i1 : isHomogeneous(ZZ)
o1 = true
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i2 : isHomogeneous(ZZ[x])
o2 = true
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i3 : isHomogeneous(ZZ[x]/(x^3-x-3))
o3 = false
|
Rings may be graded, with generators having degree 0. For example, in the ring B below, every element of A has degree 0.
i4 : A = QQ[a,b,c];
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i5 : B = A[x,y];
|
i6 : isHomogeneous B
o6 = true
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i7 : isHomogeneous ideal(a*x+y,y^3-b*x^2*y)
o7 = true
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Quotients of multigraded rings are homogeneous, if the ideal is also multigraded.
i8 : R = QQ[a,b,c,Degrees=>{{1,1},{1,0},{0,1}}];
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i9 : I = ideal(a-b*c);
o9 : Ideal of R
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i10 : isHomogeneous I
o10 = true
|
i11 : isHomogeneous(R/I)
o11 = true
|
i12 : isHomogeneous(R/(a-b))
o12 = false
|
A matrix is homogeneous if each entry is homogeneous of such a degree so that the matrix has a well-defined degree.
i13 : S = QQ[a,b];
|
i14 : F = S^{-1,2}
2
o14 = S
o14 : S-module, free, degrees {1, -2}
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i15 : isHomogeneous F
o15 = true
|
i16 : G = S^{1,2}
2
o16 = S
o16 : S-module, free, degrees {-1, -2}
|
i17 : phi = random(G,F)
o17 = {-1} | -10a2+8ab-5b2 0 |
{-2} | -9a3-7ab2-6b3 6 |
2 2
o17 : Matrix S <--- S
|
i18 : isHomogeneous phi
o18 = true
|
i19 : degree phi
o19 = {0}
o19 : List
|
Modules are homogeneous if their generator and relation matrices are homogeneous.
i20 : M = coker phi
o20 = cokernel {-1} | -10a2+8ab-5b2 0 |
{-2} | -9a3-7ab2-6b3 6 |
2
o20 : S-module, quotient of S
|
i21 : isHomogeneous(a*M)
o21 = true
|
i22 : isHomogeneous((a+1)*M)
o22 = false
|