A note on UFD

Abstract

We investigate conditions under which height-one ideals are principal. As a representative case, let $R$ be a strongly normal, almost factorial, complete intersection local ring, and let $p$ be a prime ideal of height one. We show that if $depth(R/ p)\geq dim(R)-2,$ then $p$ is principal. As an immediate application, we use elementary local cohomology techniques to reprove the celebrated Auslander--Buchsbaum theorem, thereby providing a streamlined approach to certain results of Dao and shedding new light on a problem of Samuel. As a further consequence, we prove that local rings of multiplicity at most three are hypersurfaces. We also give an affirmative answer to a question of Braun concerning reflexive ideals of finite injective dimension. Finally, we show how the reflexive hull can be recovered from the ideal transform by a simple argument, thereby providing a simplified proof of a result of Hartshorne. Then, we compute ideal transform of tensor product $D_{m}(M\otimes_RN)$ in terms of $Hom_R(M^{*},N)$.