# 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

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).