[Paper Review] Noncommutative plane curves

S. Jøndrup, Olav Arnfinn Laudal+1 more|arXiv (Cornell University)|May 18, 2004

Algebraic structures and combinatorial models12 references8 citations

TL;DR

This paper introduces noncommutative plane curves as noncommutative k-algebras whose 1-dimensional simple modules form a classical plane curve, studying extensions between simple modules and the completion problem via deformation theory. Key results include a noncommutative Jacobian criterion for 2-dimensional representations on elliptic curves and a geometric characterization of non-vanishing Ext groups via collinearity on the curve, showing that noncommutative deformations are constrained by elliptic curve addition rules.

ABSTRACT

In this paper we study noncommutative plane curves, i.e. non-commutative k-algebras for which the 1-dimensional simple modules form a plane curve. We study extensions of simple modules and we try to enlighten the completion problem, i.e. understanding the connection between simple modules of different dimension.

Motivation & Objective

• To systematically study noncommutative models of affine plane curves, particularly quadrics and cubics, over an algebraically closed field of characteristic zero.
• To understand the structure of higher-dimensional simple modules (Simp_n(A)) and their relations to 1-dimensional simple modules (Simp_1(A)) on the classical curve.
• To investigate the completion problem: how simple modules of different dimensions are connected via extensions and deformations.
• To apply deformation theory and noncommutative Jacobian techniques to classify 2-dimensional representations of noncommutative elliptic curves.
• To characterize when Ext^1 groups between simple modules are non-zero, linking this to geometric conditions on the underlying curve.

Proposed method

• Uses semiprime PI algebras and the existence of d-central polynomials to analyze extension vanishing for modules of maximal and non-maximal dimension.
• Applies the second layer link theory to relate prime ideals and study the structure of noncommutative algebras via their module categories.
• Employs formal deformation theory to construct versal families of representations, particularly for 2-dimensional modules over noncommutative plane curves.
• Utilizes the noncommutative Jacobian matrix to derive relations between matrix entries in 2×2 representations of noncommutative elliptic curves.
• Applies trace ring techniques and matrix arithmetic (Lemma 3.2) to reduce noncommutative relations to commutative equations in traces and determinants.
• Uses the Formanek element to define the simple locus as the complement of a hyperelliptic curve of genus 2 in the trace ring.

Experimental results

Research questions

• RQ1 Under what conditions do Ext^1 groups between non-isomorphic simple modules of different dimensions vanish or not vanish in noncommutative plane curves?
• RQ2 How are 2-dimensional simple representations of noncommutative elliptic curves related to the geometry of the underlying classical elliptic curve?
• RQ3 What is the role of the noncommutative Jacobian matrix in determining the existence and structure of indecomposable modules?
• RQ4 Can the completion problem—connecting simple modules of varying dimensions—be resolved via deformation-theoretic methods in noncommutative algebraic geometry?
• RQ5 What geometric constraints (e.g., collinearity) arise from the non-vanishing of Ext^1 groups in noncommutative elliptic curves?

Key findings

• Ext^1(M,V) = 0 and Ext^1(V,M) = 0 for finite length modules M not containing a maximal-dimensional simple module V, due to the action of a d-central polynomial.
• For a semiprime PI algebra of degree d, a d-central polynomial c acts as an isomorphism on d-dimensional simples and as zero on lower-dimensional simples, enabling localization arguments to prove extension vanishing.
• The 2-dimensional simple representations of the noncommutative elliptic curve algebra are parametrized by solutions to a quadratic equation derived from the noncommutative Jacobian matrix.
• The two solutions Q1 and Q2 to the Jacobian condition satisfy P = Q1 + Q2 under the elliptic curve group law, with the line through Q1 and Q2 having slope −q.
• Non-vanishing Ext^1(P,Q) and Ext^1(Q,P) implies the existence of points Q2 and P2 such that P = Q + Q2 and Q = P + P2, which leads to a contradiction unless q = 0.
• There are no 3-cycles of simple modules with all pairwise non-vanishing Ext^1 groups, as such cycles would require inconsistent vector equations in the elliptic curve group law.