Both ChainComplexMap and ComplexMap are Macaulay2 types that implement maps between chain complexes. The plan is to replace ChainComplexMap with this new type. Before this happens, this function allows interoperability between these types.
The first example is the minimal free resolution of the twisted cubic curve.
i1 : R = ZZ/32003[a..d]; |
i2 : I = monomialCurveIdeal(R, {1,2,3})
2 2
o2 = ideal (c - b*d, b*c - a*d, b - a*c)
o2 : Ideal of R
|
i3 : M = R^1/I
o3 = cokernel | c2-bd bc-ad b2-ac |
1
o3 : R-module, quotient of R
|
i4 : C = resolution M
1 3 2
o4 = R <-- R <-- R <-- 0
0 1 2 3
o4 : ChainComplex
|
i5 : f = C.dd
1 3
o5 = 0 : R <------------------------- R : 1
| b2-ac bc-ad c2-bd |
3 2
1 : R <----------------- R : 2
{2} | -c d |
{2} | b -c |
{2} | -a b |
2
2 : R <----- 0 : 3
0
o5 : ChainComplexMap
|
i6 : g = complex f
1 3
o6 = 0 : R <------------------------- R : 1
| b2-ac bc-ad c2-bd |
3 2
1 : R <----------------- R : 2
{2} | -c d |
{2} | b -c |
{2} | -a b |
o6 : ComplexMap
|
i7 : isWellDefined g o7 = true |
i8 : D = freeResolution M
1 3 2
o8 = R <-- R <-- R
0 1 2
o8 : Complex
|
i9 : assert(D.dd == g) |
The following two extension of maps between modules to maps between chain complexes agree.
i10 : J = ideal vars R o10 = ideal (a, b, c, d) o10 : Ideal of R |
i11 : C1 = resolution(R^1/J)
1 4 6 4 1
o11 = R <-- R <-- R <-- R <-- R <-- 0
0 1 2 3 4 5
o11 : ChainComplex
|
i12 : D1 = freeResolution(R^1/J)
1 4 6 4 1
o12 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o12 : Complex
|
i13 : f = extend(C1, C, matrix{{1_R}})
1 1
o13 = 0 : R <--------- R : 0
| 1 |
4 3
1 : R <-------------------- R : 1
{1} | 0 0 0 |
{1} | b 0 0 |
{1} | -a b c |
{1} | 0 -a -b |
6 2
2 : R <---------------- R : 2
{2} | 0 0 |
{2} | 0 0 |
{2} | b 0 |
{2} | 0 0 |
{2} | 0 -b |
{2} | 0 a |
4
3 : R <----- 0 : 3
0
o13 : ChainComplexMap
|
i14 : g = complex f
1 1
o14 = 0 : R <--------- R : 0
| 1 |
4 3
1 : R <-------------------- R : 1
{1} | 0 0 0 |
{1} | b 0 0 |
{1} | -a b c |
{1} | 0 -a -b |
6 2
2 : R <---------------- R : 2
{2} | 0 0 |
{2} | 0 0 |
{2} | b 0 |
{2} | 0 0 |
{2} | 0 -b |
{2} | 0 a |
o14 : ComplexMap
|
i15 : g1 = extend(D1, D, matrix{{1_R}})
1 1
o15 = 0 : R <--------- R : 0
| 1 |
4 3
1 : R <-------------------- R : 1
{1} | 0 0 0 |
{1} | b 0 0 |
{1} | -a b c |
{1} | 0 -a -b |
6 2
2 : R <---------------- R : 2
{2} | 0 0 |
{2} | 0 0 |
{2} | b 0 |
{2} | 0 0 |
{2} | 0 -b |
{2} | 0 a |
o15 : ComplexMap
|
i16 : assert(g == g1) |
This is a temporary method to allow comparisons among the data types, and will be removed once the older data structure is replaced