Koszul

Koszul refers to constructions in homological algebra named after Jean-Louis Koszul. In common usage it denotes the Koszul complex associated with a sequence in a ring, as well as the broader notions of Koszul algebras and Koszul duality. These ideas relate resolutions, syzygies, and graded algebra structures.

Koszul complex: For a ring R and a sequence x1,...,xn in R, the Koszul complex K•(x; R) is a finite free R-complex built from the exterior algebra on n generators with a differential defined by the xi. H0(K) ≅ R/(x1,...,xn). The higher homology H_i(K) measures failures of the xi to form a regular sequence; if the sequence is regular, the Koszul complex is a free resolution of R/(x), and all higher homology vanishes.

Koszul algebras: A connected graded algebra A over a field k is Koszul if k, viewed as an A-module via the augmentation A → k, has a linear graded projective resolution. Equivalently, Tor_i^A(k,k) is concentrated in degree i. Classic examples include the polynomial ring k[x1,...,xn] and the exterior algebra on a finite-dimensional vector space.

Koszul duality: For a Koszul algebra A, there is a companion quadratic algebra A^! called the Koszul dual, with duality relations between certain categories of graded modules and their homological invariants. This duality provides a powerful link between the homological properties of A and those of its dual.

Significance: Koszul theory provides efficient resolutions and clear connections between algebra, geometry, and representation theory, enabling computations of Hilbert and Poincaré series for many important algebras and informing broader structural questions in algebra.