The complex $C$ is exact if and only if the homology group $H^i(C)$ is the zero module, for all $i$. If bounds are given, then true is returned if $H^i(C) = 0$ for all $lo \le i \le hi$.
A resolution $C$ is an exact complex except in homological degree 0. The augmented complex $C'$ is exact everywhere.
i1 : S = ZZ/101[a..d]; |
i2 : I = monomialCurveIdeal(S, {1,3,4})
3 2 2 2 3 2
o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o2 : Ideal of S
|
i3 : C = freeResolution I
1 4 4 1
o3 = S <-- S <-- S <-- S
0 1 2 3
o3 : Complex
|
i4 : prune HH C
o4 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c |
0
o4 : Complex
|
i5 : assert not isExact C |
i6 : assert isExact(C, 1, infinity) |
i7 : C' = cone inducedMap(complex(S^1/I), C)[1]
1 4 4 1
o7 = cokernel | bc-ad c3-bd2 ac2-b2d b3-a2c | <-- S <-- S <-- S <-- S
-1 0 1 2 3
o7 : Complex
|
i8 : prune HH C'
o8 = 0
0
o8 : Complex
|
i9 : assert isExact C' |
The object isExact is a method function.