Anisotropy of generic Artinian reductions of reduced Gorenstein algebras

Determine whether every reduced Gorenstein standard graded algebra of Krull dimension at least its socle dimension has a generic Artinian reduction over an appropriate field extension that is anisotropic.

Background

The paper’s results concern anisotropy and p-anisotropy in Gorensteinifications of Artinian reductions of face rings associated with simplicial cycles and homology manifolds. The authors then propose a broader algebraic generalization beyond face rings.

The conjecture is deliberately stated in the most pessimistic level of generality because the authors report disagreement about which generality might hold. It asks whether generic Artinian reductions preserve anisotropy for all reduced Gorenstein standard graded algebras satisfying the stated relation between Krull dimension and socle dimension.