Regular local ring

In the mathematical branch of commutative algebra , a regular local ring is understood to be a Noetherian local ring , the maximum ideal of which can be generated by elements if the dimension of the ring denotes. Regular local rings describe the behavior of algebraic-geometric objects in points where there are no singularities such as peaks or crossings. A (not necessarily local) ring is called regular if all of its localizations are regular local rings. ${\displaystyle d}$ ${\displaystyle d}$

This article is about commutative algebra. In particular, all rings under consideration are commutative and have a one element. For more details, see Commutative Algebra .

definition

Let it be a one - dimensional Noetherian local ring with maximum ideal and residual class field . Then is called regular if one of the following equivalent conditions is met: ${\displaystyle A}$ ${\displaystyle d}$ ${\displaystyle {\mathfrak {m}}}$ ${\displaystyle k}$ ${\displaystyle A}$

• ${\displaystyle {\mathfrak {m}}}$can be generated from elements. ${\displaystyle d}$
• ${\displaystyle \dim _{k}{\mathfrak {m}}/{\mathfrak {m}}^{2}=d.}$

Any Noetherian ring is called regular if all of its local rings are regular. ${\displaystyle A}$

properties

• Regular local rings are factorial . This is the statement of the Auslands-Boxwood theorem.
• Serre's criterion : A Noetherian local ring is regular if and only if its global dimension is finite.
• From Serre's criterion it follows that localizations of regular local rings are regular again.

Examples

• Artin's local rings are regular if and only if they are solids.
• Regular local integrity rings of dimension 1 are precisely the discrete evaluation rings .