g-Elements, finite buildings and higher Cohen–Macaulay connectivity

Chari proved that if Δ is a (d−1)-dimensional simplicial complex with a convex ear decomposition, then h0⩽⋯⩽h⌊d/2⌋ [M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997) 3925–3943]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M-vector [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548]. This is proved to be true by showing that the set of pairs (ω,Θ), where Θ is a l.s.o.p. for k[Δ], the face ring of Δ, and ω is a g-element for k[Δ]/Θ, is nonempty whenever the characteristic of k is zero.Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag h-vector of such spaces similar in spirit to those examined in [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533–548] for order complexes of geometric lattices. This also leads to connections between higher Cohen–Macaulay connectivity and conditions which insure that h0<⋯