Homological Algebra and Representation Theory of Modules
Summary
Homological algebra studies algebraic structures by probing them with chain complexes and derived functors, revealing hidden layers of relationships among objects such as modules over rings. Representation theory of modules examines how rings and algebras act on these modules, using tools from category theory to classify and deform representations. Central concepts include Ext and Tor groups, which measure extensions and torsion, and derived categories, which organise complexes of modules via quasi-isomorphisms into triangulated frameworks. This interplay has driven advances in algebraic geometry, mathematical physics and combinatorics by providing universal invariants, guiding classification of algebras by representation type, and underpinning constructions in cluster algebras, mirror symmetry and coding theory.
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Homological Algebra and Representation Theory of Modules publication trend
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Technical terms
Module: A generalisation of a vector space in which scalars form a ring instead of a field.
Derived category: A construction organising complexes of modules up to quasi-isomorphism, providing a unified setting for derived functors.
T-structure: A decomposition of a triangulated category into two complementary subcategories whose intersection forms an abelian ‘heart’.
Triangulated category: A category with a shift functor and distinguished triangles encoding exactness properties of complexes.
τ-tilting theory: A modern extension of tilting theory classifying torsion classes via support τ-tilting modules, with key combinatorial and geometric applications.
References
- Bounded $t$ -structures on the category of perfect complexes. Acta Mathematica (2024).
- On triangulated orbit categories. Documenta Mathematica (2005).
- Wall and chamber structure for finite-dimensional algebras. Advances in Mathematics (2019).
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