Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid M . Our main result is that these polynomials are determined by Betti numbers associated with N0-graded minimal free resolutions of the Stanley-Reisner ideals of M and so-called elongations of M . Generalizing Greene’s theorem from coding theory, we show that ...

We show that Jn, the Stanley-Reisner ideal of the n-cycle, has a free resolution supported on the (n−3)-dimensional simplicial associahedron An. This resolution is not minimal for n > 6; in this case the Betti numbers of Jn are strictly smaller than the f -vector of An. We show that in fact the Betti numbers βd of Jn are in bijection with the number of standard Young tableaux of shape (d + 1, 2...

Let U be the set of prime ideals P completion a Stanley–Reisner ring S, such that localization at Frobenius algebra injective hull residue field S is finitely generated algebra. We give partial answer to conjecture made by M. Katzman about openness U. Specifically, we show has non-empty interior and present some sufficient conditions for principal open D(f) contained in U, intersections closed ...

Shellability is a well-known combinatorial criterion on a simplicial complex ∆ for verifying that the associated Stanley-Reisner ring k[∆] is Cohen-Macaulay. A notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, Jöllenbeck introduced a criterion on simplicial comp...

We define nice partitions of the multicomplex associated with a Stanley ideal. As the main result we show that if the monomial ideal I is a CM Stanley ideal, then I is a Stanley ideal as well, where I is the polarization of I .

Partial quotients of moment-angle complexes are topological analogues smooth, not necessarily compact toric varieties. In 1998, Buchstaber and Panov proposed a formula for the cohomology ring such partial quotient in terms torsion product involving corresponding Stanley-Reisner ring. We show that their gives correct cup if 2 is invertible chosen coefficient ring, but general. rectify this by de...

We define an action of the 0-Hecke algebra of type A on the Stanley-Reisner ring of the Boolean algebra. By studying this action we obtain a family of multivariate noncommutative symmetric functions, which specialize to the noncommutative Hall-Littlewood symmetric functions and their (q, t)-analogues introduced by Bergeron and Zabrocki. We also obtain multivariate quasisymmetric function identi...