F-Purity and Rational Singularity.

Abstract

The purpose of this dissertation is to investigate singularitieswhich are F-pure (respectively, F-pure type). A ring R of characteristic p is F-pure if for every R module M, 0 (--->) M (CRTIMES) R (--->) M (CRTIMES) ('1)R is exact where ('1)R denotes the R-algebra structure induced on R by the Frobenius map. 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 polynomial ringS, R is F-pure at the prime ideal Q if and only if (I('{p}):I) (NOT L-HOOK) Q('{p}), whereJ('{p}) denotes the ideal {a('p)(VBAR)a (ELEM) J}. Several theorems result from this criterion. If f is a quasihomogeneous hypersurface in A('n) having weights (r(,l),...,r(,n)) and an isolated singularity at the origin: (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) implies (K{X(,1),...,X(,n)}/(f))(,(X(,1),...,X(,n))) has F-pure type. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) implies (K{X(,1),...,X(,n)}/(f))(,(X(,1),...,X(,n))) does not have F-pure type. (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) remains unsolved. 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. An important conjecture is that R/fR is F-pure (type) should imply R is F-pure (type) whenever R is a Cohen-Macaulay, normal, affine local ring. It is proven that the condition Ext('1)(('1)R,R) = 0 is sufficient, though not necessary, for this to be true. Examples are given of Cohen-Macaulay, normal, affine local rings in which Ext('1)(('1)R,R) (NOT=) 0. An n-dimensional ring R of characteristic p is F-injective if H(,m)('n)(R) (--->) H(,m)('n)(('1)R) is injective. An example is contructed which is F-injective but not F-pure. From this, a counter-example 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 counter-example to the characteristic 0 version of the conjecture.