Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be finitely generated $R$-modules.

Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ and $N$ have finite projective dimension?

[Since $M\otimes_R^{\mathbf L} N$ is represented by the chain complex $M \otimes_R F_{\bullet}$ where $F_{\bullet}$ is a resolution of $N$ by finite free modules, so let me recall here that a homologically bounded below chain complex $C$ of finitely generated modules is said to have finite projective dimension if $C $ is isomorphic, in the derived category, to a bounded complex of finitely generated projective modules (free in our case) ]

I would be happy even with an explicit reference.

Thanks

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be non-zero finitely generated $R$-modules.

Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ and $N$ have finite projective dimension?

[Since $M\otimes_R^{\mathbf L} N$ is represented by the chain complex $M \otimes_R F_{\bullet}$ where $F_{\bullet}$ is a resolution of $N$ by finite free modules, so let me recall here that a homologically bounded below chain complex $C$ of finitely generated modules is said to have finite projective dimension if $C $ is isomorphic, in the derived category, to a bounded complex of finitely generated projective modules (free in our case) ]

I would be happy even with an explicit reference.

Thanks

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be finitely generated $R$-modules.

Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ and $N$ have finite projective dimension?

I would be happy even with an explicit reference.

Thanks

Let $(R, \mathfrak m)$ be a Noetherian local ring. Let $M,N$ be finitely generated $R$-modules.

Is it known that $M\otimes_R^{\mathbf L} N$ has finite projective dimension if and only if $M$ and $N$ have finite projective dimension?

[Since $M\otimes_R^{\mathbf L} N$ is represented by the chain complex $M \otimes_R F_{\bullet}$ where $F_{\bullet}$ is a resolution of $N$ by finite free modules, so let me recall here that a homologically bounded below chain complex $C$ of finitely generated modules is said to have finite projective dimension if $C $ is isomorphic, in the derived category, to a bounded complex of finitely generated projective modules (free in our case) ]

I would be happy even with an explicit reference.

Thanks