Matijevic–Roberts type theorems for F -singularities

1.1 Statement (Matijevic–Roberts type theorem (MRTT)). Let C be a class of noetherian local rings. Let R be a noetherian Z-graded ring, and P its prime ideal. Let P ∗ be the prime ideal generated by the all homogeneous elements of P . If RP ∗ ∈ C, then RP ∈ C. Clearly, the truth of the statement depends on the choice of C. Nagata conjectured the Matijevic–Roberts type theorem for the case that C is the class of Cohen–Macaulay local rings, and n = 1. Nagata’s conjecture was solved affirmatively by Hochster–Ratliff [25] and Matijevic–Roberts [29] independently. After that, due to the contribution of Aoyama–Goto [1], Avramov–Achilles [2], Cavaliere–Niesi [6], Goto–Watanabe [14], and Matijevic [28], it was proved that the Matijevic–Roberts type theorem is true for the case that C is the class of Cohen–Macaulay, Gorenstein, complete intersection, and regular local rings, for arbitrary n. After that, the result was generalized to an assertion for group actions in [17], and then M. Miyazaki and the author [20] proved the following.