Two complexes are equal if the corresponding objects and corresponding maps at each index are equal.
i1 : S = ZZ/101[a..c] o1 = S o1 : PolynomialRing |
i2 : C = freeResolution coker vars S
1 3 3 1
o2 = S <-- S <-- S <-- S
0 1 2 3
o2 : Complex
|
i3 : D = C[3][-3]
1 3 3 1
o3 = S <-- S <-- S <-- S
0 1 2 3
o3 : Complex
|
i4 : C === D o4 = false |
i5 : C == D o5 = true |
Both the maps and the objects must be equal.
i6 : (lo,hi) = concentration C o6 = (0, 3) o6 : Sequence |
i7 : E = complex for i from lo+1 to hi list 0*dd^C_i
1 3 3 1
o7 = S <-- S <-- S <-- S
0 1 2 3
o7 : Complex
|
i8 : dd^E
1 3
o8 = 0 : S <----- S : 1
0
3 3
1 : S <----- S : 2
0
3 1
2 : S <----- S : 3
0
o8 : ComplexMap
|
i9 : C == E o9 = false |
i10 : E == 0 o10 = false |
A complex is equal to zero if all the objects and maps are zero. This could require computation to determine if something that is superficially not zero is in fact zero.
i11 : f = id_C
1 1
o11 = 0 : S <--------- S : 0
| 1 |
3 3
1 : S <----------------- S : 1
{1} | 1 0 0 |
{1} | 0 1 0 |
{1} | 0 0 1 |
3 3
2 : S <----------------- S : 2
{2} | 1 0 0 |
{2} | 0 1 0 |
{2} | 0 0 1 |
1 1
3 : S <------------- S : 3
{3} | 1 |
o11 : ComplexMap
|
i12 : D = coker f
o12 = cokernel | 1 | <-- cokernel {1} | 1 0 0 | <-- cokernel {2} | 1 0 0 | <-- cokernel {3} | 1 |
{1} | 0 1 0 | {2} | 0 1 0 |
0 {1} | 0 0 1 | {2} | 0 0 1 | 3
1 2
o12 : Complex
|
i13 : D == 0 o13 = true |
i14 : C0 = complex S^0
o14 = 0
0
o14 : Complex
|
i15 : C1 = C0[4]
o15 = 0
-4
o15 : Complex
|
i16 : concentration C0 == concentration C1 o16 = false |
i17 : C0 == C1 o17 = true |
i18 : C0 == 0 o18 = true |
i19 : C1 == 0 o19 = true |
Testing for equality is not the same testing for isomorphism. In particular, different presentations of a complex need not be equal.
i20 : R = QQ[a..d]; |
i21 : f0 = matrix {{-b^2+a*c, b*c-a*d, -c^2+b*d}}
o21 = | -b2+ac bc-ad -c2+bd |
1 3
o21 : Matrix R <--- R
|
i22 : f1 = map(source f0,, {{d, c}, {c, b}, {b, a}})
o22 = {2} | d c |
{2} | c b |
{2} | b a |
3 2
o22 : Matrix R <--- R
|
i23 : C = complex {f0, f1}
1 3 2
o23 = R <-- R <-- R
0 1 2
o23 : Complex
|
i24 : HH C != complex coker f0 o24 = true |
i25 : prune HH C == complex coker f0 o25 = true |