The Buchsbaum equality conjecture for symmetric and exterior iterated Betti numbers

From papers

Let Γ\Gamma be a Buchsbaum simplicial complex, and let bi,j(Γ)b_{i,j}(\Gamma) and bi,je(Γ)b^e_{i,j}(\Gamma) denote its symmetric and exterior iterated Betti numbers. Let Δ(Γ)\Delta(\Gamma) and Δe(Γ)\Delta^e(\Gamma) be its symmetric and exterior algebraic shifts, with Stanley–Reisner ideals IΔ(Γ)I_{\Delta(\Gamma)} and IΔe(Γ)I_{\Delta^e(\Gamma)}. Buchsbaum equality conjecture. If Γ\Gamma is a Buchsbaum complex, then

bi,j(Γ)=bi,je(Γ),b_{i,j}(\Gamma)=b^e_{i,j}(\Gamma),

and hence

βi,j(IΔ(Γ))=βi,j(IΔe(Γ)).\beta_{i,j}(I_{\Delta(\Gamma)})=\beta_{i,j}(I_{\Delta^e(\Gamma)}).

The source presents this as an additional conjecture for Buchsbaum complexes; it does not state whether it has been resolved.

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Sources & referencesView supporting material

Primary source

Eric Babson, Isabella Novik and Rekha Thomas, “Symmetric iterated Betti numbers”, arXiv:math/0206063 (2002).

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