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]; |
i2 : D = simplicialComplex({a*b*c}) o2 = | abc | o2 : SimplicialComplex |
i3 : F3D = D; |
i4 : F2D = simplicialComplex({a*b,a*c,b*c}) o4 = | bc ac ab | o4 : SimplicialComplex |
i5 : F1D = simplicialComplex({a*b,c}) o5 = | c ab | o5 : SimplicialComplex |
i6 : F0D = simplicialComplex({a,b}) o6 = | b a | o6 : SimplicialComplex |
i7 : K = filteredComplex({F3D,F2D,F1D,F0D}, ReducedHomology => false) o7 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 -1 0 1 2 0 : image 0 <-- image | 1 0 | <-- image 0 <-- image 0 | 0 1 | -1 | 0 0 | 1 2 0 1 : image 0 <-- image | 1 0 0 | <-- image | 1 | <-- image 0 | 0 1 0 | | 0 | -1 | 0 0 1 | | 0 | 2 0 1 2 : image 0 <-- image | 1 0 0 | <-- image | 1 0 0 | <-- image 0 | 0 1 0 | | 0 1 0 | -1 | 0 0 1 | | 0 0 1 | 2 0 1 3 3 1 3 : image 0 <-- QQ <-- QQ <-- QQ -1 0 1 2 o7 : FilteredComplex |
i8 : E = prune spectralSequence K o8 = E o8 : SpectralSequence |
i9 : E^0 +-------+-------+-------+-------+ | 2 | 1 | | | o9 = |QQ |QQ |0 |0 | | | | | | |{0, 0} |{1, 0} |{2, 0} |{3, 0} | +-------+-------+-------+-------+ | | 1 | 2 | 1 | |0 |QQ |QQ |QQ | | | | | | |{0, -1}|{1, -1}|{2, -1}|{3, -1}| +-------+-------+-------+-------+ o9 : SpectralSequencePage |
i10 : E^0 .dd o10 = {3, -4} : 0 <----- 0 : {3, -3} 0 {-1, 0} : 0 <----- 0 : {-1, 1} 0 {3, -3} : 0 <----- 0 : {3, -2} 0 {-1, 1} : 0 <----- 0 : {-1, 2} 0 1 {3, -2} : 0 <----- QQ : {3, -1} 0 {-1, 2} : 0 <----- 0 : {-1, 3} 0 {2, -4} : 0 <----- 0 : {2, -3} 0 {2, -3} : 0 <----- 0 : {2, -2} 0 2 {2, -2} : 0 <----- QQ : {2, -1} 0 2 {2, -1} : QQ <----- 0 : {2, 0} 0 {1, -3} : 0 <----- 0 : {1, -2} 0 1 {1, -2} : 0 <----- QQ : {1, -1} 0 1 1 {1, -1} : QQ <----- QQ : {1, 0} 0 1 {1, 0} : QQ <----- 0 : {1, 1} 0 {0, -2} : 0 <----- 0 : {0, -1} 0 2 {0, -1} : 0 <----- QQ : {0, 0} 0 2 {0, 0} : QQ <----- 0 : {0, 1} 0 {0, 1} : 0 <----- 0 : {0, 2} 0 {3, -5} : 0 <----- 0 : {3, -4} 0 {-1, -1} : 0 <----- 0 : {-1, 0} 0 o10 : SpectralSequencePageMap |
i11 : E^0 +-------+-------+-------+-------+ | 2 | 1 | | | o11 = |QQ |QQ |0 |0 | | | | | | |{0, 0} |{1, 0} |{2, 0} |{3, 0} | +-------+-------+-------+-------+ | | 1 | 2 | 1 | |0 |QQ |QQ |QQ | | | | | | |{0, -1}|{1, -1}|{2, -1}|{3, -1}| +-------+-------+-------+-------+ o11 : SpectralSequencePage |
i12 : E^1 +-------+-------+-------+-------+ | 2 | 1 | | | o12 = |QQ |QQ |0 |0 | | | | | | |{0, 0} |{1, 0} |{2, 0} |{3, 0} | +-------+-------+-------+-------+ | | 1 | 2 | 1 | |0 |QQ |QQ |QQ | | | | | | |{0, -1}|{1, -1}|{2, -1}|{3, -1}| +-------+-------+-------+-------+ o12 : SpectralSequencePage |
i13 : E^0 .dd_{1,0} o13 = 0 1 1 o13 : Matrix QQ <--- QQ |
i14 : E^1 .dd o14 = {2, -3} : 0 <----- 0 : {3, -3} 0 {-2, 1} : 0 <----- 0 : {-1, 1} 0 {2, -2} : 0 <----- 0 : {3, -2} 0 {-2, 2} : 0 <----- 0 : {-1, 2} 0 2 1 {2, -1} : QQ <---------- QQ : {3, -1} | 1 | | -1 | {-2, 3} : 0 <----- 0 : {-1, 3} 0 {1, -3} : 0 <----- 0 : {2, -3} 0 {1, -2} : 0 <----- 0 : {2, -2} 0 1 2 {1, -1} : QQ <------------- QQ : {2, -1} | -1 -1 | 1 {1, 0} : QQ <----- 0 : {2, 0} 0 {0, -2} : 0 <----- 0 : {1, -2} 0 1 {0, -1} : 0 <----- QQ : {1, -1} 0 2 1 {0, 0} : QQ <---------- QQ : {1, 0} | 1 | | -1 | {0, 1} : 0 <----- 0 : {1, 1} 0 {-1, -1} : 0 <----- 0 : {0, -1} 0 2 {-1, 0} : 0 <----- QQ : {0, 0} 0 {-1, 1} : 0 <----- 0 : {0, 1} 0 {-1, 2} : 0 <----- 0 : {0, 2} 0 {2, -4} : 0 <----- 0 : {3, -4} 0 {-2, 0} : 0 <----- 0 : {-1, 0} 0 o14 : SpectralSequencePageMap |
i15 : E^1 +-------+-------+-------+-------+ | 2 | 1 | | | o15 = |QQ |QQ |0 |0 | | | | | | |{0, 0} |{1, 0} |{2, 0} |{3, 0} | +-------+-------+-------+-------+ | | 1 | 2 | 1 | |0 |QQ |QQ |QQ | | | | | | |{0, -1}|{1, -1}|{2, -1}|{3, -1}| +-------+-------+-------+-------+ o15 : SpectralSequencePage |
i16 : E^0 +-------+-------+-------+-------+ | 2 | 1 | | | o16 = |QQ |QQ |0 |0 | | | | | | |{0, 0} |{1, 0} |{2, 0} |{3, 0} | +-------+-------+-------+-------+ | | 1 | 2 | 1 | |0 |QQ |QQ |QQ | | | | | | |{0, -1}|{1, -1}|{2, -1}|{3, -1}| +-------+-------+-------+-------+ o16 : SpectralSequencePage |
i17 : E^2 +------+ | 1 | o17 = |QQ | | | |{0, 0}| +------+ o17 : SpectralSequencePage |
i18 : prune HH K_infinity o18 = -1 : 0 1 0 : QQ 1 : 0 2 : 0 o18 : GradedModule |
i19 : E^infinity +------+ | 1 | o19 = |QQ | | | |{0, 0}| +------+ o19 : Page |