We describe the most primitative way to create filtered complexes.
Let C be a chain complex and consider a list of chain complex maps {φn, φn - 1, …, φ0 } with properties that C is the target of φi, for 0 ≤i ≤n, and the image of φi-1 is a subchain complex of the image of φi, for 1 ≤i ≤n. Given this input data we produce an ascending filtered chain complex FC with the properties that Fk C = C for k ≥n + 1 and Fk C = image φk, for k = 0, …, n.
We now illustrate how this is done in two easy examples. We first make three chain complexes C, D, and E, two chain complex maps, d : D →C and e : E →C, and then compute the resulting filtration of C.
Let’s make our chain complexes C, D, and E.
i1 : R = QQ[x,y,z,w] ; |
i2 : c2 = matrix(R,{{1},{0}}) ;
2 1
o2 : Matrix R <--- R
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i3 : c1 = matrix(R,{{0,1}}) ;
1 2
o3 : Matrix R <--- R
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i4 : C = chainComplex({c1,c2})
1 2 1
o4 = R <-- R <-- R
0 1 2
o4 : ChainComplex
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i5 : D_2 = image matrix(R,{{1}});
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i6 : D_1 = image matrix(R,{{1,0},{0,0}});
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i7 : D_0 = image matrix(R,{{1}});
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i8 : D = chainComplex({inducedMap(D_0,D_1,C.dd_1),inducedMap(D_1,D_2,C.dd_2)})
o8 = image | 1 | <-- image | 1 0 | <-- image | 1 |
| 0 0 |
0 2
1
o8 : ChainComplex
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i9 : E_2 = image matrix(R,{{0}});
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i10 : E_1 = image matrix(R,{{1,0},{0,0}});
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i11 : E_0 = image matrix(R,{{1}});
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i12 : E = chainComplex({inducedMap(E_0,E_1,C.dd_1),inducedMap(E_1,E_2,C.dd_2)})
o12 = image | 1 | <-- image | 1 0 | <-- image 0
| 0 0 |
0 2
1
o12 : ChainComplex
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We now make our chain complex maps.
i13 : d = chainComplexMap(C,D,apply(spots C, i-> inducedMap(C_i,D_i,id_C _i)))
1
o13 = 0 : R <--------- image | 1 | : 0
| 1 |
2
1 : R <----------- image | 1 0 | : 1
| 1 0 | | 0 0 |
| 0 0 |
1
2 : R <--------- image | 1 | : 2
| 1 |
o13 : ChainComplexMap
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i14 : e = chainComplexMap(C,E,apply(spots C, i->inducedMap(C_i,E_i, id_C _i)))
1
o14 = 0 : R <--------- image | 1 | : 0
| 1 |
2
1 : R <----------- image | 1 0 | : 1
| 1 0 | | 0 0 |
| 0 0 |
1
2 : R <----- image 0 : 2
0
o14 : ChainComplexMap
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We can check that these are indeed chain complex maps:
i15 : isChainComplexMap d o15 = true |
i16 : isChainComplexMap e o16 = true |
Now, given the list of chain complex maps {d, e}, we obtain a filtration of C by:
i17 : K = filteredComplex({d,e})
o17 = -1 : image 0 <-- image 0 <-- image 0
0 1 2
0 : image | 1 | <-- image | 1 0 | <-- image 0
| 0 0 |
0 2
1
1 : image | 1 | <-- image | 1 0 | <-- image | 1 |
| 0 0 |
0 2
1
1 2 1
2 : R <-- R <-- R
0 1 2
o17 : FilteredComplex
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If we want to specify a minimum filtration degree we can use the Shift option.
i18 : L = filteredComplex({d,e},Shift =>1)
o18 = -2 : image 0 <-- image 0 <-- image 0
0 1 2
-1 : image | 1 | <-- image | 1 0 | <-- image 0
| 0 0 |
0 2
1
0 : image | 1 | <-- image | 1 0 | <-- image | 1 |
| 0 0 |
0 2
1
1 2 1
1 : R <-- R <-- R
0 1 2
o18 : FilteredComplex
|
i19 : M = filteredComplex({d,e},Shift =>-1)
o19 = 0 : image 0 <-- image 0 <-- image 0
0 1 2
1 : image | 1 | <-- image | 1 0 | <-- image 0
| 0 0 |
0 2
1
2 : image | 1 | <-- image | 1 0 | <-- image | 1 |
| 0 0 |
0 2
1
1 2 1
3 : R <-- R <-- R
0 1 2
o19 : FilteredComplex
|