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It is equivalent to say that the dimension of the vector space m/m2, considered as a vector space over the residue field ''k''=''A''/m of ''A'', is equal to the dimension of ''A''. See System Of Parameters . Regular local rings were originally defined by very little was known in this direction. Once such techniques were introduced in the 1950 s, Auslander and Buchsbaum proved that every regular local ring is a Unique Factorization Domain . Another property suggested by geometric intuition is that the localization of a regular local ring should again be regular. Geometrically, this corresponds to the intuition that if a surface contains a curve, and that curve is smooth, then the surface is smooth near the curve. Again, this lay unsolved until the introduction of homological techniques. However, . It is easy to show that the property of having finite global dimension is preserved under localization, and consequently that localizations of regular local rings at prime ideals are again regular. This allows us to define regularity for all rings, not just local ones: A ring ''A'' is said to be regular if its localizations at all of its prime ideals are regular local rings. It is equivalent to say that ''A'' has finite global dimension. If ''A'' is a regular ring, then it follows that the Polynomial Ring ''A'' {Link without Title} and the Formal Power Series ring ''A'' EXAMPLES # Every Field is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0. # Any Discrete Valuation Ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings. Specifically, if ''k'' is a field and ''X'' is an indeterminate, then the ring of Formal Power Series ''k'' # If ''p'' is an ordinary prime number, the ring of P-adic Integer s is an example of a discrete valuation ring, and consequently a regular local ring, which does not contain a field. # More generally, if ''k'' is a field and ''X''1, ''X''2, ..., ''X''''d'' are indeterminates, then the ring of formal power series ''k'' # If Z is the ring of integers and ''X'' is an indeterminate, the ring Z {Link without Title} (2, ''X'') is an example of a 2-dimensional regular local ring which does not contain a field. |
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