Regular locally noetherian locally factorial

Every regular local Noetherian ring is a unique factorisation domain, and hence every regular locally Noetherian scheme is locally factorial (Locally factorial scheme). This is recorded here as an external orientation fact with its source; the proof is not reproduced, and the locally factorial isomorphism proved on this page assumes local factoriality as a hypothesis rather than deriving it from regularity. In particular the remark is not a supplier for any item of this page. Normality alone does not imply local factoriality: the singular quadric cone is normal but not locally factorial, as shown on the examples companion of this page.