Balanced Cohen-Macaulay Complexes

A balanced complex of type <!-- MATH $({a_1},\ldots,{a_m})$ --> is a finite pure simplicial complex together with an ordered partition <!-- MATH $({V_1},\ldots,{V_m})$ --> of the vertices of such that card<!-- MATH $({V_i}\, \cap \,F)\, = \,{a_i}$ --> , for every maximal face F of . If <!-- MATH ${\mathbf{b}}\, = \,({b_1},\ldots,{b_m})$ --> , then define <!-- MATH ${f_\textbf{b}}(\Delta )$ --> to be the number of <!-- MATH $F\, \in \,\Delta$ --> satisfying card<!-- MATH $({V_i}\, \cap \,F)\, = \,{b_i}$ --> . The formal properties of the numbers <!-- MATH ${f_\textbf{b}}(\Delta )$ --> are investigated in analogy to the f-vector of an arbitrary simplicial complex. For a special class of balanced complexes known as balanced Cohen-Macaulay complexes, simple techniques from commutative algebra lead to very strong conditions on the numbers<!-- MATH ${f_\textbf{b}}(\Delta )$ --> . For a certain complex <!-- MATH $\Delta (P)$ --> coming from a poset P, our results are intimately related to properties of the Möbius function of P.