On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness

2025/10/08 by Ralf Schiffler, Schiffler, Ralf, Khrystyna Serhiyenko +1
Mathematics · #13F60 (Secondary) #16G50 (Primary) 16G20 #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2510.07405

openalex publication_date 2025/10/08 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga-Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or equivalently the stable categories of Cohen-Macauley modules \underlineCMP. First we show that if an algebra A is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical rad A. Next, for a 2-Calabi-Yau tilted algebra A we provide an explicit relationship between the \underlineCMP category of A and its quotient A/Ae_iA by an ideal generated by an idempotent e_i. Consequently, we obtain various equivalent characterizations of when the \underlineCMP category remains the same after passing to the quotient. We also obtain applications to two classes of algebras that are CM finite, the dimer tree algebras and their skew group algebras.

Citations

• On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras
• A geometric model for syzygies over 2-Calabi-Yau tilted algebras
• Singularity categories of Gorenstein monomial algebras
• On syzygies over 2-Calabi-Yau tilted algebras
• Singularity categories of some 2-CY-tilted algebras
• Singularity categories of skewed-gentle algebras
• The singularity categories of the Cluster-tilted algebras of Dynkin type
• The singularity category of a Nakayama algebra
• Stable categories of Cohen-Macaulay modules and cluster categories
• Cluster categories for algebras of global dimension 2 and quivers with potential
• Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

Related

• On syzygies over 2-Calabi-Yau tilted algebras
• On Gorenstein algebras of finite Cohen-Macaulay type: dimer tree algebras and their skew group algebras
• Tilting theory for finite dimensional 1-Iwanaga-Gorenstein algebras
• On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules
• Cluster-tilted algebras are Gorenstein and stably Calabi-Yau