# 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); ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | zx_2x_3 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | z x_2 | b_2 = {1} | -x_2 | {1} | z | ------------------------------------------------------------------------------------------------------------------------ phi: {z, x_2} -> {z*x_3, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 | {1} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -zx_3 0 | ------------------------------------------------------------------------------------------------------------------------ u = 1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_4z-zx_3 x_4x_2 | f_2 = | -x_4x_2 | | x_4z-zx_3 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | x_1x_2x_3x_4x_5 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | -zx_3 x_2x_4 | b_2 = {2} | -x_2x_4 | {2} | -zx_3 | ------------------------------------------------------------------------------------------------------------------------ phi: {-z*x_3, x_2*x_4} -> {x_1*x_3*x_5, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {2} | 0 | {2} | x_1x_3x_5 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -x_1x_3x_5 0 | ------------------------------------------------------------------------------------------------------------------------ u = 1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ delta(2,{z, x_2, x_3}) + delta(0,{z, x_2}) -> delta(2,{z, x_2, x_3, x_4}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | zx_2x_3 | ------------------------------------------------------------------------------------------------------------------------ res(J): b_1 = | z x_2 | b_2 = {1} | -x_2 | {1} | z | ------------------------------------------------------------------------------------------------------------------------ phi: {z, x_2} -> {z*x_3, 0} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 | {1} | zx_3 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | -zx_3 0 | ------------------------------------------------------------------------------------------------------------------------ u = 1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 ------------------------------------------------------------------------------------------------------------------------ f_1 = | x_4z-zx_3 x_4x_2 | f_2 = | -x_4x_2 | | x_4z-zx_3 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | -zx_3 x_2x_4 | a_2 = {2} | -x_2x_4 | {2} | -zx_3 | ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ phi: {x_2, x_3, z} -> {0, 0, z*x_4} ------------------------------------------------------------------------------------------------------------------------ alpha_1 = {1} | 0 x_4 | {1} | -z 0 | {1} | 0 0 | alpha_2 = {2} | 0 | {2} | 0 | {2} | zx_4 | ------------------------------------------------------------------------------------------------------------------------ beta_1 = | 0 0 -zx_4 | beta_2 = {2} | x_4 0 0 | {2} | 0 z 0 | ------------------------------------------------------------------------------------------------------------------------ u = -1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 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}) ------------------------------------------------------------------------------------------------------------------------ res(I): a_1 = | -x_1x_3x_5 x_2x_4x_6 | a_2 = {3} | -x_2x_4x_6 | {3} | -x_1x_3x_5 | ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ 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} ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------ u = 1 ------------------------------------------------------------------------------------------------------------------------ h_1 = 0 h_2 = 0 ------------------------------------------------------------------------------------------------------------------------ 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 | ------------------------------------------------------------------------------------------------------------------------```