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isCM -- whether a ring or module is Cohen-Macaulay

Synopsis

Description

This command merely checks if the I-depth of A equals the Krull dimension of A.
i1 : A = ZZ/2[x,y,z];
i2 : m = ideal(x,y,z);

o2 : Ideal of A
i3 : isCM(m,A)

o3 = true
i4 : A = ZZ/2[x,y]/(x^2,x*y);
i5 : m = ideal(x,y);

o5 : Ideal of A
i6 : isCM(m,A)

o6 = false
i7 : A =  ZZ/101[a_1,a_2,b_1,b_2,c_1]/ideal(a_1*b_1,a_2*b_2,b_1*c_1);
i8 : m = ideal vars A

o8 = ideal (a , a , b , b , c )
             1   2   1   2   1

o8 : Ideal of A
i9 : isCM(m,A)

o9 = false

This symbol is provided by the package Depth.

Caveat

Typically when one thinks of a Cohen-Macaulay ring or module, one is in the local case. Since the local case is not yet implemented into Macaulay 2, we insist that the user give an ideal for computing the depth.

Ways to use isCM :