A linear bound for Frobenius powers and an inclusion bound for tight closure (Q2491066)

In tight closure theory, computations can be extremely difficult. Often, showing that a particular element belongs to the tight closure of an ideal necessites more tools that applying the definition. The paper under review addresses this question for homogeneous elements in large classes of standard-graded rings from the viewpoint of theory of coherent sheaves and takes in consideration the degree of the homogeneous elements. Let \(R\) denote a standard-graded normal Cohen-Macaulay ring over an algebraically closed field of characteristic \(p >0\). Let \(I=(f_1,\ldots,f_n)\) be a homogeneous ideal that is \(R_{+}\)-primary, where \(R_+ = \bigoplus_{n >0} R_n\). Let \(q =p^e\) and denote \(I^{[q]} = (i^q : i \in I)\) and \(I^*\) the tight closure of \(I\). Let \(Y=\text{Proj}(R)\) and \(\omega_Y\) be its dualizing sheaf. Let \(\cdots F_2 \to F_1 \to I \to 0\) be a homogeneous complex of graded \(R\)-modules exact on \(D(R_+)\). This leads to \(\mathcal{G}_{\bullet} \to \mathcal{O}_Y \to 0\) an exact complex on \(Y\). Let \(\text{Syz}_j = \text{Ker}(\mathcal{G}_j \to \mathcal{G}_{j-1})\). For a locally free sheaf \(S\) we denote \(\overline{\mu}_{\min} (S)\) the slope of the last cofactor of the Harder-Narasimhan filtration of \(S\). Let \[ \nu = -\frac{\overline{\mu}_{\min} (\text{Syz}_t)}{\deg(Y)}, \] where \(t =\dim(Y)\). Then the author shows that \(R_{>q\nu+ \frac{\deg(\omega_Y)}{\deg(Y)}} \subseteq I^{[q]}\) which leads to \(R_{\geq \nu} \subset I^*\). This result unifies two results of similar type obtained earlier by \textit{K.~E.~Smith} [J. Math. Kyoto Univ. 37, No. 1, 35--53 (1997; Zbl 0902.13005)] and also leads to an improvement of a linear bound on \(\text{reg}(I^{[q]})\), as \(q\) varies, obtained earlier by Chardin [Regularity of ideals and their powers, preprint 2004], for the class of rings and ideals studied in this paper.