Cellular resolutions of some artinian level monomial ideals

Cellular resolutions of some artinian level monomial ideals

Thời gian: 09:30 đến 10:30 Ngày 08/02/2018

Địa điểm: B4-705, VIASM

Báo cáo viên: Augustine O'Keefe

Tóm tắt:

In this talk we will look at constructing cellular resolutions of monomial ideals in $n$ variables of the form $\mathfrak{m}^d+\langle x_i^{a_i}~|~1\leq i\leq n,~a_i\in\mathbb{N}\rangle$, where $\mathfrak{m}$ is the maximal ideal in the polynomial ring $k[x_1,\dots,x_n]$. Nagel and Reiner showed that any mixed subdivision of the $d^\text{th}$ dilation of of the $(n-1)$-dimensional simplex, $d\Delta_n$ supports a cellular resolution of $\mathfrak{m}^d$. We will discuss a topological process on a particular mixed subdivision of $d\Delta_n$ which results in a cellular resolution of $\mathfrak{m}^d+\langle x_i^{a_i}~|~1\leq i\leq n,~a_i\in\mathbb{N}\rangle$. These ideals are a generalization of Riemann-Roch monomial ideals introduced by Manjunath and Sturmfels which arise as the initial ideal of the toric ideal defined by the Laplacian of a graph.