Forcing algebra/Characteristic 0/Superheight/Remark

In characteristic zero, the plus closure behaves very differently compared with positive characteristic. If ${}R$ is a normal domain of characteristic ${}0$, then the trace map shows that the plus closure is trivial, ${}I^{+}=I$ for every ideal ${}I$. This implies also that if ${}R$ is a two-dimensional normal local ring of characteristic ${}0$ and ${}I$ an ${}{\mathfrak {m}}$-primary ideal and ${}f\in R$ an element with ${}f\notin I$, then the extended ideal ${}{\mathfrak {m}}B$ inside the forcing algebra ${}B$ has superheight ${}1$. If moreover ${}f$ belongs to the solid closure of ${}I$, then ${}D({\mathfrak {m}}B)$ is not affine and so by fact its ring of global sections is not finitely generated.