A homomorphism $f \colon M \to N$ of $R$-modules induces a morphism of chain complexes from any free resolution of $M$ to a free resolution of $N$. This method constructs this map of chain complexes.
i1 : R = QQ[a..d]; |
i2 : I = ideal(c^2-b*d, b*c-a*d, b^2-a*c)
2 2
o2 = ideal (c - b*d, b*c - a*d, b - a*c)
o2 : Ideal of R
|
i3 : J = ideal(I_0, I_1)
2
o3 = ideal (c - b*d, b*c - a*d)
o3 : Ideal of R
|
i4 : M = R^1/J
o4 = cokernel | c2-bd bc-ad |
1
o4 : R-module, quotient of R
|
i5 : N = R^1/I
o5 = cokernel | c2-bd bc-ad b2-ac |
1
o5 : R-module, quotient of R
|
i6 : f = map(N, M, 1) o6 = | 1 | o6 : Matrix |
i7 : g = freeResolution f
1 1
o7 = 0 : R <--------- R : 0
| 1 |
3 2
1 : R <--------------- R : 1
{2} | 0 0 |
{2} | 1 0 |
{2} | 0 1 |
2 1
2 : R <-------------- R : 2
{3} | -d |
{3} | -c |
o7 : ComplexMap
|
i8 : assert isWellDefined g |
i9 : assert isComplexMorphism g |
i10 : assert(source g === freeResolution M) |
i11 : assert(target g === freeResolution N) |
Taking free resolutions is a functor, up to homotopy, from the category of modules to the category of chain complexes. In the subsequent example, the composition of the induced chain maps $g$ and $g'$ happens to be equal to the induced map of the composition.
i12 : K = ideal(I_0)
2
o12 = ideal(c - b*d)
o12 : Ideal of R
|
i13 : L = R^1/K
o13 = cokernel | c2-bd |
1
o13 : R-module, quotient of R
|
i14 : f' = map(M, L, 1) o14 = | 1 | o14 : Matrix |
i15 : g' = freeResolution f'
1 1
o15 = 0 : R <--------- R : 0
| 1 |
2 1
1 : R <------------- R : 1
{2} | 0 |
{2} | 1 |
o15 : ComplexMap
|
i16 : g'' = freeResolution(f * f')
1 1
o16 = 0 : R <--------- R : 0
| 1 |
3 1
1 : R <------------- R : 1
{2} | 0 |
{2} | 0 |
{2} | 1 |
o16 : ComplexMap
|
i17 : assert(g'' === g * g') |
i18 : assert(freeResolution id_N === id_(freeResolution N)) |
Over a quotient ring, free resolutions are often infinite. Use the optional argument LengthLimit to obtain a truncation of the map between resolutions.
i19 : S = ZZ/101[a,b] o19 = S o19 : PolynomialRing |
i20 : R = S/(a^3+b^3) o20 = R o20 : QuotientRing |
i21 : f = map(R^1/(a,b), R^1/(a^2, b^2), 1) o21 = | 1 | o21 : Matrix |
i22 : g = freeResolution(f, LengthLimit => 7)
1 1
o22 = 0 : R <--------- R : 0
| 1 |
2 2
1 : R <--------------- R : 1
{1} | a 0 |
{1} | 0 b |
2 2
2 : R <---------------- R : 2
{2} | 0 ab |
{3} | 1 0 |
2 2
3 : R <--------------- R : 3
{4} | 0 b |
{4} | a 0 |
2 2
4 : R <----------------- R : 4
{5} | 0 -ab |
{6} | 1 0 |
2 2
5 : R <--------------- R : 5
{7} | 0 b |
{7} | a 0 |
2 2
6 : R <----------------- R : 6
{8} | 0 -ab |
{9} | 1 0 |
2 2
7 : R <---------------- R : 7
{10} | 0 b |
{10} | a 0 |
o22 : ComplexMap
|
i23 : assert isWellDefined g |
i24 : assert isComplexMorphism g |