Constructing resolutions using Koszul homomorphisms

This is a talk about a relative version of Koszul duality, and how it can tell us about the asymptotic homological algebra of (seemingly) non-Koszul local rings. It’s all joint work with James Cameron, Janina Letz, and Josh Pollitz.

A homomorphism f of commutative local rings, say S to R, 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) (the Tor algebra of R and k over S) is a Koszul algebra in the classical sense. I’ll explain why this is a good definition with interesting consequences, and how it is satisfied by many many examples. I might also talk about how to understand these through A-infinity structures.

The main application is the construction of explicit free resolutions over R in the presence of certain Koszul homomorphisms. 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

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