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];
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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
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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
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i5 : factor poincare A
o5 = (1 + T)(1 + 2T)(1 + 3T)(1 + 4T)
o5 : Expression of class Product
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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
|