# Depth of a module

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One of the cohomological characteristics of a module over a commutative ring. Let be a Noetherian ring, let be an ideal in it and let be an -module of finite type. Then the -depth of the module is the least integer for which The depth of a module is denoted by or by . A different definition can be given in terms of an -regular sequence, i.e. a sequence of elements of such that is not a zero divisor in the module The -depth of is equal to the length of the largest -regular sequence consisting of elements of . The maximal ideal is usually taken for in the case of a local ring . The following formula is valid: where denotes a prime ideal in , while is considered as a module over the local ring .

The concept of the depth of a module was introduced in  under the name of homological codimension. If the projective dimension of a module over a local ring is finite, then In general is not larger than the dimension of .

The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. Cohen–Macaulay ring) have been defined in terms of the depth of modules. The Serre criterion ( ) for an -module : for all prime ideals in , often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement is equivalent to saying that the local cohomology modules vanish if .

How to Cite This Entry:
Depth of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Depth_of_a_module&oldid=17353
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article