs of this general result.

It is worth noticing that in case of the special hypothesis on the separability of the action, the ring $H'$ “behaves like a universal object as far as finite generation is concerned” (see [@kn:gross14 page 231]).

[Problem 3. Hilbert’s $14^{\text{th}}$ problem.]{}

The original Hilbert’s $14^{\text{th}}$ problem examines the answers to the following question (see [@kn:hilbertpr]).

[Hilbert’s problem.]{} [*Let $A=\Bbbk[X_1,\cdots,X_n]$ be the polynomial algebra in $n$ variables, let $H$ be a subgroup $H
  \subseteq \operatorname{GL}_n(\Bbbk)$ and consider the action of $H$ on $A$ given by the restriction of the natural action of $\operatorname{GL}_n(\Bbbk)$. Is the subalgebra of $H$–invariants of $A$ finitely generated?*]{}

This problem can be generalized to the following context.

[Generalized Hilbert’s 