koszulMap -- compute the Koszul homology map whose kernel correspond the Koszulzycles for K_(p,q)(S/I)

Synopsis

Usage:
kozsulMap(I,p,q)
Inputs:
- I, an ideal
- p, an integer, homological degree
- q, an integer, internal degree
Outputs:
- a matrix, Koszul matrix over the coefficient ring

Description

Computes the Koszul matrix Λ^p V ⊗ (ring/I)_q →Λ^p-1 V ⊗ (ring/I)_q+1 for the coordinate ring mod I

i1 : kk=ZZ/101;

i2 : R=kk[x_0..x_3];

i3 : betti(I=minors(3,random(R^4,R^{2:-1,-2})));

i4 : betti res I

            0 1 2
o4 = total: 1 4 3
         0: 1 . .
         1: . . .
         2: . . .
         3: . 4 2
         4: . . 1

o4 : BettiTally

i5 : N1=koszulMap(I,2,3);

              124        120
o5 : Matrix kk    <--- kk

i6 : N2=koszulMap(I,3,2);

              120        40
o6 : Matrix kk    <--- kk

i7 : N1*N2

o7 = 0

              124        40
o7 : Matrix kk    <--- kk

i8 : rank (ker N1/image N2)

o8 = 2

Ways to use `koszulMap` :

koszulMap(Ideal,ZZ,ZZ)

koszulMap -- compute the Koszul homology map whose kernel correspond the Koszulzycles for K_(p,q)(S/I)

Synopsis

Description

Ways to use koszulMap :

Ways to use `koszulMap` :