Face number inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices
This paper studies the following two problems in the field of commutative algebra and combinatorics: (i) What can be said about the number of faces of a matroid complex? and (ii) How can we calculate the Cohen-Macaulay type of the Stanley-Reisner ring of the order complex of a finite distributive lattice? The first part of the paper considers Hilbert functions of Stanley-Reisner rings of matroid complexes. The second part of the paper proves that for a finite poset \(P\), the following numbers are equal: (a) the Cohen-Macaulay type \(type(k[\Delta(L)])\) of the Stanley-Reisner ring \(k[\Delta(L)]\) of the order complex \(\Delta(L)\) of the finite distributive lattice \(L=I(P)\) of order ideals of \(P\), (b) the number of strictly order preserving maps \(\sigma: P\to N\) such that \(\sigma^{-1}(\{i-1,i\})\) is not a clutter in \(P\) for every \(i\in P\) with \(i\geq 1\), and (c) the number of distinct equivalence classes of a certain equivalence relation defined on the set of linear extensions of the poset \(P\).
scientific article; zbMATH DE number 146275