The chain complexes together with complex morphisms forms a category. In particular, every chain complex has an identity map.
i1 : R = ZZ/101[x,y]/(x^3, y^3) o1 = R o1 : QuotientRing |
i2 : C = freeResolution(coker vars R, LengthLimit=>6)
1 2 3 4 5 6 7
o2 = R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6
o2 : Complex
|
i3 : f = id_C
1 1
o3 = 0 : R <--------- R : 0
| 1 |
2 2
1 : R <--------------- R : 1
{1} | 1 0 |
{1} | 0 1 |
3 3
2 : R <----------------- R : 2
{2} | 1 0 0 |
{3} | 0 1 0 |
{3} | 0 0 1 |
4 4
3 : R <------------------- R : 3
{4} | 1 0 0 0 |
{4} | 0 1 0 0 |
{4} | 0 0 1 0 |
{4} | 0 0 0 1 |
5 5
4 : R <--------------------- R : 4
{5} | 1 0 0 0 0 |
{5} | 0 1 0 0 0 |
{6} | 0 0 1 0 0 |
{6} | 0 0 0 1 0 |
{6} | 0 0 0 0 1 |
6 6
5 : R <----------------------- R : 5
{7} | 1 0 0 0 0 0 |
{7} | 0 1 0 0 0 0 |
{7} | 0 0 1 0 0 0 |
{7} | 0 0 0 1 0 0 |
{7} | 0 0 0 0 1 0 |
{7} | 0 0 0 0 0 1 |
7 7
6 : R <------------------------- R : 6
{8} | 1 0 0 0 0 0 0 |
{8} | 0 1 0 0 0 0 0 |
{8} | 0 0 1 0 0 0 0 |
{9} | 0 0 0 1 0 0 0 |
{9} | 0 0 0 0 1 0 0 |
{9} | 0 0 0 0 0 1 0 |
{9} | 0 0 0 0 0 0 1 |
o3 : ComplexMap
|
i4 : assert isWellDefined f |
i5 : assert isComplexMorphism f |
The identity map corresponds to an element of the Hom complex.
i6 : R = ZZ/101[a,b,c] o6 = R o6 : PolynomialRing |
i7 : I = ideal(a^2, b^2, b*c, c^3)
2 2 3
o7 = ideal (a , b , b*c, c )
o7 : Ideal of R
|
i8 : C = freeResolution I
1 4 5 2
o8 = R <-- R <-- R <-- R
0 1 2 3
o8 : Complex
|
i9 : D = Hom(C, C)
2 13 34 46 34 13 2
o9 = R <-- R <-- R <-- R <-- R <-- R <-- R
-3 -2 -1 0 1 2 3
o9 : Complex
|
i10 : homomorphism' id_C
46 1
o10 = 0 : R <-------------- R : 0
{0} | 1 |
{0} | 1 |
{0} | 0 |
{0} | 0 |
{1} | 0 |
{0} | 0 |
{0} | 1 |
{0} | 0 |
{1} | 0 |
{0} | 0 |
{0} | 0 |
{0} | 1 |
{1} | 0 |
{-1} | 0 |
{-1} | 0 |
{-1} | 0 |
{0} | 1 |
{0} | 1 |
{1} | 0 |
{1} | 0 |
{1} | 0 |
{2} | 0 |
{-1} | 0 |
{0} | 1 |
{0} | 0 |
{0} | 0 |
{1} | 0 |
{-1} | 0 |
{0} | 0 |
{0} | 1 |
{0} | 0 |
{1} | 0 |
{-1} | 0 |
{0} | 0 |
{0} | 0 |
{0} | 1 |
{1} | 0 |
{-2} | 0 |
{-1} | 0 |
{-1} | 0 |
{-1} | 0 |
{0} | 1 |
{0} | 1 |
{1} | 0 |
{-1} | 0 |
{0} | 1 |
o10 : ComplexMap
|