# Thick subcategory containment in bounded derived category vs. singularity category

Let $$R$$ be a commutative Noetherian ring, and $$D^b(\operatorname{mod } R)$$ the bounded derived category of the abelian category of finitely generated $$R$$-modules. Let me abbreviate this as $$D^b(R)$$. Consider the singularity category $$D_\text{sg}(R)$$ as the Verdier localization $$D^b(R)/{\operatorname{perf}(R)}$$, where $$\operatorname{perf}(R) :=\operatorname{thick}_{D^b(R)}(R)$$ is the subcategory of $$D^b(R)$$ of perfect complexes.

My question is the following: Given $$M,N\in D^b(R)$$, is it true that $$M\in \operatorname{thick}_{D_\text{sg}(R)}(N)$$ if and only if $$M\in \operatorname{thick}_{D^b(R)}(N\oplus R)$$ ?

Yes, this is true more generally. It follows from the following observation : for a triangulated category $$D$$, a thick subcategory $$C$$ and $$x \in D$$, $$x$$ is in $$C$$ if and only if $$x=0$$ in $$D/C$$.
Combining this with $$(D/E)/(C/E)\simeq D/C$$ with $$D= D^b(R), E = Perf(R)$$ and $$C = thick(N \oplus R)$$ for $$x = M$$ will show the result