[Paper Review] On a theorem of Stafford

Napoleon Caro, Daniel Levcovitz|arXiv (Cornell University)|May 24, 2010

Advanced Topics in Algebra1 references3 citations

TL;DR

This paper extends Stafford's classical result on the Weyl algebra by proving that every left or right ideal in the ring of differential operators over formal Laurent series $ K((x_1,\dots,x_n)) $ and over convergent Laurent series $ \mathbb{C}\{\{x_1,\dots,x_n\}\} $ is generated by two elements. The authors overcome the asymmetry between $ x $-variables and $ \partial $-variables in power series rings by adapting Stafford’s proof using Weierstrass preparation and module-theoretic techniques over noncommutative Ore domains.

ABSTRACT

Stafford proved that every left or right ideal of the Weyl algebra A_n(K) is generated by two elements. In this paper we prove that every left or right ideal of the ring of differential operators over the field of formal Laurent series K((x_1,...,x_n)) is also generated by two elements. The same is true for the ring of differential operators over the convergent Laurent series C{x_1,...,x_n}. This is in accordance with the conjecture that says that in a (noncommutative) noetherian simple ring, every left or right ideal is generated by two elements.

Motivation & Objective

• To extend Stafford’s theorem on the Weyl algebra to differential operator rings over formal Laurent series and convergent Laurent series.
• To address the lack of symmetry between $ x $-variables and $ \partial $-variables in power series rings, which breaks the duality used in Stafford’s original proof.
• To establish that every left or right ideal in $ E_n = K((x_1,\dots,x_n))\langle\partial_1,\dots,\partial_n\rangle $ is generated by two elements.
• To verify that the same result holds for the ring of differential operators over convergent Laurent series $ \mathbb{C}\{\{x_1,\dots,x_n\}\} $.
• To support the broader conjecture that in noetherian simple noncommutative rings, all left or right ideals are two-generated.

Proposed method

• Adapt the proof strategy of Stafford’s theorem from Björk’s book, modifying it for the power series setting.
• Use Weierstrass preparation theorem to put elements in a suitable normal form in $ D_n = K[[x_1,\dots,x_n]]\langle\partial_1,\dots,\partial_n\rangle $.
• Work with the ring $ R_r = F_r((x_{r+1},\dots,x_n))\langle\partial_{r+1},\dots,\partial_n\rangle $, where $ F_r $ is the quotient field of $ D_r $, to inductively reduce the problem.
• Apply module-theoretic arguments over free $ S $-modules, where $ S = T((x))\langle\partial_x\rangle $, to show that certain submodules are full.
• Use the existence of a common left multiple via the Ore condition to construct generating elements.
• Leverage the fact that $ D_n $ is a left Ore domain to lift results from $ D_n $ to the larger ring $ E_n $ via multiplication by invertible power series $ q_0 \in K[[x_1,\dots,x_n]] $.

Experimental results

Research questions

• RQ1 Can Stafford’s two-generator theorem for the Weyl algebra be extended to differential operator rings over formal Laurent series?
• RQ2 How can the asymmetry between $ x $-variables and $ \partial $-variables in power series rings be overcome in ideal generation problems?
• RQ3 Does every left or right ideal in $ K((x_1,\dots,x_n))\langle\partial_1,\dots,\partial_n\rangle $ admit a two-element generating set?
• RQ4 Is the two-generator property preserved for the ring of differential operators over convergent Laurent series $ \mathbb{C}\{\{x_1,\dots,x_n\}\} $?
• RQ5 To what extent does this result support the conjecture that all ideals in noetherian simple noncommutative rings are two-generated?

Key findings

• Every left or right ideal in $ E_n = K((x_1,\dots,x_n))\langle\partial_1,\dots,\partial_n\rangle $ is generated by two elements.
• The same two-generator property holds for the ring of differential operators over the convergent Laurent series $ \mathbb{C}\{\{x_1,\dots,x_n\}\} $.
• The result is established via an inductive argument over the number of variables, using the structure of $ R_r $-modules and the Ore condition.
• A key technical step is the construction of a common left multiple $ q_0 \in K[[x_1,\dots,x_n]] $, non-zero, such that $ q_0 c \in D_n(a + dc) + D_n(b + ec) $ for some $ d,e \in D_n $, which lifts to $ E_n $.
• The proof relies on adapting Stafford’s original argument in Björk’s book, replacing the use of symmetry between $ x $ and $ \partial $ with Weierstrass preparation and module-theoretic control over torsion modules.
• The result confirms the two-generator conjecture for two important classes of noncommutative noetherian simple rings: the differential operator rings over formal and convergent Laurent series.