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Regular local ring : ウィキペディア英語版
Regular local ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let ''A'' be a Noetherian local ring with maximal ideal m, and suppose ''a''1, ..., ''a''''n'' is a minimal set of generators of m. Then by Krull's principal ideal theorem ''n'' ≥ dim ''A'', and ''A'' is defined to be regular if ''n'' = dim ''A''.
The appellation ''regular'' is justified by the geometric meaning. A point ''x'' on an algebraic variety ''X'' is nonsingular if and only if the local ring \mathcal_ of germs at ''x'' is regular. (See also: regular scheme.) Regular local rings are ''not'' related to von Neumann regular rings.〔A local von Neumann regular ring is a division ring, so the two conditions are not very compatible.〕
For Noetherian local rings, there is the following chain of inclusions:
==Characterizations==
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if A is a Noetherian local ring with maximal ideal \mathfrak, then the following are equivalent definitions
* Let \mathfrak = (a_1, \ldots, a_n) where n is chosen as small as possible. Then A is regular if
::\mbox A = n\,,
:where the dimension is the Krull dimension. The minimal set of generators of a_1, \ldots, a_n are then called a ''regular system of parameters''.
* Let k = A / \mathfrak be the residue field of A. Then A is regular if
::\dim_k \mathfrak / \mathfrak^2 = \dim A\,,
:where the second dimension is the Krull dimension.
* Let \mbox A := \sup \|\mbox M \mboxA\mbox \} be the global dimension of A (i.e., the supremum of the projective dimensions of all A-modules.) Then A is regular if
::\mbox A < \infty\,,
:in which case, \mbox A = \dim A.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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