On numerical invariants of Noetherian local rings of characteristic \(p\)
The paper follows the concept of tight closure of an ideal \(I\) in a Noetherian ring \(R\) of characteristic \(p\), introduced by \textit{M. Hochster} and \textit{C. Huneke}. They proved that regular rings \(R\) are weakly \(F\)-regular, i.e. every ideal in \(R\) is tightly closed. Moreover they showed that weakly \(F\)-regularity implies normality. The author dives deeper into this matter by considering -- via length- functions \(\ell_R (I^*/I)\) and \(\ell_R (\overline I/I)\), where \(I^*\) denotes the tight closure and \(\overline I\) the integral closure of \(I\) -- several invariants for a local ring \((R, {\mathfrak m})\) of characteristic \(p\) related to the \({\mathfrak m}\)-primary ideals (and in particular to the parameter ideals) of \(R\). In section 2 and in section 3 he presents estimations and calculations of these invariants (see theorem 1.1). Moreover several interesting examples clarify the invariants. -- In section 4 the author describes situations where one of these invariants, \(t_0(R) : = \sup \ell_R (Q^*/Q)\), \(Q\) running over all parameter ideals of \(R\), is finite. It results (see theorem 1.2) that if \(\text{depth} R \geq 2\) and \(t_0 (R) < \infty\) then \(R\) is normal, which is somehow a generalization of the above mentioned result of Hochster and Huneke. Finally, if \((R, {\mathfrak m})\) is an equidimensional complete local ring of characteristic \(p\) with \(t_0 (R) < \infty\), then \(R\) has finite local cohomology \(H^i_{\mathfrak m} (R)\) for \(i \neq \dim R\), and \(t_0 (R_P) = 0\) for \(P \in \text{Spec} R \backslash \{{\mathfrak m}\}\) (see theorems 4.1 and 1.3). -- In section 5 polynomial extensions of \(F\)- rational rings (i.e. \(t_0 (R) = 0)\) are considered.
scientific article; zbMATH DE number 633578