Triangle Order $\leq_{\bigtriangleup}$ in Singular Categories

TL;DR

This paper proves that the triangle order in the singular category of a finite-dimensional algebra forms a partial order on its objects, clarifying the structure of these categories.

Contribution

It establishes that the triangle order $igtriangleup$ is a partial order in the singular category $D_{sg}(A)$ for finite-dimensional algebras.

Findings

Triangle order $igtriangleup$ is a partial order in $D_{sg}(A)$.

The result applies to finite-dimensional $k$-algebras.

Clarifies the structure of objects in singular categories.

Abstract

We prove that the triangle order $\leq_{\bigtriangleup}$ in the singular category $D_{sg}(A)$ defines a partial order on the set of isomorphism classes of objects in $D_{sg}(A)$ for a finite-dimensional $k$-algebra $A$.