On derived equivalences of lines, rectangles and triangles

TL;DR

This paper introduces a new method for constructing derived equivalences of algebras using tensor products, expanding the class of known equivalences and including examples related to singularity theory and quiver algebras.

Contribution

It generalizes Rickard's result by providing a tensor product-based approach to derive new tilting complexes and equivalences, especially for algebras associated with geometric shapes.

Findings

Constructed new derived equivalences using tensor products.

Identified classes of algebras with fractional Calabi-Yau properties.

Explicitly computed Calabi-Yau dimensions for various examples.

Abstract

We present a method to construct new tilting complexes from existing ones using tensor products, generalizing a result of Rickard. The endomorphism rings of these complexes are generalized matrix rings that are "componentwise" tensor products, allowing us to obtain many derived equivalences that have not been observed by using previous techniques. Particular examples include algebras generalizing the ADE-chain related to singularity theory, incidence algebras of posets and certain Auslander algebras or more generally endomorphism algebras of initial preprojective modules over path algebras of quivers. Many of these algebras are fractionally Calabi-Yau and we explicitly compute their CY dimensions. Among the quivers of these algebras one can find shapes of lines, rectangles and triangles.