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We discuss two topics related with combinatorial study of canonical modules of Stanley-Reisner rings, viz., (i) some linear inequalities on the number of faces of a matroid complex and (ii) a formula to compute the Cohen-Macaulay type of the Stanley-Reisner ring of a finite distributive lattice

Face number inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices