New Classes of Set-theoretic Complete Intersection Monomial Ideals

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 intersection. This also generalizes a result of Macchia. 1. Statement of the main theorem Let us start this note with some notions of combinatorics. A simplicial complex ∆ on the set of vertices V = {v1, . . . , vn} is a collection of subsets of V which is closed under taking subsets; that is, if F ∈ ∆ and F ′ ⊆ F , then also F ′ ∈ ∆. Every element F ∈ ∆ is called a face of ∆, and a facet of ∆ is a maximal face of ∆ with respect to inclusion. It is clear that all facets of ∆ determines it. When F1, . . . , Ft are all facets of ∆, we write ∆ = 〈F1, . . . , Ft〉. In this case, we say χ is an s-coloring of ∆ when χ is a partition of V , say V = W1 ∪ · · · ∪Ws (where the sets Wj are allowed to be empty), such that for every 1 ≤ i ≤ t and every 1 ≤ j ≤ s the inequality |Fi∩Wj| ≤ 1 holds true. Biermann and Van Tuyl [3] have defined a new simplicial complex ∆χ on the set of vertices {v1, . . . , vn, w1, . . . , ws} with faces σ∪ τ , where σ is a face of ∆ and τ is any subset of {w1, . . . , ws} such that for every wj ∈ τ , we have σ ∩Wj = ∅. It is shown in [3] that the facets of ∆χ are in the form F ∪ F ′, where F is any face of ∆ and F ′ = {wi | F ∩Wi = ∅}. The simplicial complex ∆χ is generally larger than ∆. For example, if ∆ = 〈{v1, v2, v3}, {v3, v4}, {v4, v5, v6}〉 is the simplicial complex on the set of vertices V = {v1, v2, v3, v4, v5, v6} and χ is the 3-coloring of ∆ given by V = W1 ∪W2 ∪W3, where W1 = {v1, v4}, W2 = {v2, v5}, and W3 = {v3, v6}, then we have 2000 Mathematics Subject Classification. 13F55, 13A15.