Polynomring in einer Variablen/Integraler Abschluss und Submersion/Beispiel/en

Let ${}K$ be a field and consider  ${}R=K[X]$.  Since this is a principal ideal domain, the only interesting forcing algebras (if we are only interested in the local behavior around ${}(X)$) are of the form ${}K[X,T]/\left(X^{n}T-X^{m}\right)$. For  ${}m\geq n$  this ${}K[X]$-algebra admits a section (corresponding to the fact that  ${}X^{m}\in \left(X^{n}\right)$ ), and if  ${}n\geq 1$  there exists an affine line over the maximal ideal ${}(X)$. So now assume  ${}m<n$.  If  ${}m\geq 0$  then we have a hyperbola mapping to an affine line, with the fiber over ${}(X)$ being empty, corresponding to the fact that ${}1$ does not belong to the radical of ${}\left(X^{n}\right)$ for  ${}n\geq 1$.  So assume finally  ${}1\leq m<n$.  Then ${}X^{m}$ belongs to the radical of ${}\left(X^{n}\right)$, but not to its integral closure (which is the identical closure on a one-dimensional regular ring). We can write the forcing equation as  ${}X^{n}T-X^{m}=X^{m}\left(X^{n-m}T-1\right)$.  So the spectrum of the forcing algebra consists of a (thickend) line over ${}(X)$ and of a hyperbola. The forcing morphism is surjective, but it is not a submersion. For example, the preimage of  ${}V(X)=\{(X)\}$  is a connected component hence open, but this single point is not open.