Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence
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共振、线性 syzygies、Chen 群和 Bernstein-Gelfand-Gelfand 对应
DOI：
10.1090/s0002-9947-05-03853-5
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发表时间：
2005
影响因子：
1.3
通讯作者：
Alexander I. Suciu
中科院分区：
文献类型：
--
作者：
H. Schenck;Alexander I. Suciu
如果A是一个复超平面排列，有补X，我们证明了G = π 1（X）的Chen秩等于上同调环A = H*（X，k）的最小自由分解中线性链的分次Betti数，A = H*（X，k）被看作是A上的外代数E上的一个模：θ k（G）= dim k Tor Ek-1（A，k）k，其中k ≥ 2，其中k是特征为0的域. Chen rank猜想认为，当k充分大时，θ k（G）=（k - 1）<$r≥1（r+k-1 k），其中hr是射影共振簇R1（A）的r维分支数.我们早先关于A在E上的分解和上述等式的工作证明了图形排列的猜想。利用R1（A）的几何结果和局部化论证，对任意的A建立了不等式θ k（G）≥（k - 1）<$r≥ 1hr（r + k-1 k），其中k >> 0.最后，我们证明了存在一个多项式P（t）的次数等于R1（A）的维数，使得θ k（G）= P（k），对所有k >> 0.
If A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G = π 1 (X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A = H*(X, k), viewed as a module over the exterior algebra E on A: θ k (G) = dim k Tor E k-1 (A, k) k , for k ≥ 2, where k is a field of characteristic 0. The Chen ranks conjecture asserts that, for k sufficiently large, θ k (G) = (k - 1) Σ r≥1 ( r+k-1 k ), where h r is the number of r-dimensional components of the projective resonance variety R 1 (A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R 1 (A) and a localization argument, we establish the inequality θ k (G) ≥ (k - 1) Σ r≥1 hr ( r + k-1 k ), for k >> 0, for arbitrary A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R 1 (A), such that θ k (G) = P(k), for all k >> 0.