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D.4.8.18 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 non-zero) 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
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