depth(Ideal,Ring) -- computes the depth of a ring

Synopsis

Function: depth
Usage:

d = depth(I,M)

d = depth(M)

d = depth(I,I)
Inputs:
- I, an ideal,
- M, a ring, or Module or Ideal
Outputs:
- d, an integer, the I-depth of a ring, module, or ideal

Description

The function depth(I,M) computes the I-depth of a ring, module, or ideal. In the most general setting, it does this by computing Ext^i(A^1/I,M) for an A-Module M, and noting where it does not vanish. If the ring in question is a polynomial ring over a field, then it merely computes the codimension of I.

i1 : A = QQ[x_1..x_3]/ideal(x_1^2, x_1*x_2)

o1 = A

o1 : QuotientRing

i2 : depth A

o2 = 1

If I contains a unit, then depth(I,A) outputs infinity.

i3 : depth(ideal(1_A),A)

o3 = infinity

o3 : InfiniteNumber

This symbol is provided by the package Depth.m2