[Paper Review] Homotopy categories, Leavitt path algebras and Gorenstein projective modules

Xiao‐Wu Chen, Dong Yang|arXiv (Cornell University)|Jan 2, 2013

Algebraic structures and combinatorial models3 citations

TL;DR

This paper establishes triangle equivalences between the homotopy category of acyclic complexes of injective modules over finite-dimensional algebras with radical square zero and derived categories of Leavitt path algebras, as well as stable categories of Gorenstein projective modules over trivial extension algebras. The key contribution is a unified description of the singularity category of such algebras via Leavitt path algebras and Gorenstein projective modules, generalizing prior results and linking homotopy theory, noncommutative algebraic geometry, and singularity theory.

ABSTRACT

For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic complexes of projective modules is given. Restricting these equivalences to compact objects, we obtain various descriptions of the singularity category of a finite dimensional algebra with radical square zero, which contain previous results.

Motivation & Objective

• To describe the homotopy category of acyclic complexes of injective modules over finite-dimensional algebras with radical square zero.
• To relate this category to the derived category of Leavitt path algebras and the stable category of Gorenstein projective modules.
• To provide new descriptions of the singularity category of such algebras, generalizing previous results.
• To establish equivalences between homotopy categories, derived categories of Leavitt path algebras, and Gorenstein projective stable categories.

Proposed method

• Use of differential graded algebras with trivial differential to model Leavitt path algebras.
• Application of Koszul duality techniques to relate homotopy categories of injective modules to derived categories of path algebras.
• Construction of trivial extension algebras from von Neumann regular algebras and invertible bimodules.
• Use of graded Morita equivalence and derived equivalence to relate Leavitt path algebras and their opposite algebras.
• Establishment of triangle equivalences via compact objects and restriction to singularity categories.
• Leveraging known results on Gorenstein projective modules and Frobenius categories to define stable categories.

Experimental results

Research questions

• RQ1 How are the homotopy categories of acyclic complexes of injective modules over algebras with radical square zero related to derived categories of Leavitt path algebras?
• RQ2 What is the relationship between the stable category of Gorenstein projective modules over trivial extension algebras and the homotopy categories of acyclic complexes of injective modules?
• RQ3 Can the singularity category of a finite-dimensional algebra with radical square zero be described via Leavitt path algebras and Gorenstein projective modules?
• RQ4 Under what conditions are two such algebras singularly equivalent, and how does this relate to graded Morita equivalence of their Leavitt path algebras?
• RQ5 Is there a triangle equivalence between the homotopy category of acyclic complexes of projective modules and the derived category of the opposite Leavitt path algebra?

Key findings

• There is a triangle equivalence between the homotopy category of acyclic complexes of injective modules over $kQ/J^2$ and the derived category of the opposite Leavitt path algebra $L(Q)^{ m op}$.
• The derived category of $L(Q)^{ m op}$ is triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra $\Lambda^+(Q) = L(Q)^0 \ltimes L(Q)^1$.
• The homotopy category of acyclic complexes of projective modules over $kQ/J^2$ is triangle equivalent to the derived category of $L(Q^{ m op})$ and to the stable category of Gorenstein projective modules over $\Lambda^-(Q^{ m op})$.
• The compact objects in these categories correspond to the singularity category $\mathbf{D}_{\rm sg}(kQ/J^2)$, which is equivalent to the perfect derived category $\mathrm{perf}(L(Q)^{ m op})$ and to the stable category of finitely presented Gorenstein projective $\Lambda^+(Q)$-modules.
• The equivalences restrict to compactly generated triangulated categories, and the singularity category is fully described via Leavitt path algebras and Gorenstein projective modules.
• The paper proves that two such algebras are singularly equivalent if and only if their Leavitt path algebras are graded Morita equivalent, derived equivalent, or their trivial extension algebras are Morita equivalent.