```
Macaulay2, version 1.3.1
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases, PrimaryDecomposition, ReesAlgebra, SchurRings,
TangentCone
i1 : S = QQ[x_(1,1)..x_(3,3)]
o1 = S
o1 : PolynomialRing
i2 : M = transpose genericMatrix(S,x_(1,1),3,3)
o2 = {-1} | x_(1,1) x_(1,2) x_(1,3) |
{-1} | x_(2,1) x_(2,2) x_(2,3) |
{-1} | x_(3,1) x_(3,2) x_(3,3) |
3 3
o2 : Matrix S <--- S
i3 : I = minors(2,M)
o3 = ideal (- x x + x x , - x x + x x , - x x + x x , - x x + x x , - x x + x x , -
1,2 2,1 1,1 2,2 1,2 3,1 1,1 3,2 2,2 3,1 2,1 3,2 1,3 2,1 1,1 2,3 1,3 3,1 1,1 3,3
----------------------------------------------------------------------------------------------------------------------------
x x + x x , - x x + x x , - x x + x x , - x x + x x )
2,3 3,1 2,1 3,3 1,3 2,2 1,2 2,3 1,3 3,2 1,2 3,3 2,3 3,2 2,2 3,3
o3 : Ideal of S
i4 : (res I).dd
1 9
o4 = 0 : S <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- S : 1
| x_(1,2)x_(2,1)-x_(1,1)x_(2,2) x_(1,3)x_(2,1)-x_(1,1)x_(2,3) x_(1,3)x_(2,2)-x_(1,2)x_(2,3) x_(1,2)x_(3,1)-x_(1,1)x_(3,2) x_(1,3)x_(3,1)-x_(1,1)x_(3,3) x_(2,2)x_(3,1)-x_(2,1)x_(3,2) x_(2,3)x_(3,1)-x_(2,1)x_(3,3) x_(1,3)x_(3,2)-x_(1,2)x_(3,3) x_(2,3)x_(3,2)-x_(2,2)x_(3,3) |
9 16
1 : S <----------------------------------------------------------------------------------------------------------------------------------------------------------- S : 2
{2} | -x_(1,3) x_(2,3) 0 -x_(3,1) 0 x_(3,2) 0 0 x_(3,3) 0 0 x_(3,3) 0 0 0 0 |
{2} | x_(1,2) -x_(2,2) 0 0 -x_(3,1) 0 0 x_(3,2) 0 x_(3,3) 0 -x_(3,2) 0 0 0 0 |
{2} | -x_(1,1) x_(2,1) 0 0 0 0 -x_(3,1) 0 0 0 0 0 -x_(3,2) x_(3,3) 0 0 |
{2} | 0 0 -x_(1,3) x_(2,1) 0 -x_(2,2) -x_(2,3) 0 -x_(2,3) 0 0 0 0 0 x_(3,3) 0 |
{2} | 0 0 x_(1,2) 0 x_(2,1) 0 x_(2,2) -x_(2,2) 0 -x_(2,3) 0 0 0 0 -x_(3,2) 0 |
{2} | 0 0 0 -x_(1,1) 0 x_(1,2) 0 x_(1,3) 0 0 -x_(2,3) 0 0 0 0 x_(3,3) |
{2} | 0 0 0 0 -x_(1,1) 0 0 0 x_(1,2) x_(1,3) x_(2,2) 0 0 0 0 -x_(3,2) |
{2} | 0 0 -x_(1,1) 0 0 0 0 0 0 0 0 x_(2,1) x_(2,2) -x_(2,3) x_(3,1) 0 |
{2} | 0 0 0 0 0 0 -x_(1,1) x_(1,1) -x_(1,1) 0 -x_(2,1) -x_(1,1) -x_(1,2) x_(1,3) 0 x_(3,1) |
16 9
2 : S <-------------------------------------------------------------------------------------------- S : 3
{3} | x_(3,1) -x_(3,2) 0 -x_(3,3) 0 0 0 0 0 |
{3} | 0 0 x_(3,1) 0 -x_(3,3) x_(3,2) 0 0 0 |
{3} | -x_(2,1) x_(2,2) 0 x_(2,3) 0 0 0 0 0 |
{3} | -x_(1,3) 0 x_(2,3) 0 0 0 -x_(3,3) 0 0 |
{3} | x_(1,2) 0 -x_(2,2) 0 0 0 x_(3,2) 0 0 |
{3} | 0 -x_(1,3) 0 0 0 -x_(2,3) 0 x_(3,3) 0 |
{3} | -x_(1,1) 0 x_(2,1) 0 0 0 0 x_(3,2) -x_(3,3) |
{3} | -x_(1,1) x_(1,2) 0 0 x_(2,3) 0 0 0 -x_(3,3) |
{3} | x_(1,1) 0 0 -x_(1,3) 0 x_(2,2) 0 -x_(3,2) 0 |
{3} | 0 0 0 x_(1,2) -x_(2,2) 0 0 0 x_(3,2) |
{3} | 0 0 -x_(1,1) 0 x_(1,3) -x_(1,2) 0 0 0 |
{3} | -x_(1,1) 0 0 0 x_(2,3) -x_(2,2) -x_(3,1) 0 0 |
{3} | 0 x_(1,1) 0 0 0 x_(2,1) 0 -x_(3,1) 0 |
{3} | 0 0 0 -x_(1,1) x_(2,1) 0 0 0 -x_(3,1) |
{3} | 0 0 0 0 0 0 x_(2,1) x_(2,2) -x_(2,3) |
{3} | 0 0 0 0 0 0 -x_(1,1) -x_(1,2) x_(1,3) |
9 1
3 : S <------------------------------------------ S : 4
{4} | -x_(2,3)x_(3,2)+x_(2,2)x_(3,3) |
{4} | -x_(2,3)x_(3,1)+x_(2,1)x_(3,3) |
{4} | -x_(1,3)x_(3,2)+x_(1,2)x_(3,3) |
{4} | x_(2,2)x_(3,1)-x_(2,1)x_(3,2) |
{4} | x_(1,2)x_(3,1)-x_(1,1)x_(3,2) |
{4} | x_(1,3)x_(3,1)-x_(1,1)x_(3,3) |
{4} | -x_(1,3)x_(2,2)+x_(1,2)x_(2,3) |
{4} | x_(1,3)x_(2,1)-x_(1,1)x_(2,3) |
{4} | x_(1,2)x_(2,1)-x_(1,1)x_(2,2) |
1
4 : S <----- 0 : 5
0
o4 : ChainComplexMap
```