A short exact sequence of modules is a complex \[ 0 \to L \xrightarrow{f} M \xrightarrow{g} N \to 0\] consisting of two homomorphisms of modules $f \colon L \to M$ and $g \colon M \to N$ such that $g f = 0$, $\operatorname{image} f = \operatorname{ker} g$, $\operatorname{ker} f = 0$, and $\operatorname{coker} g = 0$.
From a homomorphism $h \colon M \to N$, one obtains a short exact sequence \[ 0 \to \operatorname{image} h \to N \to \operatorname{coker} h \to 0. \]
i1 : R = ZZ/101[a,b,c]; |
i2 : h = random(R^3, R^{4:-1})
o2 = | 24a-36b-30c -22a-29b-24c -47a-39b-18c 2a+16b+22c |
| -29a+19b+19c -38a-16b+39c -13a-43b-15c 45a-34b-48c |
| -10a-29b-8c 21a+34b+19c -28a-47b+38c -47a+47b+19c |
3 4
o2 : Matrix R <--- R
|
i3 : f = inducedMap(target h, image h)
o3 = | 24a-36b-30c -22a-29b-24c -47a-39b-18c 2a+16b+22c |
| -29a+19b+19c -38a-16b+39c -13a-43b-15c 45a-34b-48c |
| -10a-29b-8c 21a+34b+19c -28a-47b+38c -47a+47b+19c |
o3 : Matrix
|
i4 : g = inducedMap(cokernel h, target h)
o4 = | 1 0 0 |
| 0 1 0 |
| 0 0 1 |
o4 : Matrix
|
i5 : C = complex {g, f}
3
o5 = cokernel | 24a-36b-30c -22a-29b-24c -47a-39b-18c 2a+16b+22c | <-- R <-- image | 24a-36b-30c -22a-29b-24c -47a-39b-18c 2a+16b+22c |
| -29a+19b+19c -38a-16b+39c -13a-43b-15c 45a-34b-48c | | -29a+19b+19c -38a-16b+39c -13a-43b-15c 45a-34b-48c |
| -10a-29b-8c 21a+34b+19c -28a-47b+38c -47a+47b+19c | 1 | -10a-29b-8c 21a+34b+19c -28a-47b+38c -47a+47b+19c |
0 2
o5 : Complex
|
i6 : isWellDefined C o6 = true |
i7 : assert isShortExactSequence C |
i8 : assert isShortExactSequence(C[10]) |
i9 : assert not isShortExactSequence(C ++ C[6]) |
i10 : D = complex(R^1, Base=>4) ++ complex(R^1, Base=>2)
1 1
o10 = R <-- 0 <-- R
2 3 4
o10 : Complex
|
i11 : assert not isShortExactSequence D |