Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other "tangible" mathematical objects. A spectral sequence is a powerful tool for this. It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.
History Homological algebra began to be studied in its most basic form in the late 19th century as a branch of topology and in the 1940s became an independent subject with the study of objects such as the ext functor and the tor functor, among others.
Chain complexes and homology
The notion of chain complex is central in homological algebra. An abstract chain complex is a sequence ( C ∙ , d ∙ ) {\displaystyle (C_{\bullet },d_{\bullet })} of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero:
C ∙ : ⋯ ⟶ C n + 1 ⟶ d n + 1 C n ⟶ d n C n − 1 ⟶ d n − 1 ⋯ , d n ∘ d n + 1 = 0. {\displaystyle C_{\bullet }:\cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots ,\quad d_{n}\circ d_{n+1}=0.}
The elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials. The chain groups Cn may be endowed with extra structure; for example, they may be vector spaces or modules over a fixed ring R. The differentials must preserve the extra structure if it exists; for example, they must be linear maps or homomorphisms of R-modules. For notational convenience, restrict attention to abelian groups (more correctly, to the category Ab of abelian groups); a celebrated theorem by Barry Mitchell implies the results will generalize to any abelian category. Every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as
B n ⊆ Z n ⊆ C n . {\displaystyle B_{n}\subseteq Z_{n}\subseteq C_{n}.}
Subgroups of abelian groups are automatically normal; therefore we can define the nth homology group Hn(C) as the factor group of the n-cycles by the n-boundaries,
H n ( C ) = Z n / B n = Ker d n / Im d n + 1 . {\displaystyle H_{n}(C)=Z_{n}/B_{n}=\operatorname {Ker} \,d_{n}/\operatorname {Im} \,d_{n+1}.}
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