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A [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808401.png" /> such that at every point the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808402.png" /> is regular (cf. [[Regular ring (in commutative algebra)|Regular ring (in commutative algebra)]]). For schemes of finite type over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808403.png" />, regularity is equivalent to the sheaf of differentials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808404.png" /> being locally free. Regular local rings are factorial (cf. [[Factorial ring|Factorial ring]]), and so any closed reduced irreducible subscheme of codimension 1 in a regular scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808405.png" /> is given locally by one equation (see [[#References|[2]]]). An important problem is the construction of a regular scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808406.png" /> with a given field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808407.png" /> of rational functions and equipped with a [[Proper morphism|proper morphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808408.png" /> onto some base scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r0808409.png" />. The solution is known in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r08084010.png" /> is the spectrum of a field of characteristic 0 (see [[#References|[3]]]), and for schemes of low dimension in the case of a prime characteristic and also in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r08084011.png" /> is the spectrum of a Dedekind domain with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r08084012.png" /> (see [[#References|[1]]]).
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A [[Scheme|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)|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|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 [[#References|[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|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 [[#References|[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 [[#References|[1]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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 {{MR|0232771}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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 {{MR|0232771}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Sometimes a regular scheme is called a [[Smooth scheme|smooth scheme]], in which case one means that the structure morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r08084013.png" /> is a [[Smooth morphism|smooth morphism]] (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080840/r08084014.png" /> is the spectrum of a field, cf. [[Spectrum of a ring|Spectrum of a ring]]).
+
Sometimes a regular scheme is called a [[Smooth scheme|smooth scheme]], in which case one means that the structure morphism $  X \rightarrow S $
 +
is a [[Smooth morphism|smooth morphism]] (where $  S $
 +
is the spectrum of a field, cf. [[Spectrum of a ring|Spectrum of a ring]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 111–115; 126 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Latest revision as of 08:10, 6 June 2020


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
How to Cite This Entry:
Regular scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Regular_scheme&oldid=48486
This article was adapted from an original article by S.G. Tankeev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article