Equality of Betti numbers over the common parameter subring for [Δ] and [SdΔ]

Determine whether, for every finite boolean complex Δ over any field, all Betti numbers in the minimal free resolution of the Stanley–Reisner ring [Δ] as a module over the common parameter subring [Θ] are equal to the corresponding Betti numbers in the minimal free resolution of the Stanley–Reisner ring [SdΔ] over [Θ].

Background

The paper studies relationships between the Stanley–Reisner ring [Δ] of a boolean complex Δ and the Stanley–Reisner ring [SdΔ] of its barycentric subdivision, focusing on their shared parameter subring [Θ]. Earlier results (Stanley, Duval, Conca–Varbaro) establish close ties between these rings (e.g., equal depth and matching extremal Betti numbers) via ASL theory and Gröbner degenerations.

Adams and Reiner formulated a stronger conjecture asserting complete equality of Betti numbers for [Δ] and [SdΔ] when the rings are resolved over the shared parameter subring [Θ], without imposing restrictions on the characteristic or additional hypotheses on Δ beyond finiteness. This paper provides evidence and context but does not resolve this conjecture.