English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We establish a general method to construct triangle equivalences between thesingularity categories of (commutative and non-commutative) Gorenstein ringsand the cluster categories of finite dimensional algebras over fields. We provethat such an equivalence exists as soon as there is a triangle equivalencebetween the graded singularity category of a Gorenstein ring and the derivedcategory of a finite dimensional algebra. This is based on two key results ondg enhancements of cluster categories and of singularity categories, which areof independent interest. First we give a characterization of dg orbitcategories among $\mathbb{Z}$-graded dg categories in terms of a generatingcondition. Secondly, we show that the canonical dg enhancements of thesingularity categories of symmetric orders have the bimodule Calabi-Yauproperty, which lifts the classical Calabi-Yau property of singularitycategories. We apply our results to such classes of rings as Gorenstein ringsof dimension at most $1$, quotient singularities, and Geigle-Lenzing completeintersections, including finite or infinite Grassmannian cluster categories, torealize their singularity categories as cluster categories of finitedimensional algebras.

Enhanced Auslander-Reiten duality and tilting theory for singularity categories