Upper bounds for dimensions of singularity categories and their annihilators

Let $R$ be a commutative noetherian ring. Denote by $\mathrm{mod}R$ the category of finitely generated $R$-modules and by $\mathrm{}(R)$ the bounded derived category of $\mathrm{mod}R$. In this paper, we first investigate localizations and annihilators of Verdier quotients of $\mathrm{}(R)$. After that, we explore upper bounds for the dimension of the singularity category of $R$ and its (strong) generators. We extend a theorem of Liu to the case where $R$ is neither an isolated singularity nor even a local ring. Some of our results are more generally stated in terms of Spanier–Whitehead category of a resolving subcategory.