超几何族的同调方法
Homological Methods for Hypergeometric Families
摘要：
我们通过提供一般情况下的秩跳跃的同调准则来分析完整系统族在复代数簇上的完整秩的行为。然后研究了由d × n整数矩阵A和参数β \in \CC^d生成的超几何系统H_A（β）的秩跳行为.为此，我们引入了\CC^d上超几何族的Euler-Koszul函子，它的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。我们证明了参数β是H_A（β）的跳秩当且仅当β位于\ZZ^d分次度α的Zapriki闭包中，其中半群环\CC [ \NN A]在其极大分次理想\frakm上支撑的局部上同调\bigoplus_{i<d} H^i_ \frakm（\CC [\NN A]）_α非零.因此，H_A（β）在\CC^d上没有秩跳跃当且仅当\CC [ \NN A]是d维Cohen-Macaulay.
英文摘要：
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(β) arising from a d x n integer matrix A and a parameter β\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 βis rank-jumping for H_A(β) if and only if βlies in the Zariski closure of the set of \ZZ^d-graded degrees αwhere the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_αof the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(β) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.
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