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# orlikSolomon(Arrangement,PolynomialRing) -- compute the defining ideal for the Orlik-Solomon algebra

## Synopsis

• Function: orlikSolomon
• Usage:
orlikSolomon(A,E)
orlikSolomon(A,k)
orlikSolomon(A,e)
orlikSolomon(A)
• Inputs:
• E, , a skew-commutative polynomial ring with one variable for each hyperplane with indexed variables, optionally, given by the symbol $e$. The user can also just specify a coefficient ring $k$.
• Optional inputs:
• HypAtInfinity => ..., default value 0
• Projective => ..., default value false
• Strategy => ..., default value Matroids
• Outputs:
• an ideal, the defining ideal of the Orlik-Solomon algebra of A

## 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.

A fundamental property is that its Hilbert series is determined by combinatorics: namely, up to a change of variables, it is the characteristic polynomial of the matroid of the arrangement.

 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 : characteristicPolynomial matroid A 3 2 o4 = x - 6x + 11x - 6 o4 : ZZ[x]

The cohomology ring of the complement of an arrangement in projective space is most naturally described as the subalgebra of the Orlik-Solomon algebra generated in degree $1$ by elements whose coefficients sum to $0$.

This is inconvenient for Macaulay2; on the other hand, one can choose a chart for projective space that places a hyperplane of the arrangement at infinity. This expresses the projective Orlik-Solomon algebra as a quotient of a polynomial ring.

By selecting the Projective option, the user can specify which hyperplane is placed at infinity. By default, the first one in order is used.

 i5 : I' = orlikSolomon(A,Projective=>true,HypAtInfinity=>2) o5 = 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 o5 : Ideal of QQ[e ..e ] 1 6 i6 : reduceHilbert hilbertSeries I' 2 1 + 5T + 6T o6 = ------------ 1 o6 : Expression of class Divide

The method caches the list of circuits of the arrangement. By default, the method uses the Matroids package to compute the Orlik-Solomon ideal. The option "Strategy=>Popescu" uses code by Sorin Popescu instead.

## Caveat

The coefficient rings of the Orlik-Solomon algebra and of the arrangement, respectively, are unrelated.