D.4.3.17 KoszulHomology

Procedure from library homolog.lib (see homolog_lib).

Compute:

A presentation of the p-th Koszul homology module H_p(f_1,...,f_k;M’), where M’=coker(M) and f_1,...,f_k are the given (ordered list of generators of the) ideal I. The computed presentation is minimized via prune. In particular, if H_p(f_1,...,f_k;M’)=0 then the return value is 0.

Return:

module H, s.th. coker(H) = H_p(f_1,...,f_k;M’).

Note:

size of input ideal has to be <= 20.

Example:

LIB "homolog.lib";
ring R=0,x(1..3),dp;
ideal x=maxideal(1);
module M=0;
KoszulHomology(x,M,0);  // H_0(x,R), x=(x_1,x_2,x_3)
→ _[1]=x(3)*gen(1)
→ _[2]=x(2)*gen(1)
→ _[3]=x(1)*gen(1)
KoszulHomology(x,M,1);  // H_1(x,R), x=(x_1,x_2,x_3)
→ _[1]=0
qring S=std(x(1)*x(2));
module M=0;
ideal x=maxideal(1);
KoszulHomology(x,M,1);
→ _[1]=-x(3)*gen(1)
→ _[2]=-x(2)*gen(1)
→ _[3]=-x(1)*gen(1)
KoszulHomology(x,M,2);
→ _[1]=0