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arXiv 1104.4236math.AC
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分次Gorenstein环的F-签名

F-signature of graded Gorenstein rings

Akiyoshi Sannai, Kei-ichi Watanabe

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8 pages
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摘要：

对于交换环$R$，Huneke和Leuschke \cite{H-L}定义了$F$ -签名.它是测量$R^{(e)}$的自由直和项的秩的不变量。这里，$R^{(e)}$是$R$本身，它通过$e$ -乘Frobenius作用$F^e$被看作是一个$R$ -模。更精确地说，对于维数为$d$的分次$F$ -有限Gorenstein环$R$，我们给出了$F$ -签名$s(R)$，$a$ -不变$a(R)$和Poincaré多项式$P(R,t)$之间的一个不等式. \[ s(R)\le\frac{(-a(R))^d}{2^{d-1}d!}\lim_{t\rightarrow 1}(1-t)^dP(R,t) \]此外，我们表明，$R^{(e)}$只有一个自由的直接被加数的任何$e$，当且仅当$R$是$F$ -纯和$a(R)=0$。这给出了这样的环的特征。

英文摘要：

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 Poincaré 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.

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