Dual of a module of finite projective dimension

Question

Let $(R, {\frak m})$ be a Noehterian local ring and $M$ be finitely generated $R$-module of finite projective dimension. Is the projective dimension of ${\rm Hom}_R(M,R)$ finite?

Answer 1

Let $R$ be a non-regular local domain. Then, picking a set of generators $x_1,\ldots, x_n$ for the maximal ideal $\mathfrak{m}$, one has an exact sequence,

$$0\to R\stackrel{(x_1,\ldots,x_n)}{\longrightarrow} R^n\to M\to 0,$$ where $M$ is defined by the sequence. Thus $M$ has projective dimension one. Dualizing, one gets an exact sequence, $$0\to \operatorname{Hom}(M, R)\to R^n\to R\to R/\mathfrak{m}=k\to 0.$$

Since $k$ has infinite projective dimension, so does $\operatorname{Hom}(M,R)$.