A classical idea originating with Hilbert and continuing through the modern work of Lazarsfeld, Voisin, and many others is that homological algebra over the polynomial ring provides many powerful tools for studying the geometry of varieties in projective space. Analogously, one wishes for similar homological tools to study the geometry of varieties embedded in spaces other than projective space. I will discuss my recent work developing such analogous results for sub-varieties of toric varieties via homological algebra over multigraded polynomial rings.
