# Singular

#### D.15.2.29 arrOrlikSolomon

Procedure from library `arr.lib` (see arr_lib).

Usage:
arrOrlikSolomon(A); arr A

Return:
[ring] exterior Algebra E as ring with Orlik-Solomon ideal as attribute I. The Orlik-Solomon ideal is generated by the differentials of dependent tuples of hyperplanes. For a complex arrangement the quotient E/I is isomorphic to the cohomology ring of the complement of the arrangement.

Note:
In order to access this ideal I activate this exterior algebra with setring.

Example:
 ```LIB "arr.lib"; ring R = 0,(x,y,z),dp; arr A = arrTypeB(3); def E = arrOrlikSolomon(A); setring E; //The generators of the Orlik-Solomon-Ideal are: I; ==> I[1]=e(7)*e(8)-e(7)*e(9)+e(8)*e(9) ==> I[2]=e(6)*e(8)-e(6)*e(9)+e(8)*e(9) ==> I[3]=e(6)*e(7)-e(6)*e(9)+e(7)*e(9) ==> I[4]=e(4)*e(5)-e(4)*e(9)+e(5)*e(9) ==> I[5]=e(3)*e(5)-e(3)*e(9)+e(5)*e(9) ==> I[6]=e(3)*e(4)-e(3)*e(9)+e(4)*e(9) ==> I[7]=e(6)*e(7)-e(6)*e(8)+e(7)*e(8) ==> I[8]=e(2)*e(5)-e(2)*e(8)+e(5)*e(8) ==> I[9]=e(1)*e(5)-e(1)*e(8)+e(5)*e(8) ==> I[10]=e(1)*e(2)-e(1)*e(8)+e(2)*e(8) ==> I[11]=e(1)*e(4)-e(1)*e(7)+e(4)*e(7) ==> I[12]=e(2)*e(3)-e(2)*e(7)+e(3)*e(7) ==> I[13]=e(2)*e(4)-e(2)*e(6)+e(4)*e(6) ==> I[14]=e(1)*e(3)-e(1)*e(6)+e(3)*e(6) ==> I[15]=e(3)*e(4)-e(3)*e(5)+e(4)*e(5) ==> I[16]=e(1)*e(2)-e(1)*e(5)+e(2)*e(5) ==> I[17]=-e(2)*e(4)*e(8)+e(2)*e(4)*e(9)-e(2)*e(8)*e(9)+e(4)*e(8)*e(9) ==> I[18]=-e(1)*e(4)*e(8)+e(1)*e(4)*e(9)-e(1)*e(8)*e(9)+e(4)*e(8)*e(9) ==> I[19]=-e(2)*e(3)*e(8)+e(2)*e(3)*e(9)-e(2)*e(8)*e(9)+e(3)*e(8)*e(9) ==> I[20]=-e(1)*e(3)*e(8)+e(1)*e(3)*e(9)-e(1)*e(8)*e(9)+e(3)*e(8)*e(9) ==> I[21]=-e(2)*e(5)*e(7)+e(2)*e(5)*e(9)-e(2)*e(7)*e(9)+e(5)*e(7)*e(9) ==> I[22]=-e(1)*e(5)*e(7)+e(1)*e(5)*e(9)-e(1)*e(7)*e(9)+e(5)*e(7)*e(9) ==> I[23]=-e(2)*e(4)*e(7)+e(2)*e(4)*e(9)-e(2)*e(7)*e(9)+e(4)*e(7)*e(9) ==> I[24]=-e(1)*e(3)*e(7)+e(1)*e(3)*e(9)-e(1)*e(7)*e(9)+e(3)*e(7)*e(9) ==> I[25]=-e(1)*e(2)*e(7)+e(1)*e(2)*e(9)-e(1)*e(7)*e(9)+e(2)*e(7)*e(9) ==> I[26]=-e(2)*e(5)*e(6)+e(2)*e(5)*e(9)-e(2)*e(6)*e(9)+e(5)*e(6)*e(9) ==> I[27]=-e(1)*e(5)*e(6)