[Paper Review] Generalization of the Correspondence about DTr-selfinjective algebras

Fan Kong|arXiv (Cornell University)|Apr 4, 2012

Algebraic structures and combinatorial models5 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between n-1-DTr-selfinjective algebras and algebras of dominant dimension and injective dimension both equal to n, for n ≥ 2. It generalizes a classical correspondence from Auslander and Solberg (1997) by linking higher-dimensional Auslander-Reiten theory with Gorenstein homological algebra, showing that the Gorenstein projective module categories of such algebras are equivalent to those of algebras with periodic DTr-orbits.

ABSTRACT

We give a correspondence between (n-1)-DTr-selfinjective algebras and algebras with dominant dimension and selinjective dimension being both n for any n bigger than 1 . Furthermore, we show the relation between the module categories of the two kinds of algebras. At last we show the sepcial condition n = 2.

Motivation & Objective

• To generalize the correspondence between 2-DTr-selfinjective algebras and algebras of dominant and injective dimension 2 established by Auslander and Solberg.
• To extend this correspondence to higher dimensions, specifically for n ≥ 2, relating (n−1)-DTr-selfinjective algebras to algebras with both dominant and injective dimension equal to n.
• To clarify the relationship between the Gorenstein projective module categories of these two classes of algebras.
• To demonstrate that periodicity in higher-dimensional DTr-orbits characterizes the Gorenstein projective module categories in this generalized setting.

Proposed method

• Define (n−1)-DTr-selfinjective algebras via a generator-cogenerator Q that is (n−2)-self-orthogonal and closed under τ^{n−1} and τ^{-(n−1)} operations.
• Construct a functor F from the class of algebras with dominant and injective dimension n to the class of (n−1)-DTr-selfinjective algebras via the opposite endomorphism ring of the minimal faithful injective module I^Λ.
• Define a functor G in the reverse direction using the endomorphism ring of the generator-cogenerator Q in the (n−1)-DTr-selfinjective algebra.
• Use Morita equivalence to define equivalence classes of algebras and generator-cogenerators, ensuring the correspondence respects algebraic structure.
• Leverage higher Auslander-Reiten theory tools, including τ^m and τ^{-m} functors, and orthogonal subcategories (⊥_n, ⊥_n) to analyze module categories.
• Establish that the Gorenstein projective modules of algebras in 𝔻_n are equivalent to those of algebras in ℬ_n via the constructed functors.

Experimental results

Research questions

• RQ1 Can the correspondence between 2-DTr-selfinjective algebras and algebras of dominant and injective dimension 2 be generalized to higher dimensions for n ≥ 2?
• RQ2 What is the precise structural condition under which an algebra of dominant dimension n is linked to an (n−1)-DTr-selfinjective algebra?
• RQ3 How do the Gorenstein projective module categories of these two classes of algebras relate under the proposed correspondence?
• RQ4 Does periodicity in higher-dimensional DTr-orbits characterize the Gorenstein projective module categories in this generalized setting?
• RQ5 Is the correspondence functorial and bijective at the level of Morita equivalence classes?

Key findings

• A one-to-one correspondence is established between the Morita equivalence classes of algebras with dominant and injective dimension n and the Morita equivalence classes of (n−1)-DTr-selfinjective algebras.
• The correspondence is explicitly constructed via F([Λ]) = [End^{op}(I^Λ), D(I^Λ)] and G([Γ,Q]) = [End^{op}(Q)], showing that the endomorphism rings of minimal faithful modules and generator-cogenerators are linked via opposite rings.
• The Gorenstein projective module categories of algebras in 𝔻_n are equivalent to those of algebras in ℬ_n, preserving the homological structure.
• The periodicity of DTr-orbits in higher dimensions (τ^{n−1}Q ∈ add Q) is shown to be the key homological invariant characterizing the correspondence.
• The construction is functorial and respects the categorical equivalence between module categories, as shown via the use of orthogonal subcategories and hom-finite properties.
• The proof relies on the existence of projective generators in the relevant abelian categories, which is established via a contradiction argument on infinite chains of non-isomorphisms.