Se p 20 07 Formulas of F - thresholds and F - jumping coefficients on toric rings ∗

F-thresholds are defined by Mustaţǎ, Takagi and Watanabe, which are invariants of the pair of ideals on rings of characteristic p > 0. In their paper, it is proved F-thresholds equal to jumping numbers for the test ideal on regular local rings. In this note, we give formulas of F-thresholds and F-jumping coefficients on toric rings. We prove that there exists an inequality between F-jumping coefficients and Fthresholds. In particular, we observe a comparison between F-pure thresholds and F-thresholds in some cases. As applications, we prove the characterization of regularity for toric rings defined by a simplicial cone, and the rationality of F-thresholds in some cases. 1 The definition of F-thresholds Throughout this paper, we assume that a ring R is reduced, and contains a perfect field k whose characteristic is p > 0. Let F : R → R be the Frobenius map which sends x ∈ R to x. For a positive integer e, the ring R viewed as an R-module via the e-times iterated Frobenius map is denoted by R. We also assume that 1R is a finite generated R-module, which is called F -finite. For an ideal J and a positive integer e, J [p ] is generated by p-th power elements of J . For example, let J be (X1,X 2 2 ) ⊂ k[X1,X2], then J [p ] is (X e 1 ,X 2p 2 ). We recall the definition and some remarks of F-thresholds, which are defined by Mustaţǎ, Takagi and Watanabe in [MTW]. These are invariants of the pair of ideals. Definition 1.1 (F-threshold [MTW, §1]). Let a and J be nonzero proper ideals of a ring R, such that a ⊆ √ J . The p-th threshold of a with respect to J is defined as ν a (p) := max{r ∈ N|a * J [pe]}. 2000 Mathematics Subject Classification. Primary 13A35; Secondary 14M25.