[Paper Review] On periodicity in bounded projective resolutions

Alex Dugas|arXiv (Cornell University)|Mar 12, 2012

Advanced Topics in Algebra6 references3 citations

TL;DR

This paper establishes that any $A$-module of complexity one with an open $\mathrm{GL}_d(k)$-orbit in the module variety $\mathcal{M}\mathrm{od}^A_d$ over a self-injective algebra $A$ over an algebraically closed field is periodic. Using geometric techniques involving orbit closures and rigidity, the authors prove that such modules must have periodic projective resolutions, and as a corollary, all simple modules of complexity one over self-injective algebras are periodic.

ABSTRACT

Let A be a self-injective algebra over an algebraically closed field k. We show that if an A-module M of complexity one has an open orbit in the variety of d-dimensional A-modules, then M is periodic. As a corollary we see that any simple A-module of complexity one must be periodic. In the course of the proof, we also show that modules with open orbits are preserved by stable equivalences of Morita type between self-injective algebras.

Motivation & Objective

• To determine conditions under which modules of complexity one over self-injective algebras are periodic.
• To establish a geometric criterion—open $\mathrm{GL}_d(k)$-orbits—for periodicity in bounded projective resolutions.
• To show that simple modules of complexity one over self-injective algebras are periodic, using orbit geometry and rigidity.
• To demonstrate that the property of having an open orbit is preserved under stable equivalences of Morita type.

Proposed method

• The authors use the affine variety $\mathcal{M}\mathrm{od}^A_d$ parametrizing $d$-dimensional $A$-modules, with $\mathrm{GL}_d(k)$ acting via base change.
• They characterize open orbits as those whose closure is an irreducible component of $\mathcal{M}\mathrm{od}^A_d$.
• They define rigidity of a module $M$ as the condition that the set of points in $\mathrm{Spec}\, R$ mapping to $M$ under base change is open for any one-dimensional noetherian domain $R$.
• They show that a module has an open orbit if and only if it is rigid in this algebraic sense.
• They use the fact that syzygies of a complexity one module with open orbit lie in a common irreducible component, forcing isomorphism of some syzygies and hence periodicity.
• They apply results from Dade on algebraic rigidity and use the structure of the enveloping algebra $A^e$ to analyze periodicity of the algebra itself.

Experimental results

Research questions

• RQ1 Under what conditions is a module of complexity one over a self-injective algebra periodic?
• RQ2 Can the geometric property of having an open $\mathrm{GL}_d(k)$-orbit in the module variety imply periodicity of the module?
• RQ3 Are all simple modules of complexity one over self-injective algebras periodic?
• RQ4 Is the property of having an open orbit preserved under stable equivalences of Morita type?
• RQ5 Can the geometry of the $A^e$-module $A$ be used to determine whether $A$ is periodic?

Key findings

• A $d$-dimensional $A$-module $M$ of complexity one with an open $\mathrm{GL}_d(k)$-orbit in $\mathcal{M}\mathrm{od}^A_d$ is periodic.
• Every simple $A$-module of complexity one over a self-injective algebra is periodic.
• Modules with open orbits are preserved under stable equivalences of Morita type between self-injective algebras.
• The orbit of a module is open if and only if it is rigid in the algebraic sense defined via base change over one-dimensional domains.
• The syzygies of a complexity one module with open orbit lie in a common irreducible component of the module variety, leading to isomorphic syzygies and hence periodicity.
• The $A^e$-module $A$ is not a proper degeneration of any other $A^e$-module, indicating a minimal geometric position in the variety $\mathcal{M}\mathrm{od}^{A^e}_d$.