koszulMaps -- compute all Koszul maps of total degree p for an artinian ideal I

i1 : setRandomSeed("ok")

o1 = 11318

i2 : R=ZZ/101[x_0..x_4];

i3 : I = ideal random(R^1,R^{7:-2});

o3 : Ideal of R

i4 : assert(dim I==0)

i5 : H=apply(3,i->hilbertFunction(i,I))

o5 = {1, 5, 8}

o5 : List

i6 : expectedSyzygies(5,H)

            0 1  2  3  4 5
o6 = total: 1 7 35 56 35 8
         0: 1 .  .  .  . .
         1: . 7  .  .  . .
         2: . . 35 56 35 8

o6 : BettiTally

i7 : cp=criticalKoszulDegrees(I)

o7 = {3}

o7 : List

i8 : (L,N)=koszulMaps(I,cp_0);L

o9 = {40, 50, 10}

o9 : List

i10 : apply(#N,j->betti N_j)

               0  1          0  1
o10 = {total: 40 50, total: 50 10}
          -1:  . 50     -1:  . 10
           0: 40  .      0: 50  .

o10 : List

i11 : rank coker N_0,rank (ker N_0/image N_1),rank ker N_1

o11 = (0, 0, 0)

o11 : Sequence

i12 : betti res I

             0 1  2  3  4 5
o12 = total: 1 7 35 56 35 8
          0: 1 .  .  .  . .
          1: . 7  .  .  . .
          2: . . 35 56 35 8

o12 : BettiTally

i13 : (L,N)=koszulMaps(I,2);L

o14 = {8, 25, 10}

o14 : List

i15 : apply(#N,j->betti N_j)

              0  1          0  1
o15 = {total: 8 25, total: 25 10}
          -1: . 25     -1:  . 10
           0: 8  .      0: 25  .

o15 : List

i16 : rank coker N_0,rank (ker N_0/image N_1),rank ker N_1

o16 = (0, 7, 0)

o16 : Sequence