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Complexes :: isShortExactSequence(Matrix,Matrix)

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

Synopsis

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)

See also