2021/12/29 by Fan, Feifei
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2112.14493
A simplicial sphere Δ is said to be generically anisotropic over a field \mathbbF if, for a certain purely transcendental field extension k of \mathbbF, a certain Artinian reduction A of the face ring k[Δ] has the following property: For every nonzero homogeneous element α∈ A of degree at most (dimΔ+1)/2, its square α2 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 \mathbbF.