This routine checks that the face/degeneracy maps of C satisfy the simplicial identities (see isSimplicialModule for a list of these identities). If there are no degeneracy maps stored, it instead checks that the naive normalization of C is a well-defined complex. Additionally, it checks that the underlying data in C is a properly formed SimplicialModule object in Macaulay2. If the variable debugLevel is set to a value greater than zero, then information about the nature of any failure is displayed.
i1 : R = QQ[a..d];
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i2 : f0 = matrix {{-b^2+a*c, b*c-a*d, -c^2+b*d}}
o2 = | -b2+ac bc-ad -c2+bd |
1 3
o2 : Matrix R <-- R
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i3 : f1 = map(source f0,, {{d, c}, {c, b}, {b, a}})
o3 = {2} | d c |
{2} | c b |
{2} | b a |
3 2
o3 : Matrix R <-- R
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i4 : C = simplicialModule(complex {f0, f1}, 3, Degeneracy => true)
1 4 9 16
o4 = R <-- R <-- R <-- R <-- ...
0 1 2 3
o4 : SimplicialModule
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i5 : isWellDefined C
o5 = true
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i6 : dd^C
1 4
o6 = (0, 0) : R <----------------------------- R : (1, 0)
| 1 -b2+ac bc-ad -c2+bd |
1 4
(0, 1) : R <--------------- R : (1, 1)
| 1 0 0 0 |
4 9
(1, 0) : R <------------------------------------------- R : (2, 0)
{0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 d c |
{2} | 0 0 0 0 0 1 0 c b |
{2} | 0 0 0 0 0 0 1 b a |
4 9
(1, 1) : R <----------------------------- R : (2, 1)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 1 0 0 0 0 |
{2} | 0 0 1 0 0 1 0 0 0 |
{2} | 0 0 0 1 0 0 1 0 0 |
4 9
(1, 2) : R <----------------------------- R : (2, 2)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
9 16
(2, 0) : R <--------------------------------------------------------- R : (3, 0)
{0} | 1 -b2+ac bc-ad -c2+bd 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 0 d c 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 0 c b 0 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 0 b a 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 1 0 0 0 0 d c 0 0 |
{2} | 0 0 0 0 0 0 0 0 1 0 0 0 c b 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 1 0 0 b a 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
9 16
(2, 1) : R <------------------------------------------- R : (3, 1)
{0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 |
9 16
(2, 2) : R <------------------------------------------- R : (3, 2)
{0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 |
9 16
(2, 3) : R <------------------------------------------- R : (3, 3)
{0} | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
o6 : SimplicialModuleMap
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i7 : ss^C
4 1
o7 = (1, 0) : R <------------- R : (0, 0)
{0} | 1 |
{2} | 0 |
{2} | 0 |
{2} | 0 |
9 4
(2, 0) : R <------------------- R : (1, 0)
{0} | 1 0 0 0 |
{2} | 0 0 0 0 |
{2} | 0 0 0 0 |
{2} | 0 0 0 0 |
{2} | 0 1 0 0 |
{2} | 0 0 1 0 |
{2} | 0 0 0 1 |
{3} | 0 0 0 0 |
{3} | 0 0 0 0 |
9 4
(2, 1) : R <------------------- R : (1, 1)
{0} | 1 0 0 0 |
{2} | 0 1 0 0 |
{2} | 0 0 1 0 |
{2} | 0 0 0 1 |
{2} | 0 0 0 0 |
{2} | 0 0 0 0 |
{2} | 0 0 0 0 |
{3} | 0 0 0 0 |
{3} | 0 0 0 0 |
16 9
(3, 0) : R <----------------------------- R : (2, 0)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
16 9
(3, 1) : R <----------------------------- R : (2, 1)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
16 9
(3, 2) : R <----------------------------- R : (2, 2)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{2} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
{3} | 0 0 0 0 0 0 0 0 0 |
o7 : SimplicialModuleMap
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i8 : dd^C*ss^C --if C is well-defined, this should be all identity maps
1 1
o8 = (0, 0) : R <--------- R : (0, 0)
| 1 |
4 4
(1, 0) : R <------------------- R : (1, 0)
{0} | 1 0 0 0 |
{2} | 0 1 0 0 |
{2} | 0 0 1 0 |
{2} | 0 0 0 1 |
4 4
(1, 1) : R <------------------- R : (1, 1)
{0} | 1 0 0 0 |
{2} | 0 1 0 0 |
{2} | 0 0 1 0 |
{2} | 0 0 0 1 |
9 9
(2, 0) : R <----------------------------- R : (2, 0)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
9 9
(2, 1) : R <----------------------------- R : (2, 1)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
9 9
(2, 2) : R <----------------------------- R : (2, 2)
{0} | 1 0 0 0 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 0 0 0 |
{2} | 0 0 1 0 0 0 0 0 0 |
{2} | 0 0 0 1 0 0 0 0 0 |
{2} | 0 0 0 0 1 0 0 0 0 |
{2} | 0 0 0 0 0 1 0 0 0 |
{2} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
o8 : SimplicialModuleMap
|