+e(1)*e(5)*e(9)-e(1)*e(6)*e(9)+e(5)*e(6)*e(9) ==> I[28]=-e(1)*e(4)*e(6)+e(1)*e(4)*e(9)-e(1)*e(6)*e(9)+e(4)*e(6)*e(9) ==> I[29]=-e(2)*e(3)*e(6)+e(2)*e(3)*e(9)-e(2)*e(6)*e(9)+e(3)*e(6)*e(9) ==> I[30]=-e(1)*e(2)*e(6)+e(1)*e(2)*e(9)-e(1)*e(6)*e(9)+e(2)*e(6)*e(9) ==> I[31]=-e(1)*e(2)*e(4)+e(1)*e(2)*e(9)-e(1)*e(4)*e(9)+e(2)*e(4)*e(9) ==> I[32]=-e(1)*e(2)*e(3)+e(1)*e(2)*e(9)-e(1)*e(3)*e(9)+e(2)*e(3)*e(9) ==> I[33]=-e(4)*e(5)*e(7)+e(4)*e(5)*e(8)-e(4)*e(7)*e(8)+e(5)*e(7)*e(8) ==> I[34]=-e(3)*e(5)*e(7)+e(3)*e(5)*e(8)-e(3)*e(7)*e(8)+e(5)*e(7)*e(8) ==> I[35]=-e(3)*e(4)*e(7)+e(3)*e(4)*e(8)-e(3)*e(7)*e(8)+e(4)*e(7)*e(8) ==> I[36]=-e(2)*e(4)*e(7)+e(2)*e(4)*e(8)-e(2)*e(7)*e(8)+e(4)*e(7)*e(8) ==> I[37]=-e(1)*e(3)*e(7)+e(1)*e(3)*e(8)-e(1)*e(7)*e(8)+e(3)*e(7)*e(8) ==> I[38]=-e(4)*e(5)*e(6)+e(4)*e(5)*e(8)-e(4)*e(6)*e(8)+e(5)*e(6)*e(8) ==> I[39]=-e(3)*e(5)*e(6)+e(3)*e(5)*e(8)-e(3)*e(6)*e(8)+e(5)*e(6)*e(8) ==> I[40]=-e(3)*e(4)*e(6)+e(3)*e(4)*e(8)-e(3)*e(6)*e(8)+e(4)*e(6)*e(8) ==> I[41]=-e(1)*e(4)*e(6)+e(1)*e(4)*e(8)-e(1)*e(6)*e(8)+e(4)*e(6)*e(8) ==> I[42]=-e(2)*e(3)*e(6)+e(2)*e(3)*e(8)-e(2)*e(6)*e(8)+e(3)*e(6)*e(8) ==> I[43]=-e(2)*e(3)*e(4)+e(2)*e(3)*e(8)-e(2)*e(4)*e(8)+e(3)*e(4)*e(8) ==> I[44]=-e(1)*e(3)*e(4)+e(1)*e(3)*e(8)-e(1)*e(4)*e(8)+e(3)*e(4)*e(8) ==> I[45]=-e(4)*e(5)*e(6)+e(4)*e(5)*e(7)-e(4)*e(6)*e(7)+e(5)*e(6)*e(7) ==> I[46]=-e(3)*e(5)*e(6)+e(3)*e(5)*e(7)-e(3)*e(6)*e(7)+e(5)*e(6)*e(7) ==> I[47]=-e(2)*e(5)*e(6)+e(2)*e(5)*e(7)-e(2)*e(6)*e(7)+e(5)*e(6)*e(7) ==> I[48]=-e(1)*e(5)*e(6)+e(1)*e(5)*e(7)-e(1)*e(6)*e(7)+e(5)*e(6)*e(7) ==> I[49]=-e(3)*e(4)*e(6)+e(3)*e(4)*e(7)-e(3)*e(6)*e(7)+e(4)*e(6)*e(7) ==> I[50]=-e(1)*e(2)*e(6)+e(1)*e(2)*e(7)-e(1)*e(6)*e(7)+e(2)*e(6)*e(7) ==> I[51]=-e(2)*e(4)*e(5)+e(2)*e(4)*e(7)-e(2)*e(5)*e(7)+e(4)*e(5)*e(7) ==> I[52]=-e(1)*e(3)*e(5)+e(1)*e(3)*e(7)-e(1)*e(5)*e(7)+e(3)*e(5)*e(7) ==> I[53]=-e(1)*e(4)*e(5)+e(1)*e(4)*e(6)-e(1)*e(5)*e(6)+e(4)*e(5)*e(6) ==> I[54]=-e(2)*e(3)*e(5)+e(2)*e(3)*e(6)-e(2)*e(5)*e(6)+e(3)*e(5)*e(6) ==> I[55]=-e(1)*e(2)*e(3)+e(1)*e(2)*e(4)-e(1)*e(3)*e(4)+e(2)*e(3)*e(4) ```