= Homological algebra
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<Homological algebra> studies <chain complexes> and <cochain complexes>, their <homology> and <cohomology>, and the functors that compare them. The identity $d^2=0$ ensures that every boundary is a cycle, so the quotient measures cycles not explained by boundaries.
A basic motivation is failure of exactness. Tensoring the inclusion $2\mathbb Z\hookrightarrow\mathbb Z$ with $\mathbb Z/2\mathbb Z$ gives a zero map between two nonzero groups, so tensoring need not preserve injections. <Tor functor> measures this failure, while <Ext functor> measures related failures for <Hom functors>. <Projective resolutions> replace a module by a complex of easier modules and make these invariants computable.
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