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

Abstract

Chari proved that if Δ is a ( d − 1 ) -dimensional simplicial complex with a convex ear decomposition, then h 0 ⩽ ⋯ ⩽ 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 h 0 < ⋯ < h i for a predetermined i.