Homological algebra mines chain complexes for hidden invariants
String modules or spaces into a sequence of maps whose consecutive composites vanish. The failure of exactness—cycles taken modulo boundaries—is homology. That bookkeeping, born from Poincaré and Hilbert, now runs through topology, geometry, and far beyond.
Homological algebra studies homology in a general algebraic setting. Its late-nineteenth-century roots lie in combinatorial topology and in module theory and syzygies, chiefly with Henri Poincaré and David Hilbert. The subject matured with category theory and with functors such as Ext and Tor in the 1940s. Chain complexes are central: sequences of objects and differentials dn with dn ∘ dn+1 = 0. Cycles are kernels, boundaries are images of the next differential, and the nth homology is cycles modulo boundaries. If all homology vanishes, the complex is exact or acyclic. Mitchell's theorem lets arguments proven for abelian groups extend to abelian categories.
Examples flood in. Singular chains on a space, simplicial chains on a complex, and presentations of abelian groups by generators and relations all yield complexes whose homology reflects the original object—singular homology being fundamental for manifolds. Homological algebra extracts invariants of rings, modules, and spaces; spectral sequences are heavy machinery for computation. Philosophically, the complex holds more information than the homology alone; technically, the subject studies how maps between objects induce maps of complexes and how different presentations yield the same homology.
Influence now spans algebraic topology, commutative algebra, algebraic geometry, number theory, the study of representations, mathematical physics, operator algebras, complex analysis, and PDEs. K-theory and Alain Connes's noncommutative geometry draw on the same methods. Cohomology theories exist for spaces, sheaves, groups, rings, Lie algebras, and C*-algebras; modern algebraic geometry is nearly unthinkable without sheaf cohomology. What began as counting holes became a universal language for measuring how algebraic sequences fail to be exact.
Source: Homological algebra
