System of parameters of a module over a local ring
Let be an
-dimensional Noetherian ring (cf. also the section "Dimension of an associative algebra" in Dimension). Then there exists an
-primary ideal generated by
elements (cf., e.g., [a1], p. 98, [a2], p. 27). If
generate such an
-primary ideal, they are said to be a system of parameters of
. The terminology comes from the situation that
is the local ring of functions at a (singular) point on an algebraic variety. The system of parameters
is a regular system of parameters if
generate
, and in that case
is a regular local ring.
More generally, if is a finitely-generated
-module of dimension
, then there are
such that
is of finite length; in that case
is called a system of parameters of
.
The ideal is called a parameter ideal.
For a semi-local ring with maximal ideals
, an ideal
is called an ideal of definition if
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for some natural number . If
is of dimension
, then any set of
elements that generates an ideal of definition is a system of parameters of
, [a3], Sect. 4.9.
References
[a1] | H. Matsumura, "Commutative ring theory" , Cambridge Univ. Press (1989) |
[a2] | M. Nagata, "Local rings" , Interscience (1962) |
[a3] | D.G. Nothcott, "Lessons on rings, modules, and multiplicities" , Cambridge Univ. Press (1968) |
System of parameters of a module over a local ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=System_of_parameters_of_a_module_over_a_local_ring&oldid=50153