# EPY -- compute the Eisenbud-Popescu-Yuzvinsky module of an arrangement

## Synopsis

• Usage:
EPY(A) or EPY(A,S) or EPY(I) or EPY(I,S)
• Inputs:
• A, , an arrangement of n hyperplanes
• I, an ideal, an ideal of the exterior algebra, the quotient by which has a linear, injective resolution
• S, , an optional polynomial ring in n variables
• Outputs:
• , The Eisenbud-Popescu-Yuzvinsky module (see below) of I or, if an arrangement is given, of its Orlik-Solomon ideal.

## Description

Let OS denote the Orlik-Solomon algebra of the arrangement A, regarded as a quotient of an exterior algebra E. The module EPY(A) is, by definition, the S-module which is BGG-dual to the linear, injective resolution of OS as an E-module.

Equivalently, EPY(A) is the single nonzero cohomology module in the Aomoto complex of A. For details, see Eisenbud-Popescu-Yuzvinsky, [TAMS 355 (2003), no 11, 4365--4383].

 i1 : R = QQ[x,y]; i2 : FA = EPY arrangement {x,y,x-y} o2 = cokernel | -X_1 X_1+X_2+X_3 X_1 | | X_2 0 X_1+X_3 | 2 o2 : QQ[X ..X ]-module, quotient of (QQ[X ..X ]) 1 3 1 3 i3 : betti res FA 0 1 2 o3 = total: 2 3 1 0: 2 3 1 o3 : BettiTally
In particular, EPY(A) has a linear free resolution over the polynomial ring, namely the Aomoto complex of A.
 i4 : A = typeA(4) o4 = {x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x , x - x } 1 2 1 3 1 4 1 5 2 3 2 4 2 5 3 4 3 5 4 5 o4 : Hyperplane Arrangement  i5 : factor poincare A o5 = (1 + T)(1 + 2T)(1 + 3T)(1 + 4T) o5 : Expression of class Product i6 : betti res EPY A 0 1 2 3 4 o6 = total: 24 50 35 10 1 0: 24 50 35 10 1 o6 : BettiTally

## Ways to use EPY :

• "EPY(Arrangement)"
• "EPY(Arrangement,PolynomialRing)"
• "EPY(Ideal)"
• "EPY(Ideal,PolynomialRing)"

## For the programmer

The object EPY is .