Backstr\"om algebras

TL;DR

This paper introduces Backström pairs and rings, explores their derived categories, and establishes bounds on their global and derived dimensions, providing new categorical and homological insights.

Contribution

It defines Backström pairs and rings, studies their derived categories, and constructs categorical resolutions and semi-orthogonal decompositions.

Findings

Derived dimension of Backström rings is at most 2

Constructed categorical resolutions for Backström rings

Established semi-orthogonal decomposition of derived categories

Abstract

We introduce Backstr\"om pairs and Backstr\"om rings, study their derived categories and construct for them a sort of categorical resolutions. For the latter we define the global dimension, construct a sort of semi-orthogonal decomposition of the derived category and deduce that the derived dimension of a Backstr\"om ring is at most $2$. Using this semi-orthogonal decomposition, we define a description of the module category as the category of elements of a special bimodule. We also construct a partial tilting for a Backstr\"om pair to a ring of triangular matrices and define the global dimension of the latter.