Alternative polarizations of Borel fixed ideals, Eliahou-Kervaire type resolution and discrete Morse theory

On CW complexes supporting Eliahou-Kervaire type resolutions of Borel fixed ideals

We prove that the Eliahou-Kervaire resolution of a Cohen-Macaulay stable monomial is supported by a regular CW complex whose underlying space is a closed ball. We also show that the modified Eliahou-Kervaire resolutions of variants of a Borel fixed ideal (e.g., a squarefree strongly stable ideal) are supported by regular CW complexes, and their underlying spaces are closed balls in the Cohen-Macaulay case.

Vector-spread monomial ideals and Eliahou-Kervaire type resolutions

We introduce the class of vector-spread monomial ideals. This notion generalizes that of $t$-spread ideals introduced by Ene, Herzog and Qureshi. In particular, we focus on vector-spread strongly stable ideals, we compute their Koszul cycles and describe their minimal free resolution. As a consequence the graded Betti numbers and the Poincaré series are determined. Finally, we consider a generalization of algebraic shifting theory for such a class of ideals.

Eliahou-Kervaire 分解在斜多项式环上的推广

The Eliahou-Kervaire resolution over a skew polynomial ring

本文将Eliahou和Kervaire在多项式环中为稳定理想构造的最小分解推广到斜多项式环，证明其同调性质仍成立，并给出推广的乘积结构。

A Minimal Poset Resolution of Stable Ideals

We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.

单项式理想的 Morse 消解与 Betti 分裂

Morse resolutions of monomial ideals and Betti splittings

本文利用离散 Morse 理论与分裂技巧研究单项式理想的自由消解，证明剪枝消解对稳定理想和线性商理想具有极小性，并统一了 Eliahou-Kervaire 与 Herzog-Takayama 等经典构造，同时提出约化极小性研究的方法及幂理想剪枝消解变体。

斜多项式环上的 Herzog–Takayama 分解

The Herzog-Takayama resolution over a skew polynomial ring

本文将 Herzog 与 Takayama 针对具有线性商的单项式理想构造的极小自由分解，从普通多项式环推广到斜多项式环。