If Q = R[e,f] and M is an exact couple, then f annihilates the even degrees of M and e^2 annihilates the odd degrees of M. (In this context, even and odd are determined by the functions Q.isEvenDegree and Q.isOddDegree).
i1 : Q = coupleRing(ZZ,1,e,f,Degrees=>{{-1,0},{2,-2}})
o1 = Q
o1 : PolynomialRing
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A free Q module is not a free couple because the tautological couple relations do not hold
i2 : expectExactCouple Q^{{0,0},{-1,0},{-2,0}}
error: e^2 fails to annihilate aux
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To obtain the free couple as a quotient of the free module, use enforceCoupleRelations
i2 : C = enforceCoupleRelations Q^{{0,0},{-1,0},{-2,0}}
o2 = cokernel {0, 0} | e_1^3 f_1 0 0 0 0 |
{1, 0} | 0 0 e_1^2 e_1f_1 0 0 |
{2, 0} | 0 0 0 0 e_1^3 f_1 |
3
o2 : Q-module, quotient of Q
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i3 : expectExactCouple C |
The object enforceCoupleRelations is a method function.