[Verse 1]
In the kernel lives the secret, where the boundary maps dissolve
Quotient spaces tell the story that differential forms evolve
H sub n of our complex equals kernel over image space
Where d of n plus one below cuts the cycles we can trace
[Chorus]
Kernel over image, that's the homological way
D squared equals zero, fundamental truth today
Ext and Tor at zero, homotopy's embrace
Short exact goes long exact, chase the diagram's grace
[Verse 2]
When you compose the boundary twice, something magical occurs
Zero is your answer, no matter how the sequence stirs
Differential squared to nothing, chain complexes love this rule
Exactness flows from this foundation, algebraic jewel
[Chorus]
Kernel over image, that's the homological way
D squared equals zero, fundamental truth today
Ext and Tor at zero, homotopy's embrace
Short exact goes long exact, chase the diagram's grace
[Bridge]
Extension functors start with homomorphisms plain
Ext of zero gives you back where morphisms reign
Torsion products begin with tensor, pure and clean
Tor at zero level shows what multiplication means
[Verse 3]
Five terms break to infinity when exact sequences bloom
Long and winding homology fills up every room
Connecting morphisms weave between the cohomology grades
Snake lemma guides the journey through these algebraic trades
[Chorus]
Kernel over image, that's the homological way
D squared equals zero, fundamental truth today
Ext and Tor at zero, homotopy's embrace
Short exact goes long exact, chase the diagram's grace
[Outro]
Four equations bind the future of commutative dreams
Homological algebra flowing through these abstract streams