# MathSciDoc: An Archive for Mathematician ∫

#### TBDmathscidoc:1701.333168

Arkiv for Matematik, 48, (2), 211-229, 2008.11
Let$α$_{1},$α$_{2},…,$α$_{$m$}be linear forms defined on ℂ^{$n$}and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where$V$($α$_{$i$})={$p$∈ℂ^{$n$}$:$$α$_{$i$}($p$)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic$A$_{$n$}-module, where$A$_{$n$}is the$n$th Weyl algebra and since holonomic$A$_{$n$}-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this$A$_{$n$}-module which is also holonomic. Define M_{$α$}^{$β$}to be the free rank-1 ℂ[$x$]_{$α$}-module on the generator$α$^{$β$}(thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index$β$=($β$_{1},…,$β$_{$m$})∈ℂ^{$m$}. Our main result is the computation of the number of decomposition factors of M_{$α$}^{$β$}and their description when$n$=2.
@inproceedings{tilahun2008decomposition,
title={Decomposition of$D$-modules over a hyperplane arrangement in the plane},

Tilahun Abebaw, and Rikard Bøgvad. Decomposition of$D$-modules over a hyperplane arrangement in the plane. 2008. Vol. 48. In Arkiv for Matematik. pp.211-229. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203627151712977.