A non-vanishing result on the singularity category

TL;DR

This paper establishes a non-vanishing property in the singularity category related to virtually periodic objects, impacting the understanding of silting subcategories and cohomology in algebraic structures.

Contribution

It proves a non-vanishing result for Hom groups in the singularity category and verifies the Singular Presilting Conjecture for specific classes of algebras.

Findings

Singularity category has no silting subcategory for certain algebras.

Differential graded Leavitt algebra has non-vanishing cohomology in all degrees.

Established a trichotomy for the Hom-finiteness of cohomology.

Abstract

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomology of differential graded Leavitt algebras.