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

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 th...

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Bibliographic Details
Published in:arXiv.org 2011-06
Main Authors: Nakashima, Norihiro, Okuyama, Go, Saito, Mutsumi
Format: Article
Language:English
Subjects:
Differential equations
Hyperplanes
Mathematical analysis
Modules
Operators (mathematics)
Polynomials
Rings (mathematics)
Vector space
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Summary: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.
ISSN:2331-8422