Comparing the face rings of a boolean complex and its barycentric subdivision.

(Joint work with Ben Blum Smith, Johns Hopkins University, USA) The barycentric subdivision of a boolean complex preserves many combinatorial and topological properties, but its effect on the associated Stanley–Reisner ring is more subtle. In this talk, I will discuss the problem of comparing the face ring of a boolean complex with that of its barycentric subdivision in an equivariant setting. I will first explain why equivariant isomorphisms do not exist in general, presenting a counterexample in characteristic 2. I will then describe a positive rigidity result showing that, for Cohen–Macaulay boolean complexes and in characteristic coprime to the automorphism group, such equivariant isomorphisms do exist. The proof combines generalized forms of Garsia transfer with linear-algebraic characterizations of Cohen–Macaulayness.