Homological AlgebraEdit

Homological algebra is a branch of mathematics that uses algebraic methods to study structures arising in topology, geometry, and algebra itself. At its heart are chain complexes and functors that preserve or reflect the delicate patterns these complexes encode. By translating geometric or topological problems into algebraic data, homological algebra provides invariants and computational tools that help distinguish, classify, and relate mathematical objects across diverse areas such as algebraic topology, algebraic geometry, and representation theory.

Historically, the subject grew out of questions in topology about how spaces can be built from simple pieces and how those pieces interact. Early work by Eilenberg and Steenrod formalized homology theories and axioms that guided computations. Over time, the language shifted toward category theory, with Grothendieck and others promoting abelian categories as a natural setting for homological methods. This move allowed one to treat chain complexes, exact sequences, and derived constructions in a unifying framework. In the later decades, innovations like derived categorys, DG categorys, and modern approaches based on ∞-category theory extended the reach of homological ideas far beyond their origins, into areas such as derived algebraic geometry and modern representation theory.

History

Origins in algebraic topology, where chain complexes and their homology groups were introduced to capture information about spaces.
The advent of abelian categories provided a flexible and robust setting for homological methods, enabling a more systematic treatment of exact sequences and functors.
The development of derived functors, including the functors Ext and Tor, gave precise tools for measuring how far a functor is from preserving exactness.
Verdier’s introduction of derived categories offered a powerful, though abstract, language for working with complexes up to quasi-isomorphism, catalyzing advances in many areas.
Contemporary developments emphasize DG categories and ∞-categories, which address limitations of triangulated categories and broaden the scope of homological techniques in geometry and mathematical physics.

Core ideas

Chain complexes: A chain complex is a sequence of objects connected by differentials with the property that composing two successive maps gives zero. The homology of a chain complex measures the failure of exactness and captures invariants of the underlying object or space. See Chain complex and Homology.
Homology and cohomology: Homology groups (and cohomology groups) are algebraic invariants that arise from chain complexes and functorial constructions. They are central to distinguishing different spaces, maps, and algebraic structures. See Homology and Cohomology.
Exact sequences and derived functors: Exact sequences reveal where algebraic information is preserved or lost under maps. Derived functors extend classical functors to measure their failure to be exact; key examples are the right derived functors of left exact functors, and the left derived functors of right exact functors. See Exact sequence and Derived functor.
Resolutions: To compute derived functors, one often replaces an object by a projective or injective resolution, a chain complex that is easier to handle yet represents the same homological information. See Projective resolution and Injective resolution.
Ext and Tor: These are the most familiar derived functors; Ext measures extensions of modules, while Tor detects tensor-torsion phenomena. See Ext and Tor.
Derived categories and beyond: The derived category refines the notion of homological equivalence by inverting quasi-isomorphisms, while modern trends explore DG categories and ∞-categories to capture more nuanced higher-homotopical information. See Derived category, DG category and ∞-category.

Constructions and tools

Chain complexes and homology: Building blocks for many invariants; computations often proceed by choosing convenient resolutions or filtrations.
Mapping cones and homotopy: Tools for comparing complexes and understanding when two maps induce the same effect on homology.
Spectral sequences: A computational device that arises from filtrations and double complexes, allowing gradual approximation of homology and cohomology groups. See Spectral sequence.
Derived functors and resolutions: Systematic ways to extend functors beyond exactness; practical computations rely on choosing projective or injective resolutions when available. See Derived functor, Projective resolution, Injective resolution.
Derived categories and triangulated structure: A framework in which objects are chain complexes up to quasi-isomorphism, with morphisms localized accordingly; triangulated structure encodes long exact sequences and cones. See Derived category and Triangulated category (if you are exploring the topic in depth).

Examples and applications

Singular homology of topological spaces: The singular chain complex encodes information about a space, and its homology groups are fundamental invariants. See Singular chain complex and Singular homology.
Group homology and cohomology: When a group acts on a module, derived functor methods yield invariants that reflect the action; these play a role in topology and number theory. See Group cohomology.
Sheaf cohomology and algebraic geometry: Derived functors of global sections recover global geometric information; this is central to the study of sheaves on schemes and complex manifolds. See Sheaf cohomology and Algebraic geometry.
Representation theory and categorification: Derived categories and higher-categorical approaches illuminate relationships between representations, algebras, and geometric objects; this connects to areas such as Hochschild cohomology and Derived category of coherent sheaves.

Foundations and perspectives

Homological algebra sits at the intersection of algebra, topology, and geometry, and its methods are framed inside various foundational settings. One often works in an abelian category with enough projectives or injectives, ensuring that resolutions exist and that derived functors can be computed. The choice of foundations—whether to emphasize classical abelian categories, triangulated categories, DG categories, or ∞-categories—shapes the kinds of questions that are easiest to pose and answer. This has led to fruitful debates in the mathematical community about the most natural or powerful language for advanced homological methods, with different schools of thought favoring different frameworks depending on the problems at hand. See Abelian category, Category theory, DG category, ∞-category.

Contemporary work often combines multiple viewpoints. For instance, derived categories provide a compact way to encode information about complexes, while DG categories and ∞-categories retain refinements that can be essential in geometric and physical applications. The ongoing development of derived algebraic geometry and related areas illustrates how homological techniques continue to evolve in response to new mathematical challenges. See Derived category, DG category, and Infinity category.