[Paper Review] Fabric idempotents and homological dimensions

Jordan McMahon|arXiv (Cornell University)|Mar 19, 2018

Algebraic structures and combinatorial models25 references4 citations

TL;DR

This paper introduces fabric idempotents—special idempotent ideals in finite-dimensional algebras that control homological dimensions and enable the classification of singularity categories. By generalizing Chen-Ye's work on Nakayama algebras, it proves that the singularity category of a higher Nakayama algebra is equivalent to the stable category of a self-injective higher Nakayama algebra, extending the classification to higher-dimensional analogues.

ABSTRACT

Over a finite-dimensonal algbera $A$, simple $A$-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings, and they have also appeared in work concerning the finitistic dimension conjecture. If we however work in a $d$-cluster-tilting subcategory, then not all simples are contained in this subcategory. In this context, a replacement might be to work with idempotent ideals instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localising modules studied by Chen-Krause, and to illustrate the theory we show that they provide rich combinatorial properties. An application is to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.

Motivation & Objective

• To extend the classification of singularity categories of Nakayama algebras to higher Nakayama algebras using homological invariants.
• To define and study fabric idempotents as a generalization of localising modules for algebras with projective dimension one modules.
• To provide a framework for tracking projective generators in resolutions of Gorenstein injective modules via fabric dimension.
• To establish equivalences between singularity categories and stable module categories of self-injective algebras through successive idempotent contractions.

Proposed method

• Introduce fabric idempotents as idempotents f in a finite-dimensional algebra A such that the quotient A/⟨f⟩ has projective dimension at most one and satisfies dual conditions on Auslander-Reiten translations.
• Define the fabric dimension of a projective A/⟨f⟩-module as the minimal n such that it arises as the n-th syzygy of an injective A-module.
• Use the Auslander-Platzeck-Todorov theory to relate fabric idempotents to homological dimensions and Gorenstein injective modules.
• Apply successive contractions along fabric idempotents to reduce higher Nakayama algebras to self-injective ones, preserving singularity categories.
• Construct explicit quiver and relation reductions to show that D_sg(A) ≅ D_sg(fAf) and eventually D_sg(A) ≅ D_sg(B) for a self-injective B.
• Verify that the final contracted algebra f′fAff′ is self-injective, so D_sg(f′fAff′) ≅ ⁡mod(B).

Experimental results

Research questions

• RQ1 Can the classification of singularity categories of Nakayama algebras be extended to higher Nakayama algebras?
• RQ2 What structural conditions on an algebra allow for a fabric idempotent to exist and control homological dimensions?
• RQ3 How does the fabric dimension of a module determine the change in projective generators in its resolution?
• RQ4 Under what conditions does contracting along a fabric idempotent preserve the singularity category?
• RQ5 When does the repeated application of fabric idempotent reduction lead to a self-injective algebra?

Key findings

• For a finite-dimensional higher Nakayama algebra A, there exists a self-injective higher Nakayama algebra B such that D_sg(A) ≅ D_sg(B) ≅ ⁡mod(B).
• The fabric dimension of A/⟨f⟩ is 5 for the 6-Iwanaga-Gorenstein algebra in Example 5, indicating that the first six syzygies of injective modules are generated by A(e + e_14), and beyond that by Af.
• In Example 5, Ω²(I_03) = P_14, Ω⁶(I_13) = P_12, and Ω⁶(I_23) = P_13, confirming the fabric dimension and generator shifts.
• The algebra fAf in Example 5 is isomorphic to the higher Nakayama algebra A^{(2)}_{∝l′} with l′ = (4,3,3,3), and fAf is not self-injective, but f′fAff′ is self-injective.
• The final algebra f′fAff′ has quiver with vertices 12,13,23,25,35,36 and mesh relations, and its singularity category is equivalent to the stable module category.
• The construction ensures that D_sg(A) ≅ D_sg(fAf) ≅ D_sg(f′fAff′) ≅ ⁡mod(f′fAff′), completing the classification.