# Finitely generated module, which is a virtually small complex, embeds into a module of finite projective dimension?

Let $$R$$ be a commutative Noetherian ring, and let $$\text{mod } R$$ denote the abelian category of finitely generated $$R$$-module. Consider the bounded derived category $$D^b(\text{mod } R)$$ which is a triangulated category. Every object $$X \in \text{mod } R$$ can be naturally identified in $$D^b(\text{mod } R)$$. For an object $$X \in D^b(\text{mod } R)$$, let $$\text{Thick}_{D^b(\text{mod } R)} X$$ denote the intersection of all thick subcategories (https://ncatlab.org/nlab/show/thick+subcategory) of $$D^b(\text{mod } R)$$ containing $$X$$. For example, note that $$\text{Thick}_{D^b(\text{mod } R)} R$$ is the collection of all perfect complexes, hence $$M \in \text{mod } R$$ belongs to $$\text{Thick}_{D^b(\text{mod } R)} R$$ if and only if $$M$$ has finite projective dimension.

Following 4.1 of https://doi.org/10.4171/cmh/56, we say $$X\in D^b(\text{mod } R)$$ is Virtually small if there exists a non-exact complex $$Y \in \text{Thick}_{D^b(\text{mod } R)} X \cap \text{Thick}_{D^b(\text{mod } R)} R$$.

My question is: If $$M \in \text{mod } R$$ is Virtually small in $$D^b(\text{mod } R)$$, then does $$M$$ embed into a finitely generated module of finite projective dimension?

For every $$M$$, $$M\oplus R$$ is virtually small, so your question is equivalent to the question: Does every finitely generated $$R$$-module embed in a finitely generated module of finite projective dimension?
The answer is no. For example, let $$R$$ be a local commutative finite dimensional algebra over a field $$k$$, that is not self-injective, such as $$R=k[x,y]/(x^2,xy,y^2)$$. Then the only finitely generated modules of finite projective dimension are the finitely generated projective modules, and the injective generator $$\operatorname{Hom}_k(R,k)$$ does not embed in a projective module.