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

## Synopsis

• Usage:
K = trueKoszul ff
• Inputs:
• ff, , Matrix with the elements on which to build the Koszul complex
• Outputs:
• K, ,

## 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

• koszul -- Koszul complex or specific matrix in the Koszul complex

## Ways to use trueKoszul :

• "trueKoszul(Matrix)"

## For the programmer

The object trueKoszul is .