Finite Length and Pure-Injective Modules over a Ring of Differential Operators

✍ Scribed by Gennadi Puninski

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 123 KB
• Volume: 231
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2000.8322

✦ Synopsis

ww xx Let k be an algebraically closed field of characteristic zero, O O s k x , . . . , x n 1 n the ring of formal power series over k, and D D the ring of differential operators n over O O . Suppose that is a prime ideal of O O of height n y 1; i.e., A s O O r is a n n n curve. We prove that every indecomposable finite length module over D D with n support on is uniserial with isomorphic or alternating composition factors. For Ž . the ring D D A of differential operators over A we also classify indecomposable pure-injective modules and show that the Cantor᎐Bendixson rank of the Ziegler Ž . spectrum over D D A is equal to 2. ᮊ 2000 Academic Press 1 singularity.

In this paper we investigate and completely classify the finite length Ž . equivalently holonomic D D -modules. In particular we prove that every 1 such module is uniserial and either homogeneous or with alternating 546