Returns a new complex which drops (sets to zero) all modules outside the given range.
i1 : R = ZZ/101[a,b,c,d,e]; |
i2 : I = intersect(ideal(a,b),ideal(c,d,e)) o2 = ideal (b*e, a*e, b*d, a*d, b*c, a*c) o2 : Ideal of R |
i3 : C = freeResolution I
1 6 9 5 1
o3 = R <-- R <-- R <-- R <-- R
0 1 2 3 4
o3 : Complex
|
i4 : naiveTruncation(C, 1, 2)
6 9
o4 = R <-- R
1 2
o4 : Complex
|
i5 : C16 = naiveTruncation(C, 1, 6)
6 9 5 1
o5 = R <-- R <-- R <-- R
1 2 3 4
o5 : Complex
|
i6 : isWellDefined C16 o6 = true |
i7 : naiveTruncation(C, 1, infinity)
6 9 5 1
o7 = R <-- R <-- R <-- R
1 2 3 4
o7 : Complex
|
i8 : naiveTruncation(C, -13, 2)
1 6 9
o8 = R <-- R <-- R
0 1 2
o8 : Complex
|
i9 : naiveTruncation(C, -infinity, 2)
1 6 9
o9 = R <-- R <-- R
0 1 2
o9 : Complex
|
i10 : naiveTruncation(C, , 2)
1 6 9
o10 = R <-- R <-- R
0 1 2
o10 : Complex
|
Truncation gives rise to a natural short exact sequence of complexes.
i11 : C' = naiveTruncation(C,, 1)
1 6
o11 = R <-- R
0 1
o11 : Complex
|
i12 : C'' = naiveTruncation(C, 2, infinity)
9 5 1
o12 = R <-- R <-- R
2 3 4
o12 : Complex
|
i13 : f = inducedMap(C, C')
1 1
o13 = 0 : R <--------- R : 0
| 1 |
6 6
1 : R <----------------------- R : 1
{2} | 1 0 0 0 0 0 |
{2} | 0 1 0 0 0 0 |
{2} | 0 0 1 0 0 0 |
{2} | 0 0 0 1 0 0 |
{2} | 0 0 0 0 1 0 |
{2} | 0 0 0 0 0 1 |
o13 : ComplexMap
|
i14 : g = inducedMap(C'', C)
9 9
o14 = 2 : R <----------------------------- R : 2
{3} | 1 0 0 0 0 0 0 0 0 |
{3} | 0 1 0 0 0 0 0 0 0 |
{3} | 0 0 1 0 0 0 0 0 0 |
{3} | 0 0 0 1 0 0 0 0 0 |
{3} | 0 0 0 0 1 0 0 0 0 |
{3} | 0 0 0 0 0 1 0 0 0 |
{3} | 0 0 0 0 0 0 1 0 0 |
{3} | 0 0 0 0 0 0 0 1 0 |
{3} | 0 0 0 0 0 0 0 0 1 |
5 5
3 : R <--------------------- R : 3
{4} | 1 0 0 0 0 |
{4} | 0 1 0 0 0 |
{4} | 0 0 1 0 0 |
{4} | 0 0 0 1 0 |
{4} | 0 0 0 0 1 |
1 1
4 : R <------------- R : 4
{5} | 1 |
o14 : ComplexMap
|
i15 : assert isShortExactSequence(g,f) |
There is another type of truncation, canonicalTruncation, which induces an isomorphism on (a range) of homology.