𝐹-purity and rational singularity

Abstract

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