open access

https://hdl.handle.net/2115/48291

The freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement

英語

Jet module

2012-02-01

Elsevier

Journal of Algebra

351(1)

294-318

Let A be a generic hyperplane arrangement composed of r hyperplanes in an n-dimensional vector space, and S the polynomial ring in n variables. We consider the S-submodule D(m)(A) of the nth Weyl algebra of homogeneous differential operators of order m preserving the defining ideal of A. We prove that if n ≥ 3, r > n, m > r - n + 1, then D(m)(A) is free (Holm's conjecture). Combining this with some results by Holm, we see that D(m)(A) is free unless n ≥ 3, r > n, m < r - n + 1. In the remaining case, we construct a minimal free resolution of D(m)(A) by generalizing Yuzvinsky's construction for m = 1. In addition, we construct a minimal free resolution of the transpose of the m-jet module, which generalizes a result by Rose and Terao for m = 1.

学術雑誌論文

Accepted Manuscript

0021-8693