Shedding vertices and Ass-decomposable monomial ideals

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner ideals of vertex decomposable simplicial complexes. The recursive structure of ass-decomposable monomial ideals allows us to find a simple formula for the depth, and in squarefree case, an upper bound for the regularity of such ideals.Comment: 13 page

The saturation number of monomial ideals

Let $S=\mathbb{K}[x_1,\ldots, x_n]$ be the polynomial ring over a field $\mathbb{K}$ and $\mathfrak{m}= (x_1, \ldots, x_n)$ be the irredundant maximal ideal of $S$. For an ideal $I \subset S$, let $\mathrm{sat}(I)$ be the minimum number $k$ for which $I \colon \mathfrak{m}^k = I \colon \mathfrak{m}^{k+1}$. In this paper, we compute the saturation number of irreducible monomial ideals and their powers. We apply this result to find the saturation number of the ordinary powers and symbolic powers of some families of monomial ideals in terms of the saturation number of irreducible components appearing in an irreducible decomposition of these ideals. Moreover, we give an explicit formula for the saturation number of monomial ideals in two variables

Gr\"obner basis and Krull dimension of Lov\'asz-Saks-Sherijver ideal associated to a tree

Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $\Gamma =([n], E)$ be a simple graph, where $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal $$L_\Gamma^\mathbb{K}(2) = \left( x_{i}x_{j} + y_{i}y_{j} \colon \quad \{i, j\} \in E(\Gamma)\right) \subset S,$$ is called the Lov\'{a}sz-Saks-Schrijver ideal, LSS-ideal for short, of $\Gamma$ with respect to $\mathbb{K}$. In the present paper, we compute a Gr\"obner basis of this ideal with respect to lexicographic ordering induced by $x_1>\cdots>x_n>y_1>\cdots>y_n$ when $\Gamma=T$ is a tree. As a result, we show that it is independent of the choice of the ground field $\mathbb{K}$ and compute the Hilbert series of $L_T^\mathbb{K}(2)$. Finally, we present concrete combinatorial formulas to obtain the Krull dimension of $S/L_T^\mathbb{K}(2)$ as well as lower and upper bounds for Krull dimension.Comment: 23 pages, 1 figur