Let k = |deg| be the total degree and G ⊂Fk the summand spanned by the generators of Fk in degree d, H ⊂Fk+1 the summand spanned by generators of degree d’ with 0 ≤d-d’ ≤n. The function returns the corresponding submatrix m: H -> G of the differential.
i1 : n={1,2};
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i2 : (S,E) = productOfProjectiveSpaces n o2 = (S, E) o2 : Sequence |
i3 : F=dual res((ker transpose vars E)**E^{{ 2,3}},LengthLimit=>4)
70 35 15 5 1
o3 = E <-- E <-- E <-- E <-- E
-4 -3 -2 -1 0
o3 : ChainComplex
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i4 : betti F
-4 -3 -2 -1 0
o4 = total: 70 35 15 5 1
0: 70 35 15 5 1
o4 : BettiTally
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i5 : tallyDegrees F
o5 = (Tally{{-1, -3} => 20}, Tally{{-1, -2} => 12}, Tally{{-1, -1} => 6},
{-2, -2} => 18 {-2, -1} => 9 {-2, 0} => 3
{-3, -1} => 12 {-3, 0} => 4 {0, -2} => 6
{-4, 0} => 5 {0, -3} => 10
{0, -4} => 15
------------------------------------------------------------------------
Tally{{-1, 0} => 2}, Tally{{0, 0} => 1})
{0, -1} => 3
o5 : Sequence
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i6 : deg={2,1}
o6 = {2, 1}
o6 : List
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i7 : m=lowerCorner(F,deg);
9 9
o7 : Matrix E <--- E
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i8 : tally degrees target m, tally degrees source m
o8 = (Tally{{-2, -1} => 9}, Tally{{-1, -1} => 6})
{-2, 0} => 3
o8 : Sequence
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i9 : Fm=(res(coker m,LengthLimit=>7))[sum deg]
9 9 8 15 32 57 91 137
o9 = E <-- E <-- E <-- E <-- E <-- E <-- E <-- E
-3 -2 -1 0 1 2 3 4
o9 : ChainComplex
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i10 : betti Fm
-3 -2 -1 0 1 2 3 4
o10 = total: 9 9 8 15 32 57 91 137
0: 9 9 5 1 . . . .
1: . . . 3 7 11 15 19
2: . . 3 11 25 45 71 103
3: . . . . . 1 5 15
o10 : BettiTally
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i11 : cohomologyMatrix(Fm,-{3,3},{4,4})
o11 = | 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 6h 3h 0 3 6 9 0 0 |
| 2h h 0 1 2 3 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 2h3 h3 0 h2 2h2 3h2 0 0 |
8 8
o11 : Matrix (ZZ[h, k]) <--- (ZZ[h, k])
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