- Regular local ring
In
commutative algebra , a regular local ring is aNoetherian local ring having the property that the minimal number of generators of itsmaximal ideal is exactly the same as itsKrull dimension . The minimal number of generators of the maximal ideal is always bounded below by the Krull dimension. In symbols, let "A" be a local ring with maximal ideal m, and suppose that m is generated by "a"1, ..., "a""n". Then in general "n" ≥ dim "A", and "A" is defined to be regular if and only if "n" = dim "A".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
Wolfgang Krull , but they first became prominent in the work ofOscar Zariski , who showed that geometrically, a regular local ring corresponds to a smooth point on analgebraic variety . Let "Y" be analgebraic variety contained in affine "n"-space, and suppose that "Y" is the vanishing locus of the polynomials "f1",...,"fm". "Y" is nonsingular at "P" if "Y" satisfies aJacobian condition: If "M" = (∂"fi"/∂"xj") is the matrix of partial derivatives of the defining equations of the variety, then the rank of the matrix found by evaluating "M" at "P" is "n" − dim "Y". Zariski proved that "Y" is nonsingular at "P" if and only if the local ring of "Y" at "P" is regular. This implies that smoothness is an intrinsic property of the variety, in other words it does not depend on where or how the variety is embedded in affine space. It also suggests that regular local rings should have good properties, but before the introduction of techniques fromhomological algebra very little was known in this direction. Once such techniques were introduced in the1950 s, Auslander and Buchsbaum proved that every regular local ring is aunique 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,
Jean-Pierre Serre found a homological characterization of regular local rings: A local ring "A" is regular if and only if "A" has finiteglobal dimension . 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" ["x"] and theformal power series ring "A" "x"are both regular. 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.
# Anydiscrete 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 offormal power series "k" "X"is a regular local ring having (Krull) dimension 1.
# If "p" is an ordinary prime number, the ring ofp-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" "X"1, "X"2, ..., "X""d"is a regular local ring having (Krull) dimension "d".
# If Z is the ring of integers and "X" is an indeterminate, the ring Z ["X"] (2, "X") is an example of a 2-dimensional regular local ring which does not contain a field.References
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Jean-Pierre Serre , "Local algebra",Springer-Verlag , 2000, ISBN 3-540-66641-9. Chap.IV.D.
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