# Regular scheme

A scheme $ ( X , {\mathcal O} _ {X} ) $
such that at every point the local ring $ {\mathcal O} _ {X,x} $
is regular (cf. Regular ring (in commutative algebra)). For schemes of finite type over an algebraically closed field $ k $,
regularity is equivalent to the sheaf of differentials $ \Omega _ {X/k} ^ {1} $
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 $ ( X , {\mathcal O} _ {X} ) $
is given locally by one equation (see [2]). An important problem is the construction of a regular scheme $ ( X , {\mathcal O} _ {X} ) $
with a given field $ K $
of rational functions and equipped with a proper morphism $ X \rightarrow S $
onto some base scheme $ S $.
The solution is known in the case when $ S $
is the spectrum of a field of characteristic 0 (see [3]), and for schemes of low dimension in the case of a prime characteristic and also in the case when $ S $
is the spectrum of a Dedekind domain with $ \mathop{\rm dim} X / S \leq 1 $(
see [1]).

#### References

[1] | S.S. Abhyankar, "On the problem of resolution of singularities" I.G. Petrovskii (ed.) , Proc. Internat. Congress Mathematicians Moscow, 1966 , Moscow (1968) pp. 469–481 MR0232771 |

[2] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |

[3] | H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. , 79 (1964) pp. 109–203; 205–326 MR0199184 Zbl 0122.38603 |

#### Comments

Sometimes a regular scheme is called a smooth scheme, in which case one means that the structure morphism $ X \rightarrow S $ is a smooth morphism (where $ S $ is the spectrum of a field, cf. Spectrum of a ring).

#### References

[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 MR0463157 Zbl 0367.14001 |

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Regular scheme.

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