The Fourier transform is the automorphism of the Weyl algebra that sends xi to -Di and Di to xi. In order to compute the Fourier transform of the finitely generated module M, we compute the Fourier transform of the matrix A of which M is the cokernel.
i1 : makeWA(QQ[x,y])
o1 = QQ[x..y, dx, dy]
o1 : PolynomialRing, 2 differential variable(s)
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i2 : A = matrix{{2*x^2+1,y*dy},{9*x*dx, x*y*dx^2}}
o2 = | 2x2+1 ydy |
| 9xdx xydx^2 |
2 2
o2 : Matrix (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy])
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i3 : Fourier A
o3 = | 2dx^2+1 -ydy-1 |
| -9xdx-9 x2dxdy+2xdy |
2 2
o3 : Matrix (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy])
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i4 : L = x^2*dy + y*dy^2 + 3*dx^5*dy
5 2 2
o4 = 3dx dy + x dy + y*dy
o4 : QQ[x..y, dx, dy]
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i5 : Fourier L
5 2 2
o5 = 3x y + y*dx - y dy - 2y
o5 : QQ[x..y, dx, dy]
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i6 : I = ideal(8*x*y*dy^3+y^5, dx^7+5)
5 3 7
o6 = ideal (y + 8x*y*dy , dx + 5)
o6 : Ideal of QQ[x..y, dx, dy]
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i7 : Fourier I
3 5 2 7
o7 = ideal (8y dx*dy - dy + 24y dx, x + 5)
o7 : Ideal of QQ[x..y, dx, dy]
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i8 : C = chainComplex{matrix{{x*dx, y^2+dx}},matrix{{dx*dy},{y^2*dy^3}}}
1 2 1
o8 = (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy])
0 1 2
o8 : ChainComplex
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i9 : FC = Fourier C
1 2 1
o9 = (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy]) <-- (QQ[x..y, dx, dy])
0 1 2
o9 : ChainComplex
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i10 : FC.dd
1 2
o10 = 0 : (QQ[x..y, dx, dy]) <--------------------- (QQ[x..y, dx, dy]) : 1
| -xdx-1 dy^2+x |
2 1
1 : (QQ[x..y, dx, dy]) <--------------------------- (QQ[x..y, dx, dy]) : 2
{2} | xy |
{2} | y3dy^2+6y2dy+6y |
o10 : ChainComplexMap
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