The Stanley–Reisner Ideals of Polygons as Set-Theoretic Complete Intersections

We show that the Stanley–Reisner ideal of the one-dimensional simplicial complex whose diagram is an n-gon is always a set-theoretic complete intersection in any positive characteristic.     

Interval partitions and Stanley depth

In this paper, we answer a question posed by Herzog, Vladoiu, and Zheng. Their motivation involves a 1982 conjecture of Richard Stanley concerning what is now called the Stanley depth of a module. The question of Herzog et al., concerns partitions of the non-empty subsets of {1,2,…,n} into intervals. Specifically, given a positive integer n, they asked whether there exists a partition P(n) of the non-empty subsets of {1,2,…,n} into intervals, so that |B|?n/2 for each interval [A,B] in P(n). We answer this question in the affirmative by first embedding it in a stronger result. We then provide two alternative proofs of this second result. The two proofs use entirely different methods and yield non-isomorphic partitions. As a consequence, we establish that the Stanley depth of the ideal (x₁,…,xn)⊆K[x₁,…,xn] (K a field) is ⌈n/2⌉.     

An elementary approach to containment relations between symbolic and ordinary powers of certain monomial ideals

We provide an elementary explanation of a surprising result of Ein–Lazarsfeld–Smith and Hochster–Huneke on the containment between symbolic and ordinary powers of ideals for a certain class of simple monomial ideals.     

Bases in the mod p Steenrod algebra, β included

In [3] well known results of Wall and Arnon on the monomial bases in the mod 2 Steenrod algebra (see [9], [1]) were generalized to the subalgebra ${\stackrel{￣}{A}}_p$ of the mod p Steenrod algebra, $p>2$, generated by the reduced powers. In the present paper we considered the case of the full Steenrod algebra $A_p$. We constructed βX-, βZ-, βC-, ZA-, and XC-bases. We proved extremal properties of the βX-, βZ-, ZA-, and XC-bases. Also we constructed a new polynomial generators of the ring $H^{?}(K(\mathbb{Z}/p,n),\mathbb{Z}/p)$ in terms of the βC-basis.