论文标题
无平方单体理想力量的简单决议
Simplicial Resolutions of Powers of Square-free Monomial Ideals
论文作者
论文摘要
即使在无平方的情况下，泰勒的分辨率几乎绝不是单一理想的力量。在本文中，我们引入了任何无平方单一理想的每个功率的较小分辨率，这仅取决于理想的发电机数量。更确切地说，对于每对固定整数$ r $和$ Q $，我们构建了一个简单的复合体，该复合体支持任何具有$ Q $生成器的任何无方便单元理想的$ r $ th power。最终的分辨率明显小于泰勒分辨率，并且对于特殊情况而言，分辨率最少。考虑到固定理想的发电机的关系，我们可以进一步缩小这些决议。我们还介绍了一种称为“极端理想”的理想类别，并表明所有无正方形的单一理想的贝蒂数量都受到贝蒂（Betti）数量的极端理想的数量界定。我们的结果导致对任何无方便的单样理想的贝蒂数量上的上限，这些幂大大改善了泰勒分辨率提供的二项式界限。
The Taylor resolution is almost never minimal for powers of monomial ideals, even in the square-free case. In this paper we introduce a smaller resolution for each power of any square-free monomial ideal, which depends only on the number of generators of the ideal. More precisely, for every pair of fixed integers $r$ and $q$, we construct a simplicial complex that supports a free resolution of the $r$-th power of any square-free monomial ideal with $q$ generators. The resulting resolution is significantly smaller than the Taylor resolution, and is minimal for special cases. Considering the relations on the generators of a fixed ideal allows us to further shrink these resolutions. We also introduce a class of ideals called "extremal ideals", and show that the Betti numbers of powers of all square-free monomial ideals are bounded by Betti numbers of powers of extremal ideals. Our results lead to upper bounds on Betti numbers of powers of any square-free monomial ideal that greatly improve the binomial bounds offered by the Taylor resolution.
