A scheme such that at every point the local ring is regular (cf. Regular ring (in commutative algebra)). For schemes of finite type over an algebraically closed field , regularity is equivalent to the sheaf of differentials being locally free. Regular local rings are factorial (cf. Factorial ring), and so any closed reduced irreducible subscheme of codimension 1 in a regular scheme is given locally by one equation (see ). An important problem is the construction of a regular scheme with a given field of rational functions and equipped with a proper morphism onto some base scheme . The solution is known in the case when is the spectrum of a field of characteristic 0 (see ), and for schemes of low dimension in the case of a prime characteristic and also in the case when is the spectrum of a Dedekind domain with (see ).
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Regular scheme. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Regular_scheme&oldid=23954