Differential operators, anisotropy, and simplicial spheres

TL;DR

This paper establishes new identities involving differential operators in the algebraic structures of simplicial spheres, generalizing previous characteristic 2 results and applying them to prove anisotropy and weak Lefschetz properties.

Contribution

It introduces generalized identities for differential operators in Stanley-Reisner rings of simplicial spheres, extending prior characteristic-specific results and enabling new algebraic proofs.

Findings

Proved identities involving differential operators in simplicial spheres.

Established anisotropy of certain forms on the Stanley-Reisner ring.

Proved weak Lefschetz properties for these algebraic structures.

Abstract

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.