In 1971, McMullen conjectured a characterization of the face numbers of convex simplicial polytopes. This conjecture, dubbed the “g-conjecture”, was resolved over the following ten years by work of Stanley and Billera–Lee. The extension of this conjecture to simplicial spheres remained open much longer. We will discuss the ingenious characteristic 2 proof given by Papadakis–Petrotou in 2020 and provide a unifying framework for it in commutative algebra. This is joint work in progress with Adiprasito, Papadakis, Petrotou, and Oba.