Abstract:

In 2020, Papadakis and Petrotou gave a proof of the g-theorem, a characterization of the face numbers of simplicial spheres, using a surprising identity in characteristic 2. This identity implies a generic Hard Lefschetz theorem for face rings of simplicial spheres.  In this talk, based on joint work with Adiprasito, Oba, Papadakis, and Petrotou, we reformulate these results in commutative algebra, relating the Papadakis–Petrotou identity to the Frobenius endomorphism. We obtain a condition that guarantees a Hard Lefschetz theorem and relate it to F-purity. This gives immediate proofs of the g-theorem and of the Hibi–Ohsugi conjecture on the unimodality of the h^*-polynomial of Gorenstein IDP polytopes.