A non-vanishing result on the singularity category

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.