A non-vanishing result on the singularity category (Q6583751)

The singularity category is a fundamental homological invariant for a ring with infinite global dimension. Let \(d\ge 1\). An object \(M\) in an abelian category \(A\) is called virtually \(d\)-periodic provided that \(M\) has infinite projective dimension and that the \(d\)-th syzygy \(\Omega^d(M)\) lies in \(\langle M\rangle\), where \(\langle M\rangle\) denotes the sub-additive category which contains \(M\) and all projective objects in \(A\), and is closed under extensions, direct summands.\N\NThe present authors 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. There are several nice consequences: 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. They authors also verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras.