# orlikSolomon -- defining ideal for the Orlik-Solomon algebra

## Synopsis

• Usage:
orlikSolomon(A) or orlikSolomon(A,E) or orlikSolomon(A,e)
• Inputs:
• E, a ring, a skew-commutative polynomial ring with one variable for each hyperplane
• e, , a name for an indexed variable
• Optional inputs:
• HypAtInfinity => ..., default value 0, hyperplane at infinity
• Projective => ..., default value false, specify projective complement
• Outputs:
• an ideal, the defining ideal of the Orlik-Solomon algebra of A
• Consequences:

## Description

The Orlik-Solomon algebra is the cohomology ring of the complement of the hyperplanes, either in complex projective or affine space. The optional Boolean argument Projective specifies which. The code for this method was written by Sorin Popescu.
 i1 : A = typeA(3) o1 = {x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 2 3 2 4 3 4 o1 : Hyperplane Arrangement  i2 : I = orlikSolomon(A,e) o2 = ideal (e e - e e + e e , e e - e e + e e , e e - e e + e e , e e 4 5 4 6 5 6 2 3 2 6 3 6 1 3 1 5 3 5 1 2 ------------------------------------------------------------------------ - e e + e e ) 1 4 2 4 o2 : Ideal of QQ[e ..e ] 1 6 i3 : reduceHilbert hilbertSeries I 2 3 1 + 6T + 11T + 6T o3 = ------------------- 1 o3 : Expression of class Divide i4 : I' = orlikSolomon(A,Projective=>true,HypAtInfinity=>2) o4 = ideal (e e - e e + e e , e e - e e + e e , e e - e e + e e , e e 4 5 4 6 5 6 2 3 2 6 3 6 1 3 1 5 3 5 1 2 ------------------------------------------------------------------------ - e e + e e , e ) 1 4 2 4 3 o4 : Ideal of QQ[e ..e ] 1 6 i5 : reduceHilbert hilbertSeries I' 2 1 + 5T + 6T o5 = ------------ 1 o5 : Expression of class Divide

The code for orlikSolomon was contributed by Sorin Popescu.

## Ways to use orlikSolomon :

• "orlikSolomon(Arrangement)"
• "orlikSolomon(Arrangement,Ring)"
• "orlikSolomon(Arrangement,Symbol)"
• orlikSolomon(Arrangement,PolynomialRing) (missing documentation)
• orlikSolomon(CentralArrangement,PolynomialRing) (missing documentation)

## For the programmer

The object orlikSolomon is .