[Verse 1]
When Hom and tensor break the chain
Exact sequences feel the strain
Not every functor plays it clean
We need a tool to read between
Derived functors catch the drift
Measuring where the gaps persist
Right side climbing, left side down
Homology wears the crown
[Chorus]
Ext and Tor, they tell the story
Of what's lost in algebraic glory
Ext extends what Hom can't see
Tor reveals the tensor debris
Projective resolution's the key
Apply and count what's left to be
Zero, one, two up the ladder
Cohomology that matters
[Verse 2]
Ext to the nth from Hom derived
Take your module, get it primed
Projective cover, strip it bare
Apply Hom and see what's there
Ext of zero equals Hom
Ext of one classifies the bomb
Extensions trapped between the lines
Zero arrow N to E to M defines
[Chorus]
Ext and Tor, they tell the story
Of what's lost in algebraic glory
Ext extends what Hom can't see
Tor reveals the tensor debris
Projective resolution's the key
Apply and count what's left to be
Zero, one, two up the ladder
Cohomology that matters
[Bridge]
When all your Ext just disappears
Projective modules volunteer
When Tor evaporates away
Your module's flat, hooray hooray
[Verse 3]
Tor descends from tensor's throne
Left derived, stands alone
Tensor M with N in place
Resolution shows the chase
Tor of cyclic groups collide
Z mod m with n beside
Gives you gcd wrapped tight
Greatest common divisor's might
[Final Chorus]
Ext and Tor, the derived pair
Measuring functors everywhere
When exactness starts to fray
Homological saves the day
Resolution, tensor, count
That's what derived's about
Category theory's finest art
Algebra's beating heart