Let \(R\) be a Noetherian ring, \(P\) be a prime ideal of \(R\) such that \(R_P\) is Cohen-Macaulay of dimension \(h\), \(\omega\) be a finitely generated \(R\)-module such that \(\omega_P\) is a canonical module for \(R_P\), and \(W\) be a subset of \(R\) that naturally maps onto the set of nonzero elements of \(R/P\). We show that for every finitely generated $R$-module $M$, there exists \(g \in W\) such that \(H_P^j(M)_g \cong Hom(Ext_R^{h-j}(M, \omega), H_P^h(\omega))_g\), which gives the well-known local duality when we localize at \(P\). Moreover, each \(H_P^j(M)_g\) has an ascending filtration such that all the factors are free over \(R/P\). We use this result to study the purity exponents in Noetherian rings of prime characteristic \(p\). In the case where \(R\) is excellent Cohen-Macaulay (this assumption can be weakened), we establish an upper semicontinuity result for the purity exponent considered as a function on the spectrum of \(R\). 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 excellent Cohen-Macaulay ring. This is joint work with Mel Hochster.