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trueKoszul -- "Makes Koszul complex, with bases sorted in lex"

Synopsis

Description

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

See also

Ways to use trueKoszul :

For the programmer

The object trueKoszul is a method function.