Depth of a module

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 [1] 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 .

References

[1]	M. Auslander, D.A. Buchsbaum, "Homological dimension in Noetherian rings" Proc. Nat. Acad. Sci. USA , 42 (1956) pp. 36–38
[2]	J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
[3]	A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , IHES (1965)