This work studies the application of non-commutative projective geometry to the ring of differential operators on a smooth complex variety, or more generally a Lie algebroid on such a variety. Many classical results true about complex projective space have analogs which are proven, including Serre Finiteness, Serre Vanishing, Serre Duality, the Gorenstein property, the Koszulness property, and the Beilinson equivalence. Applications to the study of ideals, projective modules and the Grothendieck group are explored.