Commutative Algebra pp 227-245
A Characterization of F-Regularity in Terms of F-Purity
In recent years, some very interesting theorems have been proven independently using complex analytic techniques or, alternatively, using reduction to characteristic p techniques (relying on special properties of the Frobenius homomorphism). In particular, Hochster and Roberts  proved that the ring RG of invariants of a group G acting on a regular ring R is necessarily Cohen-Macaulay by an argument which exploits the fact that RG is a direct summand of R in characteristic 0 and that, therefore, after reduction to characteristic p, the Frobenius homomorphism is especially well-behaved for “almost all p”. Not long after, using the Grauert-Riemenschneider vanishing theorem, Boutôt  proved an even stronger result— in the affine and analytic cases, a direct summand (in characteristic 0) of a ring with rational singularity necessarily has a rational singularity.
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