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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 $M\in \operatorname{thick}_{D^b(R)}(N\oplus R)$ ?

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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

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