$F$-purity and rational singularity in graded complete intersection rings

Abstract:

A simple criterion is given for determining “almost completely” whether the positively graded complete intersection ring $R = K[{X_1}, \ldots , {X_{n + t}}]/({G_i}, \ldots , {G_t})$, of dimension $n$, has an $F$-pure type singularity at $m = ({X_1}, \ldots , {X_{n + t}})$. Specifically, if $\operatorname {deg} ({X_i}) = {\alpha _i} > 0$ for $1 \leq i \leq n + t$ and $\operatorname {deg} ({G_i}) = {d_i} > 0$ for $i \leq i \leq t$, then there exists an integer $\delta$ determined by the singular locus of $R$ such that: (1) $R$ has $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} < \delta$. (2) $R$ does not have $F$-pure type if $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0$. The characterization given by this theorem is particularly effective if the singularity of $R$ at $m$ is isolated. In that case, $\delta = 0$ so that only the condition $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$ is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if $R$ has an isolated rational singularity, then $R$ has $F$-pure type. The converse is also “almost true” with the only exception being the case where $\Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0$. In proving this criterion, a weak but more stable form of $F$-purity, called $F$-contractedness, is defined and explored. $R$ is $F$-contracted (in characteristic $p > 0$) if every system of parameters for $m$ is contracted with respect to the Frobenius map $F: R \to R$. Just as for $F$-purity, the notion of $F$-contracted type is defined in characteristic 0 by reduction to characteristic $p$. The two notions of $F$-pure (type) and $F$-contracted (type) coincide when $R$ is Gorenstein; whence, in particular, when $R$ is a complete intersection ring.