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-- Copyright 1999-2002 by Anton Leykin and Harrison Tsai
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
-- These routines compute the dual of a holonomic D-module M, which is
-- the transposition of the right module Ext^n_D(M,D)
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
Ddual = method()
Ddual Ideal := I -> (
Ddual ((ring I)^1/I)
)
Ddual Module := M -> (
pInfo(1, "ENTERING Ddual ... ");
W := ring M;
if W.monoid.Options.WeylAlgebra === {} then
error "expected an element of a Weyl algebra";
pInfo (1, "Ddual: holonomicity check ...");
if not isHolonomic M then
error "expected a holonomic module";
createDpairs W;
n := #W.dpairVars#0;
outputList := {};
C := Dres(M, LengthLimit => n+1);
m0 := transpose Dtransposition C.dd#n;
m1 := transpose Dtransposition C.dd#(n+1);
F0 := map(C#n, C#(n-1), m0);
F1 := map(C#(n+1), C#n, m1);
pres := presentation homology(F1,F0);
dualM := cokernel zeroize Dprune pres;
dualM
)
TEST ///
-- dual of an Appell F1
I = AppellF1({2,-3,-2,5});
assert(ideal relations Ddual I ==
substitute(AppellF1({-1,4,2,-3}), vars ring I));
-- dual of gkz associated to quadratic equation
A = matrix{{1,1,1},{0,1,2}};
b = {8/3,9/17};
I = gkz(A,b);
assert(ideal relations Ddual I == substitute(gkz(A,-b-{1,1}), vars ring I));
///
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