isShortExactSequence(Matrix,Matrix) -- whether a pair of matrices forms a short exact sequence

Synopsis

Function: isShortExactSequence
Usage:

isShortExactSequence(g, f)
Inputs:
- f, a matrix,
- g, a matrix,
Outputs:
- a Boolean value, that is true if these form a short exact sequence

Description

A short exact sequence of modules \[ 0 \to L \xrightarrow{f} M \xrightarrow{g} N \to 0\] consists 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 : assert isShortExactSequence(g,f)

Ideal quotients also give rise to short exact sequences.

i6 : I = ideal(a^3, b^3, c^3)

             3   3   3
o6 = ideal (a , b , c )

o6 : Ideal of R

i7 : J = I + ideal(a*b*c)

             3   3   3
o7 = ideal (a , b , c , a*b*c)

o7 : Ideal of R

i8 : K = I : ideal(a*b*c)

             2   2   2
o8 = ideal (c , b , a )

o8 : Ideal of R

i9 : g = map(comodule J, comodule I, 1)

o9 = | 1 |

o9 : Matrix

i10 : f = map(comodule I, (comodule K) ** R^{-3}, {{a*b*c}})

o10 = | abc |

o10 : Matrix

i11 : assert isShortExactSequence(g,f)

Ways to use this method: