Generic Local Duality and Purity Exponents

We prove a form of generic local duality that generalizes a result of Karen E. Smith. Specifically, let R be a Noetherian ring, let P be a prime ideal of R of height h, let $A:=R/P$, and W be a subset of R that maps onto A \0\. Suppose that RP is Cohen-Macaulay, and that $ω$ is a finitely generated R-module such that ωP is a canonical module for RP. Let E:=HʰP(ω). We show that for every finitely generated R-module M there exists g ∈ W such that for all j≥ 0, HPʲ(M)g HomR(ExtRʰ⁻ʲ(M,\, ω),\, E)g, and that, moreover, every HPʲ(M)g has an ascending filtration by a countable sequence of finitely generated submodules such that the factors are finitely generated free Ag-modules. In fact, this sequence may be taken to be ʲ(M)gPⁿ\ₙ. We use this result to study the purity exponent for a nonzerodivisor c in a reduced excellent Noetherian ring R of prime characteristic p, which is the least e ∈ N such that the map R → R1/pᵉ with 1 ↦ c1/pᵉ is pure. In particular, in the case where R is a homomorphic image of an excellent Cohen-Macaulay ring and is S₂, we establish an upper semicontinuity result for the function ec:Spec(R) → N, where ec(P) is the purity exponent for the image of c in RP. This result enables us to prove that excellent strongly F-regular rings are very strongly F-regular (also called F-pure regular). Another consequence is that the F-pure locus is open in an S₂ ring that is a homomorphic image of an excellent Cohen-Macxaulay ring.