Tight closure/Generische Schranke/Parametersituation/Beispiel/en

Suppose that  $n=d+1$  in the situation of Fakt. Then the generic elements $f_1,\dots ,f_{d+1}$ are parameters. In the polynomial ring  $P=K[x_0,x_1,\dots ,x_d]$  we have for parameters of degree $a_1,\dots ,a_{d+1}$ the inclusion

$P_{\ge \sum _{i=0}^d{a}_i-d}\subseteq (f_1,\dots ,f_{d+1}),$

because the graded Koszul resolution ends in $R(-\sum _{i=0}^d{a}_i)$ and

$(H_{𝔪}^{d+1}(P){)}_k=0\text{ for }k\ge -d.$

So the theorem implies for a graded ring $R$ finite over $P$ that  $\subseteq (f_1,\dots ,f_{d+1}{)}^{\ast }$  holds for generic elements. But by the graded Briançon-Skoda Theorem (see Fakt) this holds for parameters even without the generic assumption.