We show how to use pageModule to study the differentials in a spectral sequence. The following lines construct the homological Serre spectral sequence for the Hopf fibration S^3 \to S^2.
i1 : Q = coupleRing(ZZ,1,e,f,Degrees=>{{-1,0},{2,-2}})
o1 = Q
o1 : PolynomialRing
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i2 : declareCouple(Q, {z => {4,0}}, {x => {1,0}, y => {1,2}, w => {5,2}})
o2 = cokernel {5, 2} | e_1^2 e_1f_1 0 0 0 0 0 0 |
{1, 0} | 0 0 e_1^2 e_1f_1 0 0 0 0 |
{1, 2} | 0 0 0 0 e_1^2 e_1f_1 0 0 |
{4, 0} | 0 0 0 0 0 0 e_1^3 f_1 |
4
o2 : Q-module, quotient of Q
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i3 : C = cospan(e_1*z-f_1*y)
o3 = cokernel {5, 2} | 0 e_1^2 e_1f_1 0 0 0 0 0 0 |
{1, 0} | 0 0 0 e_1^2 e_1f_1 0 0 0 0 |
{1, 2} | -f_1 0 0 0 0 e_1^2 e_1f_1 0 0 |
{4, 0} | e_1 0 0 0 0 0 0 e_1^3 f_1 |
4
o3 : Q-module, quotient of Q
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i4 : isHomogeneous C o4 = true |
i5 : expectExactCouple C |
We extract the E^1 page as a module for D_1
i6 : prune pageModule(1,D,C)
o6 = cokernel {2, 1} | D_1 0 0 0 |
{0, 0} | 0 0 D_1 0 |
{0, 1} | 0 0 0 D_1 |
{2, 0} | 0 D_1 0 0 |
ZZ[D ] /ZZ[D ]\
1 | 1 |4
o6 : -------module, quotient of |------|
2 | 2 |
D | D |
1 \ 1 /
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Note that the differential annihilates all four generators. We now extract the E^1 page with its differential, D_2:
i7 : E2 = prune pageModule(2,D,C)
o7 = cokernel {2, 0} | 0 0 |
{2, 1} | D_2 0 |
{0, 0} | 0 D_2 |
ZZ[D ] /ZZ[D ]\
2 | 2 |3
o7 : -------module, quotient of |------|
2 | 2 |
D | D |
2 \ 2 /
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i8 : degree D_2
o8 = {-2, 1}
o8 : List
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This time, the generator in degree $(2,0)$ maps via D_2 to a nontrivial element in degree $(0,1)$. Since the module has no additional generators in that degree, the differential is an isomorphism between these two degrees. We can also compute the action of this differential directly:
i9 : structureMap({2,0}, {0,1}, D_2, pageModule(2,D,C))
o9 = | 1 |
1 1
o9 : Matrix ZZ <--- ZZ
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Computing the next page shows the cancellation:
i10 : prune pageModule(3,D,C)
o10 = cokernel {2, 1} | D_3 0 |
{0, 0} | 0 D_3 |
ZZ[D ] /ZZ[D ]\
3 | 3 |2
o10 : -------module, quotient of |------|
2 | 2 |
D | D |
3 \ 3 /
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The object pageModule is a method function.