Free Resolutions and Sparse Determinantal Ideals
Free Resolutions and Sparse Determinantal Ideals
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自由分辨率和稀疏决定性理想
DOI：
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发表时间：
2011
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通讯作者：
Adam Boocher
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作者：
Adam Boocher
稀疏通用矩阵是其元素是不同变量和零的矩阵。这样的矩阵进行了研究朱斯蒂和梅尔谁计算一些不变量的理想最大未成年人。在本文中，我们通过计算所有这些稀疏行列式理想的最小自由分辨率来扩展这些结果。我们这样做，通过引入一种技术，修剪最小的自由决议时，变量的一个子集被设置为零。我们的技术正确地计算了一个最小的自由决议在两种情况下的利益：决议的单项式理想，理想解决的Weson-Northcott复杂。作为结果，我们可以证明稀疏行列式理想在整数上有线性分解，并且投影维数仅取决于矩阵中同为零的列数。最后，我们证明了所有这样的理想的性质，无论选择的术语顺序，理想和它的初始理想的贝蒂数是相同的。特别是这些理想的非零生成元形成了一个普遍的Grobner基。
A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over the integers, and that the projective dimension depends only on the number of columns of the matrix which are identically zero. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Grobner basis.