In this paper we study the Betti numbers of Stanley-Reisner ideals generated in degree 2. We show that the first six Betti numbers do not depend on the characteristic of the ground field. We also show that, if the number of variables n is at most 10, all Betti numbers are independent of the ground field. For n = 11, there exists precisely 4 examples in which the Betti numbers depend on the grou...

Let ∆ be a simplicial complex and χ be an s-coloring of ∆. Biermann and Van Tuyl have introduced the simplicial complex ∆χ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of ∆χ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of ∆χ over a field is set-theoretic complete interse...

We develop combinatorial tools to study the relationship between the Stanley depth of a monomial ideal I and the Stanley depth of its compliment, S/I. Using these results we are able to prove that if S is a polynomial ring with at most 5 indeterminates and I is a square-free monomial ideal, then the Stanley depth of S/I is strictly larger than the Stanley depth of I. Using a computer search, we...

Following Johnsen and Verdure (2013), we can associate to any linear code $C$ an abstract simplicial complex in turn, a Stanley-Reisner ring $R_C$. The $R_C$ is standard graded algebra over field its projective dimension precisely the of $C$. Thus admits minimal free resolution resulting Betti numbers are known determine generalized Hamming weights question purity was considered by Ghorpade Sin...

Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: • constructing the minimal free resolutions of special monomial ideals, cf. [AHH, BPS] • constructing non-minimal free resolutions; for example, Taylor’s resolution (cf. [Ei, p. 439]) and the cellular resolutions •...

We have introduced the Janet's algorithm for the Stanley decomposition of a monomial ideal I ⊂ S = K[x 1 , ..., x n ] and prove that Janet's algorithm gives the squarefree Stanley decomposition of S/I for a squarefree monomial ideal I. We have also shown that the Janet's algorithm gives a partition of a simplicial complex.

The antiprism triangulation provides a natural way to subdivide simplicial complex \(\Delta \), similar barycentric subdivision, which appeared independently in combinatorial algebraic topology and computer science. It can be defined as the of chains multi-pointed faces from point view, by successively applying construction, or balanced stellar subdivisions, on geometric view. This paper studie...