 $R:=K[[u,v]]$ be the power series ring in 2 variables. Note that $R$ is a 2-dimensional regular local ring, but it is not essentially of finite type over $k$. Define a derivation of $K$ to $R$ by $$D_K:= \sum_i v(x_{i+1}u+y_{i+1}v)\frac{\partial}{\partial x_i} +
u(x_{i+1}u+y_{i+1}v)\frac{\partial}{\partial y_i}.$$ This extends to a derivation of $R$ to $R$ by setting $D_R|_K=D_K$ and $D_R(u)=D_R(v)=0$. Note that $D_R\circ D_R=0$, thus $\sigma: r\mapsto r+D_R(r)$ is an order 2 automorphism of $R$. We claim that the ring of invariants $R^{\sigma}$ is not Noetherian.

To see this, note first that $x_{i}u+y_{i}v\in R^{\sigma}$ for every $i$.

[*Claim.*]{} For every $n$, $x_{n+1}u+y_{n+1}v\not\in \bigl(x_{1}u+y_{1}v, \dots, x_{n}u+y_{n}v\bigr)
R^{\sigma}$.

Proof. Assume the contrary and write $$x_{n+1}u+y_{n+1}v=\sum_{i\leq n} r_i\bigl(x_{i}u+y_{i}v\bigr)
{\quad\mbox{where $r_i\in R^{\sigma}$.}\quad}$$ Working modulo $(u,v)^2$ and gathering the terms involving $u$, we get an equality $$x_{n+1}\equiv \sum_{i\leq n}  r_ix_{i} \mod R^{\sigma}\cap (u,v)R.$$ Applying $D_R$ and again gathering the terms involving $u$ we obtain that $x_{n+2}
\equiv\sum_{i\leq n}  r_ix_{i+1}$ modulo $R^{\sigma}\cap (u,v)R$. Repeating this $s$ times gives $$x_{n+s+1}=\sum_{i\leq n}  \bar r_ix_{i+s}{\quad\mbox{where $\bar r_i\in K$.}\quad}$$ Since the $\bar r_i$ involve only finitely many variables, we get a contradiction for large $s$. Thus $$(p_1)\subset (p_1,p_2)\subset (p_1,p_2,p_3)\subset \cdots$$ is an infinite increasing sequence of ideals in $R^{\sigma}$.

The next examples show that, if $S$ is a smooth projective surface, then a geometric quotient $S/R$ can be nonprojective (but proper) and if $X$ is a smooth proper 3-fold, $X/R$ can be an algebraic space which is not a scheme.

\[nonqpglue.exmp\] 1. Let $C,D$ be smooth projective curves and $S$ the blow up of $C\times D$ at a point $(c,d)$. Let $C_1\subset S$ be the birational transform of $C\times \{d\}$, $C_2:=C\times \{d'\}$ for some $d'\neq d$ and ${{\mathbb P}}^1\cong E\subset 