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A codimension 3 example with details -- Constructing a minimal resolution for a codimension 3 cyclic polytopes with details.

We give all the details of the unprojection construction of a Buchsbaum-Eisenbud case.

Δ(4,7):

i1 : R=QQ[x_1..x_7];
i2 : cc=cycRes(4,R,verbose=>2,UseBuchsbaumEisenbud=>false);

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delta(2,{z, x_2, x_3})   +   delta(0,{z, x_2})   ->   delta(2,{z, x_2, x_3, x_4})

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res(I): 

a_1 = | zx_2x_3 |


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res(J): 

b_1 = | z x_2 |

b_2 = {1} | -x_2 |
      {1} | z    |

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phi: {z, x_2} -> {z*x_3, 0}

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alpha_1 = {1} | 0    |
          {1} | zx_3 |


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beta_1 = | -zx_3 0 |


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u = 1

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h_1 = 0


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f_1 = | x_4z-zx_3 x_4x_2 |

f_2 = | -x_4x_2   |
      | x_4z-zx_3 |


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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(4,{x_1, x_2, x_3, x_4, x_5})   +   delta(2,{z, x_2, x_3, x_4})   ->   delta(4,{x_1, x_2, x_3, x_4, x_5, x_6})

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res(I): 

a_1 = | x_1x_2x_3x_4x_5 |


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res(J): 

b_1 = | -zx_3 x_2x_4 |

b_2 = {2} | -x_2x_4 |
      {2} | -zx_3   |

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phi: {-z*x_3, x_2*x_4} -> {x_1*x_3*x_5, 0}

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alpha_1 = {2} | 0         |
          {2} | x_1x_3x_5 |


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beta_1 = | -x_1x_3x_5 0 |


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u = 1

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h_1 = 0


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f_1 = | -x_6zx_3-x_1x_3x_5 x_6x_2x_4 |

f_2 = | -x_6x_2x_4         |
      | -x_6zx_3-x_1x_3x_5 |


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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(2,{z, x_2, x_3})   +   delta(0,{z, x_2})   ->   delta(2,{z, x_2, x_3, x_4})

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res(I): 

a_1 = | zx_2x_3 |


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res(J): 

b_1 = | z x_2 |

b_2 = {1} | -x_2 |
      {1} | z    |

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phi: {z, x_2} -> {z*x_3, 0}

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alpha_1 = {1} | 0    |
          {1} | zx_3 |


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beta_1 = | -zx_3 0 |


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u = 1

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h_1 = 0


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f_1 = | x_4z-zx_3 x_4x_2 |

f_2 = | -x_4x_2   |
      | x_4z-zx_3 |


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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(2,{z, x_2, x_3, x_4})   +   delta(0,{z, x_2, x_3})   ->   delta(2,{z, x_2, x_3, x_4, x_5})

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res(I): 

a_1 = | -zx_3 x_2x_4 |

a_2 = {2} | -x_2x_4 |
      {2} | -zx_3   |


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res(J): 

b_1 = | x_2 x_3 z |

b_2 = {1} | 0   z    -x_3 |
      {1} | -z  0    x_2  |
      {1} | x_3 -x_2 0    |

b_3 = {2} | x_2 |
      {2} | x_3 |
      {2} | z   |

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phi: {x_2, x_3, z} -> {0, 0, z*x_4}

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alpha_1 = {1} | 0  x_4 |
          {1} | -z 0   |
          {1} | 0  0   |

alpha_2 = {2} | 0    |
          {2} | 0    |
          {2} | zx_4 |


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beta_1 = | 0 0 -zx_4 |

beta_2 = {2} | x_4 0 0 |
         {2} | 0   z 0 |


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u = -1

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h_1 = 0

h_2 = 0


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f_1 = | -zx_3 x_2x_4 x_5x_2 x_5x_3 x_5z-zx_4 |

f_2 = | x_4  0   0    x_5 0    |
      | 0    z   0    0   x_5  |
      | 0    -z  x_3  0   -x_4 |
      | z    0   -x_2 z   0    |
      | -x_3 x_2 0    0   0    |

f_3 = | x_5x_2    |
      | x_5x_3    |
      | x_5z-zx_4 |
      | -x_2x_4   |
      | -zx_3     |


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++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

delta(4,{x_1, x_2, x_3, x_4, x_5, x_6})   +   delta(2,{z, x_2, x_3, x_4, x_5})   ->   delta(4,{x_1, x_2, x_3, x_4, x_5, x_6, x_7})

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res(I): 

a_1 = | -x_1x_3x_5 x_2x_4x_6 |

a_2 = {3} | -x_2x_4x_6 |
      {3} | -x_1x_3x_5 |


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res(J): 

b_1 = | -zx_3 x_2x_4 x_2x_5 x_3x_5 -zx_4 |

b_2 = {2} | x_4  0   0    x_5 0    |
      {2} | 0    z   0    0   x_5  |
      {2} | 0    0   x_3  0   -x_4 |
      {2} | 0    0   -x_2 z   0    |
      {2} | -x_3 x_2 0    0   0    |

b_3 = {3} | x_2x_5  |
      {3} | x_3x_5  |
      {3} | -zx_4   |
      {3} | -x_2x_4 |
      {3} | -zx_3   |

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phi: {-z*x_3, x_2*x_4, x_2*x_5, x_3*x_5, -z*x_4} -> {x_1*x_3*x_6, 0, 0, 0, x_1*x_4*x_6}

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alpha_1 = {2} | 0    0   |
          {2} | 0    x_6 |
          {2} | 0    0   |
          {2} | -x_1 0   |
          {2} | 0    0   |

alpha_2 = {3} | 0          |
          {3} | 0          |
          {3} | -x_1x_4x_6 |
          {3} | 0          |
          {3} | -x_1x_3x_6 |


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beta_1 = | -x_1x_3x_6 0 0 0 -x_1x_4x_6 |

beta_2 = {3} | 0 0    0 x_6 0 |
         {3} | 0 -x_1 0 0   0 |


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u = 1

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h_1 = 0

h_2 = 0


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f_1 = | -x_1x_3x_5 x_2x_4x_6 -x_7zx_3-x_1x_3x_6 x_7x_2x_4 x_7x_2x_5 x_7x_3x_5 -x_7zx_4-x_1x_4x_6 |

f_2 = | 0    0    0    x_6  0    x_7 0    |
      | 0    -x_1 0    0    0    0   x_7  |
      | -x_4 0    0    -x_5 0    0   0    |
      | 0    -z   0    0    -x_5 0   -x_6 |
      | 0    0    -x_3 0    x_4  0   0    |
      | 0    0    x_2  -z   0    x_1 0    |
      | x_3  -x_2 0    0    0    0   0    |

f_3 = | -x_7x_2x_5        |
      | -x_7x_3x_5        |
      | x_7zx_4+x_1x_4x_6 |
      | x_7x_2x_4         |
      | x_7zx_3+x_1x_3x_6 |
      | -x_2x_4x_6        |
      | -x_1x_3x_5        |


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