[Paper Review] The singularity category of a quadratic monomial algebra

Xiao‐Wu Chen|arXiv (Cornell University)|Feb 7, 2015

Algebraic structures and combinatorial models24 references3 citations

TL;DR

This paper establishes a singular equivalence between a quadratic monomial algebra and an algebra with radical square zero via pre-triangle equivalences between their stable module categories. The key contribution is a concrete description of the singularity category of a quadratic monomial algebra using graded projective modules over Leavitt path algebras of associated quivers.

ABSTRACT

We exploit singular equivalences between artin algebras, that are induced from certain functors between the stable module categories. Such functors are called pre-triangle equivalences. We construct two pre-triangle equivalences connecting the stable module category over a quadratic monomial algebra and the one over an algebra with radical square zero. Consequently, we obtain an explicit singular equivalence between the two algebras.

Motivation & Objective

• To establish a singular equivalence between a quadratic monomial algebra and an algebra with radical square zero.
• To introduce and utilize the concept of pre-triangle equivalences between stable module categories to induce singular equivalences.
• To describe the singularity category and Gorenstein defect category of a quadratic monomial algebra in terms of graded projective modules over Leavitt path algebras.
• To generalize prior results on gentle algebras to the broader class of quadratic monomial algebras.
• To provide explicit triangle equivalences linking the singularity category, Gorenstein-projective modules, and the Gorenstein defect category of such algebras.

Proposed method

• Define pre-triangle equivalences as functors between stable module categories that induce triangle equivalences after stabilization.
• Construct two explicit pre-triangle equivalences connecting the stable module category of a quadratic monomial algebra to that of a radical square zero algebra.
• Use the stabilization process to relate the singularity category of the original algebra to the stabilized stable module category.
• Leverage known results on algebras with radical square zero to describe their singularity categories via graded projective modules over Leavitt path algebras.
• Apply the theory of looped categories and stabilization to relate the singularity category to the Gorenstein defect category and the category of graded projective modules.
• Utilize the relation quiver of the algebra to classify perfect and defect components, which determine the structure of the singularity category.

Experimental results

Research questions

• RQ1 How can singular equivalences be constructed between quadratic monomial algebras and algebras with radical square zero?
• RQ2 What is the role of pre-triangle equivalences in inducing singular equivalences between stable module categories?
• RQ3 How does the singularity category of a quadratic monomial algebra decompose in terms of Leavitt path algebras?
• RQ4 What is the relationship between the Gorenstein defect category and the singularity category of such algebras?
• RQ5 Under what conditions does the singularity category of a quadratic monomial algebra become equivalent to a product of triangulated categories?

Key findings

• The singularity category of a quadratic monomial algebra is triangle equivalent to the product of triangulated categories $\mathcal{T}_{d_1} \times \mathcal{T}_{d_2} \times \cdots \times \mathcal{T}_{d_m}$, where $d_i$ are the sizes of the perfect components of the relation quiver.
• The Gorenstein defect category of a quadratic monomial algebra is triangle equivalent to the category of graded projective modules over the Leavitt path algebra of the defect quiver $\mathcal{R}_A^{\rm def}$ with a shift automorphism.
• When the algebra is Gorenstein, the Gorenstein defect category is trivial, and the singularity category is equivalent to the product of triangulated categories $\mathcal{T}_{d_1} \times \cdots \times \mathcal{T}_{d_m}$.
• For a quadratic monomial algebra with no perfect components in its relation quiver, the singularity category is equivalent to the Gorenstein defect category, which is itself equivalent to the category of graded projective modules over the Leavitt path algebra of the defect quiver.
• The equivalence between the stable module category of the quadratic monomial algebra and that of the radical square zero algebra is induced by a zigzag of pre-triangle equivalences, leading to a full singular equivalence.
• Explicit triangle equivalences are established between $A$-mod, $B$-mod, and the categories $\mathcal{T}_{d_i}$, showing that the singularity category decomposes as a product of triangulated categories based on the structure of the relation quiver.