Editing Regular local ring (section)

==Origin of basic notions==
{{see also|smooth scheme}}
Regular local rings were originally defined by [[Wolfgang Krull]] in 1937,<ref>{{Citation | last1=Krull | first1=Wolfgang | author1-link= Wolfgang Krull | title=Beiträge zur Arithmetik kommutativer Integritätsbereiche III | journal=Math. Z. | year=1937 | volume=42 | pages=745–766 | doi = 10.1007/BF01160110}}</ref> but they first became prominent in the work of [[Oscar Zariski]] a few years later,<ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=Algebraic varieties over ground fields of characteristic 0 | journal=Amer. J. Math. | year=1940 | volume=62 | pages=187–221 | doi=10.2307/2371447| jstor=2371447 }}</ref><ref>{{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=The concept of a simple point of an abstract algebraic variety | journal=Trans. Amer. Math. Soc. | year=1947 | volume=62 | pages=1–52 | doi=10.1090/s0002-9947-1947-0021694-1| doi-access=free }}</ref> who showed that geometrically, a regular local ring corresponds to a smooth point on an [[algebraic variety]].  Let ''Y'' be an [[algebraic variety]] contained in affine ''n''-space over a [[perfect field]], and suppose that ''Y'' is the vanishing locus of the polynomials ''f<sub>1</sub>'',...,''f<sub>m</sub>''.  ''Y'' is nonsingular at ''P'' if ''Y'' satisfies a [[Jacobian variety|Jacobian condition]]: If ''M'' = (∂''f<sub>i</sub>''/∂''x<sub>j</sub>'') 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'' &minus; dim ''Y''.  Zariski proved that ''Y'' is nonsingular at ''P'' if and only if the local ring of ''Y'' at ''P'' is regular. (Zariski observed that this can fail over non-perfect fields.)  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 from [[homological algebra]] very little was known in this direction.  Once such techniques were introduced in the 1950s, 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.   Again, this lay unsolved until the introduction of homological techniques.  It was [[Jean-Pierre Serre]] who found a homological characterization of regular local rings: A local ring ''A'' is regular if and only if ''A'' has finite [[global dimension]], i.e. if every ''A''-module has a projective resolution of finite length.  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 justifies the definition of ''regularity'' for non-local commutative rings given in the next section.