| Gröbner Bases |
| Syzygies |
|
Resolutions |
| Quantum Alg. |
| Left Maximality |
| Max. Twosided |
Let us compute the normal free resolution of the ideal, generated by e2, f , then minimize it and compute the minimal resolution with the particular algorithm.
ideal j=e2,f; j=std(j); j;
==>
j[1]=f
j[2]=h2+h
j[3]=eh+e
j[4]=e2
Step 1 The normal resolution:
resolution Nj=nres(j,0);
Nj;
==>
1 4 4 1 0
r <-- r <-- r <-- r <-- r
0 1 2 3 4
print(matrix(Nj[1]));
==>
f,h2+h,eh+e,e2
print(matrix(Nj[2]));
==>
0, h2+5h+6,eh+3e,e2,
0, -f, -1, 0,
e, 0, -f, -2,
-h+3,0, 0, -f
print(matrix(Nj[3]));
==>
f2,
-e,
ef,
-fh+f
Step 2 The minimized normal resolution:
resolution MiNj=minres(Nj);
MiNj;
==>
1 4 4 1 0
r <-- r <-- r <-- r <-- r
0 1 2 3 4
print(matrix(MiNj[1]));
==>
f,e2
print(matrix(MiNj[2]));
==>
e3, e2f2-6efh-6ef+6h2+18h+12,
-ef-2h+6,-f3
print(matrix(MiNj[3]));
==>
f2,
-e
Step 3 The minimal resolution:
resolution Mj=mres(j,0);
Mj;
==>
1 2 2 1 0
r <-- r <-- r <-- r <-- r
0 1 2 3 4
print(matrix(MiNj[1]));
==>
f,e2
print(matrix(MiNj[2]));
==>
e3, e2f2-6efh-6ef+6h2+18h+12,
-ef-2h+6,-f3
print(matrix(MiNj[3]));
==>
f2,
-e
As we can see, the modules in results of the 2nd and 3rd steps are the same. Back to the parent