Let S be the subring R[e^2, f]. Homogeneous elements of S are restricted to an index-two subgroup of the bidegrees of M; as an S-module, M splits as a direct sum of its even part and its odd part. We write E for the odd part and A for the even part. Multiplication by e induces maps from E to A and back again. We say that M is exact if
image(f : A --> A) = kernel(e : A --> E)
image(e : A --> E) = kernel(e : E --> A)
image(e : E --> A) = kernel(f : A --> A).
i1 : R = QQ[d,t,Degrees=>{{0,1},{1,0}}]/d^2;
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i2 : declareGenerators(R,{a=>{0,0},b=>{0,0},c=>{0,0},ab=>{0,1},ac=>{0,1},bc=>{0,1}});
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i3 : M = cospan(d*a+ab+ac, d*b-ab+bc, d*c-ac-bc, d*ab, d*ac, d*bc,
t*bc, t^2*ac, t^3*ab, t^4*c, t^5*b, t^6*a);
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i4 : netList table(7,4,(i,j)->hilbertFunction({6-i,j},M))
+-+-+-+-+
o4 = |0|0|0|0|
+-+-+-+-+
|1|0|0|0|
+-+-+-+-+
|2|0|0|0|
+-+-+-+-+
|3|0|0|0|
+-+-+-+-+
|3|1|0|0|
+-+-+-+-+
|3|2|0|0|
+-+-+-+-+
|3|3|0|0|
+-+-+-+-+
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i5 : Q = QQ[e_1,f_1,Degrees=>{{-1,1},{2,0}}];
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i6 : E1 = exactCouple(Q,M); |
i7 : expectExactCouple E1; -- No error |
i8 : E1' = E1 / E1_0; -- but expectExactCouple E1' would give the error "failure of exactness at page: ker e != im e." |
The object expectExactCouple is a method function.