As a first example, consider two Koszul complexes $C$ and $D$. From a random map $f : R^1 \to Hom(C, D)$, we construct the corresponding map of chain complexes $g : C \to D$.
i1 : R = ZZ/101[a,b,c] o1 = R o1 : PolynomialRing |
i2 : C = freeResolution ideal"a,b,c"
1 3 3 1
o2 = R <-- R <-- R <-- R
0 1 2 3
o2 : Complex
|
i3 : D = freeResolution ideal"a2,b2,c2"
1 3 3 1
o3 = R <-- R <-- R <-- R
0 1 2 3
o3 : Complex
|
i4 : g = randomComplexMap(D, C, InternalDegree => 2)
1 1
o4 = 0 : R <------------------------------------- R : 0
| 24a2-36ab-29b2-30ac+19bc+19c2 |
3 3
1 : R <-------------------------------------------------- R : 1
{2} | -10a-29b-8c 21a+34b+19c -28a-47b+38c |
{2} | -22a-29b-24c -47a-39b-18c 2a+16b+22c |
{2} | -38a-16b+39c -13a-43b-15c 45a-34b-48c |
3 3
2 : R <----------------------- R : 2
{4} | -47 -16 -23 |
{4} | 47 7 39 |
{4} | 19 15 43 |
1 1
3 : R <----- R : 3
0
o4 : ComplexMap
|
i5 : isWellDefined g o5 = true |
i6 : f = homomorphism' g
20 1
o6 = 0 : R <----------------------------------------- R : 0
{0} | 24a2-36ab-29b2-30ac+19bc+19c2 |
{1} | -10a-29b-8c |
{1} | -22a-29b-24c |
{1} | -38a-16b+39c |
{1} | 21a+34b+19c |
{1} | -47a-39b-18c |
{1} | -13a-43b-15c |
{1} | -28a-47b+38c |
{1} | 2a+16b+22c |
{1} | 45a-34b-48c |
{2} | -47 |
{2} | 47 |
{2} | 19 |
{2} | -16 |
{2} | 7 |
{2} | 15 |
{2} | -23 |
{2} | 39 |
{2} | 43 |
{3} | 0 |
o6 : ComplexMap
|
i7 : isWellDefined f o7 = true |
The map $g : C \to D$ corresponding to a random map into $Hom(C,D)$ does not generally commute with the differentials. However, if the element of $Hom(C,D)$ is a cycle, then the corresponding map does commute.
i8 : g = randomComplexMap(D, C, Cycle => true, InternalDegree => 3)
1 1
o8 = 0 : R <---------------------------------------------------------------- R : 0
| -17a3-11a2b+36ab2+35b3+48a2c+2abc+11b2c-38ac2+33bc2+40c3 |
3 3
1 : R <------------------------------------------------------------------------------------------------- R : 1
{2} | -17a2-11ab-11b2+48ac+2bc-46c2 -17ab-12b2+48bc-22c2 3b2-17ac-11bc-6c2 |
{2} | 47a2+35ab+11ac+28c2 a2+36ab+35b2+2ac+11bc+23c2 -3a2+36ac+35bc+18c2 |
{2} | 8a2+33ab-28b2+40ac 22a2-38ab+10b2+40bc -47a2+2ab-7b2-38ac+33bc+40c2 |
3 3
2 : R <------------------------------------------ R : 2
{4} | a-11b+2c -3a-11c -3b-c |
{4} | 22a-46b -47a+2b-46c -47b-22c |
{4} | -23a+28b -7a+28c 2a-7b+23c |
1 1
3 : R <------------- R : 3
{6} | 2 |
o8 : ComplexMap
|
i9 : isWellDefined g o9 = true |
i10 : f = homomorphism' g
20 1
o10 = 0 : R <-------------------------------------------------------------------- R : 0
{0} | -17a3-11a2b+36ab2+35b3+48a2c+2abc+11b2c-38ac2+33bc2+40c3 |
{1} | -17a2-11ab-11b2+48ac+2bc-46c2 |
{1} | 47a2+35ab+11ac+28c2 |
{1} | 8a2+33ab-28b2+40ac |
{1} | -17ab-12b2+48bc-22c2 |
{1} | a2+36ab+35b2+2ac+11bc+23c2 |
{1} | 22a2-38ab+10b2+40bc |
{1} | 3b2-17ac-11bc-6c2 |
{1} | -3a2+36ac+35bc+18c2 |
{1} | -47a2+2ab-7b2-38ac+33bc+40c2 |
{2} | a-11b+2c |
{2} | 22a-46b |
{2} | -23a+28b |
{2} | -3a-11c |
{2} | -47a+2b-46c |
{2} | -7a+28c |
{2} | -3b-c |
{2} | -47b-22c |
{2} | 2a-7b+23c |
{3} | 2 |
o10 : ComplexMap
|
i11 : isWellDefined f o11 = true |
i12 : assert isCommutative g |
i13 : assert(degree f === 0) |
i14 : assert(source f == complex(R^{-3}))
|
i15 : assert(target g === D) |
i16 : assert(homomorphism f == g) |