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System of parameters of a module over a local ring

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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

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)
How to Cite This Entry:
System of parameters of a module over a local ring. M. Hazewinkel (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=System_of_parameters_of_a_module_over_a_local_ring&oldid=18255
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098