$F$-purity and rational singularity

Abstract:

We investigate singularities which are $F$-pure (respectively $F$-pure type). A ring $R$ of characteristic $p$ is $F$-pure if for every $R$-module $M,0 \to M \otimes R \to M \otimes ^1R$ is exact where $^1R$ denotes the $R$-algebra structure induced on $R$ via the Frobenius map (if $r \in R$ and $s \in ^{1}R$, then $r \cdot s = {r^p}s$ in $^1R$). $F$-pure type is defined in characteristic $0$ by reducing to characteristic $p$. It is proven that when $R = S/I$ is the quotient of a regular local ring $S$, $R$ is $F$-pure at the prime ideal $Q$ if and only if $({I^{[p]}}:I) \not \subset {Q^{[p]}}$. Here, ${J^{[p]}}$ denotes the ideal $\{ {a^p}|a \in J\}$. Several theorems result from this criterion. If $f$ is a quasihomogeneous hypersurface having weights $({r_1},\ldots ,{r_n})$ and an isolated singularity at the origin: (1) $\sum \nolimits _{i = 1}^n {{r_i} > 1}$ implies $K[{X_1},\ldots ,{X_n}]/(f)$ has $F$-pure type at $m = ({X_1},\ldots ,{X_n})$. (2) $\sum \nolimits _{i = 1}^n {{r_i} < 1}$ implies $K[{X_1},\ldots ,{X_n}]/(f)$ does not have $F$-pure type at $m$. (3) $\sum \nolimits _{i = 1}^n {{r_i} = 1}$ remains unsolved, but does connect with a problem that number theorists have studied for many years. This theorem parallels known results about rational singularities. It is also proven that classifying $F$-pure singularities for complete intersection ideals can be reduced to classifying such singularities for hypersurfaces, and that the $F$-pure locus in the maximal spectrum of $K[{X_1},\ldots ,{X_n}]/I$, where $K$ is a perfect field of characteristic $P$, is Zariski open. An important conjecture is that $R/fR$ is $F$-pure (type) should imply $R$ is $F$-pure (type) whenever $R$ is a Cohen-Macauley, normal local ring. It is proven that $\operatorname {Ext}^1{(^1}R,R) = 0$ is a sufficient, though not necessary, condition. A local ring $(R,m)$ of characteristic $p$ is $F$-injective if the Frobenius map induces an injection on the local cohomology modules $H_m^i(R) \to H_m^i{(^1}R)$. An example is constructed which is $F$-injective but not $F$-pure. From this a counterexample to the conjecture that $R/fR$ is $F$-pure implies $R$ is $F$-pure is constructed. However, it is not a domain, much less normal. Moreover, it does not lead to a counterexample to the characteristic $0$ version of the conjecture.