Here we provide an easy example of a filtered simplicial complex and the resulting spectral sequence. This example is small enough that all aspects of it can be explicitly computed by hand.
i1 : A = QQ[a,b,c,d]; |
i2 : D = simplicialComplex {a*d*c, a*b, a*c, b*c}; |
i3 : F2D = D o3 = | acd bc ab | o3 : SimplicialComplex |
i4 : F1D = simplicialComplex {a*c, d} o4 = | d ac | o4 : SimplicialComplex |
i5 : F0D = simplicialComplex {a,d} o5 = | d a | o5 : SimplicialComplex |
i6 : K= filteredComplex({F2D, F1D, F0D},ReducedHomology => false) o6 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 -1 0 1 2 0 : image 0 <-- image | 1 0 | <-- image 0 <-- image 0 | 0 0 | -1 | 0 0 | 1 2 | 0 1 | 0 1 : image 0 <-- image | 1 0 0 | <-- image | 0 | <-- image 0 | 0 0 0 | | 1 | -1 | 0 1 0 | | 0 | 2 | 0 0 1 | | 0 | | 0 | 0 1 4 5 1 2 : image 0 <-- QQ <-- QQ <-- QQ -1 0 1 2 o6 : FilteredComplex |
i7 : E = prune spectralSequence(K) o7 = E o7 : SpectralSequence |
i8 : E^0 +-------+-------+-------+ | 2 | 1 | 1 | o8 = |QQ |QQ |QQ | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | 1 | 4 | |0 |QQ |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -2}|{1, -2}|{2, -2}| +-------+-------+-------+ o8 : SpectralSequencePage |
i9 : E^1 +-------+-------+-------+ | 2 | | | o9 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 2 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o9 : SpectralSequencePage |
i10 : E^2 +-------+-------+-------+ | 2 | | | o10 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 2 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o10 : SpectralSequencePage |
i11 : E^3 +-------+-------+-------+ | 1 | | | o11 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o11 : SpectralSequencePage |
i12 : E^infinity +-------+-------+-------+ | 1 | | | o12 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o12 : Page |
i13 : C = K_infinity 4 5 1 o13 = image 0 <-- QQ <-- QQ <-- QQ -1 0 1 2 o13 : ChainComplex |
i14 : prune HH C o14 = -1 : 0 1 0 : QQ 1 1 : QQ 2 : 0 o14 : GradedModule |
i15 : E^2 .dd o15 = {-3, 2} : 0 <----- 0 : {-1, 1} 0 {-3, 3} : 0 <----- 0 : {-1, 2} 0 {-3, 4} : 0 <----- 0 : {-1, 3} 0 {0, -2} : 0 <----- 0 : {2, -3} 0 {0, -1} : 0 <----- 0 : {2, -2} 0 2 2 {0, 0} : QQ <------------ QQ : {2, -1} | 0 1 | | 0 -1 | {0, 1} : 0 <----- 0 : {2, 0} 0 {-1, -1} : 0 <----- 0 : {1, -2} 0 {-1, 0} : 0 <----- 0 : {1, -1} 0 {-1, 1} : 0 <----- 0 : {1, 0} 0 {-1, 2} : 0 <----- 0 : {1, 1} 0 {-2, 0} : 0 <----- 0 : {0, -1} 0 2 {-2, 1} : 0 <----- QQ : {0, 0} 0 {-2, 2} : 0 <----- 0 : {0, 1} 0 {-2, 3} : 0 <----- 0 : {0, 2} 0 {-3, 1} : 0 <----- 0 : {-1, 0} 0 o15 : SpectralSequencePageMap |
Considering the $E^2$ and $E^3$ pages of the spectral sequence we conclude that the map $d^2_{2,-1}$ must have a $1$-dimensional image and a $1$-dimensinal kernel. This can be verified easily:
i16 : rank ker E^2 .dd_{2,-1} o16 = 1 |
i17 : rank image E^2 .dd_{2,-1} o17 = 1 |