A central idea in Boij-Soederberg Theory is that there is a connection between free resolutions over the polynomial ring and sheaf cohomology on projective space. This idea emerged from Eisenbud and Schreyer's proof of the Boij-Soederberg conjectures.

In joint work with David Eisenbud, we give a new perspective on this duality. I will outline the resulting duality between syzygies and sheaf cohomology, and I will explain how our duality pairing enables us to extend the scope of Boij--Soederberg theory in several directions.