Frobenius categories, Gorenstein algebras and rational surface singularities

Abstract

We give sufficient conditions for a Frobenius category to be equivalent to the category of Gorenstein projective modules over an Iwanaga-Gorenstein ring. We then apply this result to the Frobenius category of special Cohen-Macaulay modules over a rational surface singularity, where we show that the associated stable category is triangle equivalent to the singularity category of a certain discrepant partial resolution of the given rational singularity. In particular, this produces uncountably many Iwanaga-Gorenstein rings of finite GP type. We also apply our method to representation theory, obtaining Auslander-Solberg and Kong type results.

27 pages, to appear in Comp. Math

Cited by in corpus (14)

• Noncommutative deformations and flops
• Relative singularity categories I: Auslander resolutions
• A categorification of Grassmannian cluster algebras
• Classifying exact categories via Wakamatsu tilting
• Internally Calabi-Yau algebras and cluster-tilting objects
• Relative singularity categories II: DG models
• Mutation of frozen Jacobian algebras
• Flops and Clusters in the Homological Minimal Model Program
• Singularity categories via the derived quotient
• A new equivalence between singularity categories of commutative algebras
• Perfect matching modules, dimer partition functions and cluster characters
• Idempotent completions of $n$-exangulated categories
• Quiver representations over a quasi-Frobenius ring and Gorenstein-projective modules
• Singularity categories via higher McKay quivers with potential