论文标题
稀疏对称矩阵的次最大未成年人的理想
Ideals of submaximal minors of sparse symmetric matrices
论文作者
论文摘要
我们研究稀疏通用对称基质的次最大未成年人理想的代数和同源特性。该理想是由所有$（N-1）$ - 对称$ n \ times n $矩阵的未成年人产生的，其上三角的条目是不同的变量或零，并且只允许零位置。尚存的非对角线条目被编码为带有$ n $顶点的简单图形$ g $。我们证明，通过简单的修剪过程，从情况下从情况下获得了最小的自由分辨率，从而扩展了Boocher的方法。这使我们能够根据$ n $和$ g $的单个不变性计算所有分级的贝蒂号。此外，事实证明，当且仅当$ g $连接或根本没有边缘时，这些理想总是激进的，并且具有cohen-macaulay商。关键输入是一些与$ g $相关的非对角性期限订单的新格格纳基础结果。
We study algebraic and homological properties of the ideal of submaximal minors of a sparse generic symmetric matrix. This ideal is generated by all $(n-1)$-minors of a symmetric $n \times n$ matrix whose entries in the upper triangle are distinct variables or zeros and the zeros are only allowed at off-diagonal places. The surviving off-diagonal entries are encoded as a simple graph $G$ with $n$ vertices. We prove that the minimal free resolution of this ideal is obtained from the case without any zeros via a simple pruning procedure, extending methods of Boocher. This allows us to compute all graded Betti numbers in terms of $n$ and a single invariant of $G$. Moreover, it turns out that these ideals are always radical and have Cohen--Macaulay quotients if and only if $G$ is either connected or has no edges at all. The key input are some new Gröbner basis results with respect to non-diagonal term orders associated to $G$.
