Homological methods for hypergeometric families
Homological methods for hypergeometric families
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DOI：
10.1090/s0894-0347-05-00488-1
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发表时间：
2004-06
影响因子：
3.9
通讯作者：
L. Matusevich;Ezra Miller;U. Walther
中科院分区：
文献类型：
--
作者：
L. Matusevich;Ezra Miller;U. Walther
我们分析了完整系统族中完整秩在复代数簇上的行为，通过提供一般情况下的秩跳跃的同调判据。然后我们研究了由d × n整数矩阵A和参数\beta \in \CC^d生成的超几何系统H_A（\beta）的秩跳行为.为此，我们引入了\CC^d上超几何族的Euler-Koszul函子，其同调推广了超几何系统的概念，并且我们证明了与我们上面的一般同调构造的同调同构。我们证明了H_A（\beta）的参数\beta是秩跳的当且仅当\beta位于\ZZ^d-分次度\alpha的Zapriki闭包中，其中半群环\CC[\NN A]在其极大分次理想\frakm处支撑的局部上同调\bigoplus_{i<d}H^i_\frakm（\CC[\NN A]）_\alpha非零.因此，H_A（\beta）在\CC^d上没有秩跳跃当且仅当\CC[\NN A]是d维的Cohen-Macaulay。
We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems H_A(\beta) arising from a d x n integer matrix A and a parameter \beta \in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^d, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter \beta is rank-jumping for H_A(\beta) if and only if \beta lies in the Zariski closure of the set of \ZZ^d-graded degrees \alpha where the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_\alpha of the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(\beta) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.