Rings of Frobenius operators (Q2877367)

Let \(R\) be a local ring of prime characteristic and \(E\) be the injective hull of its residue field. Let \(\mathcal F (E)\) be the ring of Frobenius operators of \(E\) over \(R\). This paper studies the finite generation of \(\mathcal F (E)\) over \(\mathcal F^0(E) = \text{Hom}_R(E,E)\) as a ring extension.NEWLINENEWLINEThe authors introduce a twisted construction \(\mathcal T (\mathcal R)\) associated to an \(\mathbb{N}\)-graded commutative ring \(\mathcal R\). They prove that, when \(R\) is normal and complete, the ring of Frobenius operators is isomorphic to the twisted construction associated to the anticanonical cover of \(R\). Using this result, they give an alternate proof to a result by \textit{K. Schwede} [Trans. Am. Math. Soc. 363, No. 11, 5925--5941 (2011; Zbl 1276.13008)] that states that, for a normal \(\mathbb{Q}\)-Gorenstein ring \(R\), if the order of the canonical module in the divisor class group is relatively prime to \(p\), then \(\mathcal F(E)\) is a finitely generated ring extension of \(\mathcal F^0(E)\). A conjecture is stated that asserts the converse.NEWLINENEWLINEThe paper also proves a conjecture of \textit{M. Katzman} [Proc. Am. Math. Soc. 138, No. 7, 2381--2383 (2010; Zbl 1192.13004)] confirming that determinantal ring obtained by moding out size \(2\) minors of a \(2 \times 3\) matrix of indeterminates has an infinitely generated ring of Frobenius operators over \(\mathcal F^0(E)\). The last two sections contain applications to the \(F\)-jumping numbers of ideals in a ring. Let \(R\) be a normal \(\mathbb{N}\)-graded that is finitely generated over an \(F\)-finite ring \(R_0\). Assume that the anticanonical cover of \(R\) is finitely generated as an \(R\)-algebra. It is shown in the paper that the set of \(F\)-jumping numbers of ideals in \(R\) is a subset of the real numbers with no limit points. In the last section, the authors consider a standard graded polynomial ring \(A\) over an \(F\)-finite field of prime characteristic and \(I\) a homogeneous ideal of \(A\). The authors prove that, if \(R=A/I\) has finite graded \(F\)-representation type, then the regularity of \(A/(I+J^{[p^e]})\) has linear growth with respect to \(p^e\), for each homogeneous ideal \(J\). In particular, the Cartier algebra of \(R=A/I\) is gauge bounded and the set of \(F\)-jumping numbers for ideals in \(R\) is a subset of the real numbers with no limit points.