Generating set of a module: Difference between revisions

Revision as of 01:07, 29 May 2022

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Let ''R'' be a [[local ring]] with [[maximal ideal]] ''m'' and [[residue field]] ''k'' and ''M'' finitely generated module. Then [[Nakayama's lemma]] says that ''M'' has a minimal generating set whose cardinality is <math>\dim_k M / mM = \dim_k M \otimes_R k</math>. If ''M'' is [[flat module|flat]], then this minimal generating set is [[linearly independent]] (so ''M'' is free). See also: [[minimal resolution (algebra)|Minimal resolution]].

A more refined information is obtained if one considers the relations between the generators; ''[[cf.]] [[Free presentation of a module]]''.

== See also ==