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) = TorS(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.
This is all joint with James Cameron, Janina Letz, and Josh Pollitz.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
