The usual Koszul command produces a complex with the basis sorted in revlex. The sort in lex matches the sort of the monomials in the exterior algebra.
i1 : S = ZZ/101[a,b,c,d] o1 = S o1 : PolynomialRing |
i2 : ff = matrix{{a,b,c,d}}
o2 = | a b c d |
1 4
o2 : Matrix S <--- S
|
i3 : (koszul ff).dd_2
o3 = {1} | -b -c 0 -d 0 0 |
{1} | a 0 -c 0 -d 0 |
{1} | 0 a b 0 0 -d |
{1} | 0 0 0 a b c |
4 6
o3 : Matrix S <--- S
|
i4 : (trueKoszul ff).dd_2
o4 = {1} | -b -c -d 0 0 0 |
{1} | a 0 0 -c -d 0 |
{1} | 0 a 0 b 0 -d |
{1} | 0 0 a 0 b c |
4 6
o4 : Matrix S <--- S
|
i5 : basis(2,(ZZ/101[a,b,c,d, SkewCommutative => true]))
o5 = | ab ac ad bc bd cd |
ZZ 1 ZZ 6
o5 : Matrix (---[a..d]) <--- (---[a..d])
101 101
|
The object trueKoszul is a method function.