Differential operators, anisotropy, and simplicial spheres

We find identities involving differential operators in the generic artinian reduction of the Stanley-Reisner ring of a simplicial sphere in any positive characteristic. These identities generalize the characteristic 2 identities used by Papadakis and Petrotou to give a proof of the algebraic g-conjecture. We show that these identities are a shadow of an identity on the degree map, and we use them to prove the anisotropy of certain forms on the generic artinian reduction of the Stanley–Reisner ring and to prove weak Lefschetz results.