Let $$S={\textsf {k}}[X_1,\dots , X_n]$$ S = k [ X 1 , ⋯ , X n ] be a polynomial ring, where $${\textsf {k}}$$ k is a field. This article deals with the defining ideal of the Rees algebra of a squarefree monomial ideal generated in degree $$n-2$$ n - 2 . As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of the tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.

中文翻译：

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某些覆盖理想的正则性、里斯代数和 Betti 数

令 $$S={\textsf {k}}[X_1,\dots , X_n]$$ S = k [ X 1 ,⋯ , X n ] 是一个多项式环，其中 $${\textsf {k}}$ $ k 是一个字段。本文讨论以 $$n-2$$n - 2 次生成的无平方单项式理想的里斯代数的定义理想。因此，我们证明了树的补图的覆盖理想的 Betti 幂数不依赖于树的选择。此外，我们研究了与某些图相关的覆盖理想的幂的规律性和 Betti 数。