Koszul duality has been observed in many forms across algebra, geometry and topology. This is a talk about commutative algebra, and I'll try to explain everything from the beginning.

A homomorphism f: R -> S of commutative local rings has a derived fibre F - this is 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^R(S,k) is a Koszul algebra in the classical sense. I'll explain why this is a very good definition, how it is satisfied by many examples, and what you can do with it.

The main application is the construction of explicit free resolutions over S in the presence of a Koszul homomorphism. These are infinite free resolutions described by a finite amount of data, and so they tell you about the asymptotic homological algebra of S. This construction simultaneously generalises 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.

This is all joint with James Cameron, Janina Letz, and Josh Pollitz.