Module of finite projective dimension and base change with respect to the completion

Question

Let $(R,\mathfrak{m})$ be a local ring and let $M$ be an $\hat{R}$-module, where $\hat{R}$ denotes the $\mathfrak{m}$-adic completion of $R$. Suppose that $\operatorname{pd}_R(M)< \infty$. Does this imply $\operatorname{pd}_{\hat{R}}(M)< \infty$ ?

Answer 1

The answer is yes. If $\operatorname{pd}_R(M)<\infty$, using flatness, one has $\operatorname{pd}_{\hat{R}}(M\otimes_R\hat{R})<\infty$. Tensoring $R\to \hat{R}$ by $M$, one has a map $M\to M\otimes_R\hat{R}$ and using the map $M\times \hat{R}\to M$ by $(m,a)\mapsto am$, we get a map $M\otimes_R\hat{R}\to M$, which gives a splitting $M\otimes_R\hat{R}=M\oplus N$ for some $N$. Since direct summand of a module of finite projective dimension must have finite projective dimension, we are done.