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 $A\setminus \{0\}$. Suppose that $R_P$ is Cohen-Macaulay, and that $\omega$ is a finitely generated $R$-module such that $\omega_P$ is a canonical module for $R_P$. Let $E:=H^h_P(\omega)$. We show that for every finitely generated $R$-module $M$ there exists $g \in W$ such that for all $j\geq 0$, $H_P^j(M)_g \cong \mathrm{Hom}_R(\mathrm{Ext}_R^{h-j}(M,\, \omega),\, E)_g$, and that, moreover, every $H_P^j(M)_g$ has an ascending filtration by a countable sequence of finitely generated submodules such that the factors are finitely generated free $A_g$-modules. In fact, this sequence may be taken to be $\{\mathrm{Ann}_{H_P^j(M)_g}P^n\}_n$. 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 $e \in \mathbb{N}$ such that the map $R \to R^{1/p^e}$ with $1 \mapsto c^{1/p^e}$ is pure. In particular, in the case where $R$ is a homomorphic image of an excellent Cohen-Macaulay ring and is S$_2$, we establish an upper semicontinuity result for the function $\mathfrak{e}_c:\mathrm{Spec}(R) \to \mathbb{N}$, where $\mathfrak{e}_c(P)$ is the purity exponent for the image of $c$ in $R_P$. 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 S$_2$ ring that is a homomorphic image of an excellent Cohen-Macxaulay ring.