Singular Manual: arrOrlikSolomon

D.15.1.29 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)