Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $D^b(R)/\text{thick}_{D^b(R)}(R)$.
If $r\in R$ and $M\in D^b(R)$ are such that the morphism $M\xrightarrow{r} M$ is $0$ in $D_{sg}(R)$, then in $D^b(R)$, does there exist a perfect complex $P$ and morphisms $f:M\to P$ and $g:P\to M$ such that $g\circ f=r\cdot id_M$ ?
Yes.
More generally, if $\mathcal{T}$ is a triangulated category and $\mathcal{S}$ is a thick subcategory, then any morphism $\varphi:M\to N$ of $\mathcal{T}$ that becomes zero in $\mathcal{T}/\mathcal{S}$ factors through an object of $\mathcal{S}$.
By the usual "calculus of fractions" description of morphisms of $\mathcal{T}/\mathcal{S}$, if $\varphi$ is zero in $\mathcal{T}/\mathcal{S}$, then in $\mathcal{T}$ there is a morphism $s:L\to M$ such that $\varphi\circ s=0$ and a distinguished triangle $L\xrightarrow{s}M\to P\to L[1]$ with $P$ in $\mathcal{S}$. But then $\varphi$ factors through $P$.
