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Homological Algebra

Field: Category Theory

The study of homological algebra.

Sequence of Expressions

Let A\mathcal{A} be an abelian category (e.g., the category of RR-modules, ModR\text{Mod}_R). A **chain complex** C∙C^{\bullet} in A\mathcal{A} is a sequence of objects and morphisms:\n⋯→Cn−1→dn−1Cn→dnCn+1→…\dots \to C^{n-1} \xrightarrow{d^{n-1}} C^n \xrightarrow{d^n} C^{n+1} \to \dots \nsuch that the composition of consecutive boundary maps is zero for all nn: dn+1∘dn=0d^{n+1} \circ d^n = 0. \n\nThe **nn-th homology object** of the complex C∙C^{\bullet}, denoted Hn(C∙)H^n(C^{\bullet}), is defined as the quotient object:\nHn(C∙)=ker⁡(dn)/im(dn−1).H^n(C^{\bullet}) = \ker(d^n) / \text{im}(d^{n-1}). \nHomological Algebra studies the properties of these homology objects, particularly how they behave under derived functors (such as Tor\text{Tor} and Ext\text{Ext}), which measure the failure of exactness in A\mathcal{A}.