hilbertBurchMatrices -- Check the depth conditions for the exactness of the 1,4,4,1 subcomplex of the 1,6,8,3 subcomplex

• Usage:

  A = hilbertBurchMatrices L

  answer = hilbertBurchConditions L

  d = depthCondition1 L

  answer = hasExactSubcomplex L

• Inputs:
  • L, a list, generators of a semigroup with 4 generators
• Outputs:
  • A, a list, a List of Hilbert Burch submatrices of the 2nd syzygy matrix
  • answer, a Boolean value, the answer d:ZZ the depth of the 2x2 minors of the 2x4 submatrix of the 3rd syzygy matrix

i1 : L = {6,9,13,16}

o1 = {6, 9, 13, 16}

o1 : List

i2 : satisfiesDegreeCondition1 L and satisfiesDegreeCondition2 L

o2 = true

i3 : hilbertBurchMatrices L

o3 = {{22} | -x_4 -x_0x_1 0    0    |, {18} | -x_1 -x_4   0   0   |}
      {25} | -x_1 -x_4    0    0    |  {22} | -x_3 -x_0^2 0   0   |
      {32} | x_0  x_3     0    0    |  {25} | x_0  x_3    0   0   |
      {18} | 0    0       -x_1 -x_4 |  {32} | 0    0      x_0 x_3 |

o3 : List

i4 : hilbertBurchConditions L

o4 = true

i5 : depthCondition1 L

o5 = 2

i6 : hasExactSubcomplex L

o6 = true