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D.4.11.4 depth
Procedure from library homolog.lib (see homolog_lib).
- Usage:
- depth(M,[I]); M module, I ideal
- Return:
- int,
- if called with 1 argument: the depth of M'=coker(M) w.r.t. the
ideal generated by the variables in the basering
(the maximal ideal, if the ring is local)
- if called with 2 arguments: the depth of M'=coker(M) w.r.t. the
ideal I.
- Note:
- Not checked: if I*M'==M', depth is infinity.
- Note:
- procedure makes use of KoszulHomology.
Example:
| | LIB "homolog.lib";
ring R=0,(x,y,z),dp;
ideal I=x2,xy,yz;
module M=0;
depth(M,I); // depth(<x2,xy,yz>,Q[x,y,z])
==> 2
M=[1];
depth(M); // depth(0)
==> 3
ring r=0,(x,y,z),ds; // local ring
matrix M[2][2]=x,xy,1+yz,0;
print(M);
==> x, xy,
==> 1+yz,0
depth(M); // depth(maxideal,coker(M))
==> 2
ideal I=x;
depth(M,I); // depth(<x>,coker(M))
==> 0
I=x+z;
depth(M,I); // depth(<x+z>,coker(M))
==> 1
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