Description
This method determines if the given R-module is projective with constant rank by considering the ideal of minors of its presentation matrix. In particular, if \phi is the presentation matrix of the module
M, let I_t(\phi) be the ideal in R generated by the t \times\ t minors of \phi. If there exists an r such that I_r(\phi) = R and I_{r+1}(\phi) = 0, then we know that
M is necessarily projective of constant rank (see Proposition 1.4.10 of Bruns-Herzog below). The method
isProjective calls on
maxMinors to compute the ideal of minors I_r(\phi) such that I_r(\phi) \neq 0 and I_{r+1}(\phi) = 0. If I_r(\phi) is the whole ring, then the module
M is projective with constant rank.
Reference:
-
W. Bruns and J. Herzog. Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN: 0-521-41068-1.
i1 : R = QQ[x,y,z]
o1 = R
o1 : PolynomialRing
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i2 : P = matrix{{x^2*y+1,x+y-2,2*x*y}}
o2 = | x2y+1 x+y-2 2xy |
1 3
o2 : Matrix R <-- R
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i3 : isProjective ker P
o3 = true
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i4 : M = matrix{{-y,-z^2,0},{x,0,-z^2},{0,x^2,x*y}}
o4 = | -y -z2 0 |
| x 0 -z2 |
| 0 x2 xy |
3 3
o4 : Matrix R <-- R
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i5 : isProjective cokernel M
o5 = false
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i6 : I = ideal(x^2,x*y,z^2)
2 2
o6 = ideal (x , x*y, z )
o6 : Ideal of R
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i7 : isProjective module I
o7 = false
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i8 : isProjective R^3
o8 = true
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i9 : isProjective module ideal x
o9 = true
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