Given a complex $C$, this method produces the natural quasi-isomorphism from a complex $F$ all of whose terms are free modules to the complex $C$. The algorithm used minimizes the ranks of the free modules in $F$.
i1 : R = ZZ/101[a,b,c]; |
i2 : I = ideal(a^2, a*b, b*c)
2
o2 = ideal (a , a*b, b*c)
o2 : Ideal of R
|
i3 : C = Hom(freeResolution I, R^1/I)
o3 = cokernel {-3} | bc ab a2 0 0 0 | <-- cokernel {-2} | bc ab a2 0 0 0 0 0 0 | <-- cokernel | bc ab a2 |
{-3} | 0 0 0 bc ab a2 | {-2} | 0 0 0 bc ab a2 0 0 0 |
{-2} | 0 0 0 0 0 0 bc ab a2 | 0
-2
-1
o3 : Complex
|
i4 : assert all(min C .. max C, i -> not isFreeModule C_i) |
i5 : fC = resolutionMap C
2
o5 = -2 : cokernel {-3} | bc ab a2 0 0 0 | <------------------ R : -2
{-3} | 0 0 0 bc ab a2 | {-3} | -1 0 |
{-3} | 0 -1 |
9
-1 : cokernel {-2} | bc ab a2 0 0 0 0 0 0 | <-------------------------------------- R : -1
{-2} | 0 0 0 bc ab a2 0 0 0 | {-2} | 1 0 0 -c -a 0 0 0 0 |
{-2} | 0 0 0 0 0 0 bc ab a2 | {-2} | 0 -1 0 0 0 -b -a 0 0 |
{-2} | 0 0 -1 0 0 0 -c -b -a |
14
0 : cokernel | bc ab a2 | <------------------------------------ R : 0
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 |
o5 : ComplexMap
|
i6 : FC = resolution C
2 9 14 9 2
o6 = R <-- R <-- R <-- R <-- R
-2 -1 0 1 2
o6 : Complex
|
i7 : prune HH FC
o7 = cokernel {-3} | b a 0 | <-- cokernel {-1} | b a 0 0 0 0 0 0 0 0 0 0 | <-- cokernel | bc ab a2 |
{-3} | 0 -c a | {-1} | 0 -c b a 0 0 0 0 0 0 0 0 |
{-1} | 0 0 0 0 c a 0 0 0 0 0 0 | 0
-2 {-1} | 0 0 0 0 0 0 b 0 0 a 0 0 |
{-1} | 0 0 0 0 0 0 0 c a 0 0 0 |
{-1} | 0 0 0 0 0 0 0 0 0 -c b a |
-1
o7 : Complex
|
i8 : assert isQuasiIsomorphism fC |
i9 : assert isFree FC |
i10 : assert isWellDefined fC |
i11 : assert(0 == coker fC) -- showing that fC is surjective. |
The resolution of a short exact sequence is simply the zero complex.
i12 : J = ideal(a,b) o12 = ideal (a, b) o12 : Ideal of R |
i13 : K = ideal(b^2,c)
2
o13 = ideal (b , c)
o13 : Ideal of R
|
i14 : g1 = map(R^1/(J+K), R^1/J ++ R^1/K, {{1,-1}})
o14 = | 1 -1 |
o14 : Matrix
|
i15 : g2 = map(R^1/J ++ R^1/K, R^1/(intersect(J,K)), {{1},{1}})
o15 = | 1 |
| 1 |
o15 : Matrix
|
i16 : D = complex{g1, g2}
o16 = cokernel | a b b2 c | <-- cokernel | a b 0 0 | <-- cokernel | bc ac b2 |
| 0 0 b2 c |
0 2
1
o16 : Complex
|
i17 : assert isWellDefined D |
i18 : assert isShortExactSequence(g1,g2) |
i19 : fD = resolutionMap D o19 = 0 o19 : ComplexMap |
i20 : assert isWellDefined fD |
i21 : assert isQuasiIsomorphism fD |
i22 : assert(0 == source fD) -- so fD is certainly not surjective! |
To avoid recomputation, this method caches its value.
The object resolutionMap is a method function with options.