On the anisotropy and Lefschetz property for PL-spheres

A simplicial sphere $Δ$ is said to be generically anisotropic over a field $\mathbb{F}$ if, for a certain purely transcendental field extension $\mathbf{k}$ of $\mathbb{F}$, a certain Artinian reduction $A$ of the face ring $\mathbf{k}[Δ]$ has the following property: For every nonzero homogeneous element $α\in A$ of degree at most $(\dimΔ+1)/2$, its square $α^2$ is also nonzero. The importance of this property is that the hard Lefschetz property for simplicial spheres can be derived from it. A recent result of Papadakis and Petrotou shows that every simplicial sphere is generically anisotropic over any field of characteristic $2$. In this paper, we give an equivalent condition of being generically anisotropic, and use it to present a simplified proof of Papadakis-Petrotou theorem for PL-spheres. We also prove that the simplicial spheres of dimension $2$ are generically anisotropic over any field $\mathbb{F}$.

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