short.txt

A short exact sequence is a sequence of maps:
       f     g     h     k
    0 --> A --> B --> C --> 0
s.t.
 Im f = Ker g
 Im g = Ker h
 Im h = Ker k

Example:

       f         g         h         k
    0 --> I ∩ J --> I ⊕ J --> I + J --> 0

    Choose I = aℤ  i.e. multiples of a
           J = bℤ       multiples of b

    I ∩ J = multiples of both a and b

    I ⊕ J = pairs (am, bn) with m, n ∈ ℤ (defn. of ⊕)

    I + J = all integers of the form am + bn
          = multiples of HCF(a, b) by Euclidean algorithm.

Define:

 f 0 = 0
 g x = (x, -x)
 h (x, y) = x + y
 k x = 0

Im f = Ker k
------------
 Im f = {(0, 0)}
 The only solution to g(m, n) = 0 is (0, 0).

Im g = Ker h
------------
 Elements of I ⊕ J are (am, bn) where m, n ∈ ℤ
 Ker h = {(am, bn) s.t. am = bn}
 I.e. pairs (x, x) where x is a multiple of a and x is also a multiple of b.
 = Im g

Im h = Ker k
------------
 Im h = { am + bn | m, n ∈ ℤ }
      = all multiples of HCF(a, b) by Euclidean algorithm
      = all of I + J
      = Ker k