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HyperplaneArrangements :: EPY

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

Synopsis

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 :

For the programmer

The object EPY is a method function.