The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement

Details The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement The Freeness and Minimal Free Resolutions of Modules of Differential Operators of a Generic Hyperplane Arrangement

2011-06-09

arxiv.org/abs/1106.1759v1

Mathematics

Combinatorics

20 pages

Scientific paper

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 \geq 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 \geq 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.

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