Koszul duality – MathVideos.org

Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of ∞-categories) between augmented dg algebras and conilpotent dg-coalgebras. One should think of this as a non-commutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I'll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to non-commutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra.

This is a talk about the situation in commutative algebra. A homomorphism f: S → R of commutative local rings has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) = Tor^S(R,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition and how it is satisfied by many many examples.

The main application is the construction of explicit free resolutions over R in the presence of a Koszul homomorphism. These tell you about the asymptotic homological algebra of R, and so the structure of the derived category of R. This construction simultaneously generalizes the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.

Tag - Koszul duality

Matt Booth: Global Koszul duality

Benjamin Briggs: Koszul homomorphisms and resolutions in commutative algebra