F-signature of graded Gorenstein rings

For a commutative ring $R$, the $F$-signature was defined by Huneke and Leuschke \cite{H-L}. It is an invariant that measures the order of the rank of the free direct summand of $R^{(e)}$. Here, $R^{(e)}$ is $R$ itself, regarded as an $R$-module through $e$-times Frobenius action $F^e$.In this paper, we show a connection of the F-signature of a graded ring with other invariants. More precisely, for a graded $F$-finite Gorenstein ring $R$ of dimension $d$, we give an inequality among the $F$-signature $s(R)$, $a$-invariant $a(R)$ and Poincar\'{e} polynomial $P(R,t)$. $$s(R)\le\frac{(-a(R))^d}{2^{d-1}d!}\lim_{t\rightarrow 1}(1-t)^dP(R,t)$$Moreover, we show that $R^{(e)}$ has only one free direct summand for any $e$, if and only if $R$ is $F$-pure and $a(R)=0$. This gives a characterization of such rings.Comment: 8 page

On F-pure thresholds

Using the Frobenius map, we introduce a new invariant for a pair (R,\a) of a ring $R$ and an ideal \a \subset R, which we call the F-pure threshold \mathrm{c}(\a) of \a, and study its properties. We see that the F-pure threshold characterizes several ring theoretic properties. By virtue of Hara and Yoshida's result, the F-pure threshold \mathrm{c}(\a) in characteristic zero corresponds to the log canonical threshold \mathrm{lc}(\a) which is an important invariant in birational geometry. Using the F-pure threshold, we prove some ring theoretic properties of three-dimensional terminal singularities of characteristic zero. Also, in fixed prime characteristic, we establish several properties of F-pure threshold similar to those of the log canonical threshold with quite simple proofs.Comment: 19 pages; v.2: minor changes, to appear in J. Algebr