A computation on the decomposition factors of D -modules over a hyperplane arrangement in space

  • Smegnsh Demelash Addis Ababa University
  • Tilahun Ababaw Addis Ababa University
Keywords: D-modules; Decomposition Factors; Support of a D-module; Length of a D-module; No-broken Circuits

Abstract

Let m be a positive integer,α_i:C^n⟶C^n, for i=1,2,…,m be linear forms and H_i={P∈C^n:α_i (P)=0} be the corresponding hyperplane for each i=1,2,…,m . The linear forms α_1,α_2,…,α_m define a hyperplane arrangement and X=C^n\V(α), where α=∏_(i=1)^m α_i and "V"(α)={P∈C^n:α(P)=0}. The coordinate ring O_X of X is the localization 〖C[x_1,…,x_n]〗_αand the ring O_X=〖C[x_1,…,x_n]〗_α is a holonomic A_n-module, where A_n is the n-th Weyl algebra, hence it has finite length. In this work, we will compute the number of decomposition factors of the A_3-module 〖C[x]〗_α, where α defines a central hyperplane arrangement in space, in terms of the no-broken circuits and describe the decomposition factors in terms of their supports.

Author Biographies

Smegnsh Demelash, Addis Ababa University

Department of Mathematics

Tilahun Ababaw, Addis Ababa University

Department of Mathematics

Published
2020-11-21
Section
Research articles

Journal Identifiers


eISSN: 2520–7997
print ISSN: 0379-2897