Torsion-free Cohen-Macaulay modules We say an $R$-module $M$ over integral domain $R$ is a torsion-free module if zero is the only element annihilated by some non-zero element of the ring $R$. Let $R=K[[x_1,\dots ,x_d]]$, $d>1$, be the formal power series ring over a field $K$. It is clear that $R$ is a Cohen-Macaulay ring of dimension $d$.

Is there any Cohen-Macaulay $R$-module $M$ of dimension $d-1$ such that $M$ is also torsion-free as an $R$-module?


For every finitely generated module $\dim M=\dim R/\mathrm{Ann}_R(M)$. Since $M$ is torsion-free, $\mathrm{Ann}_R(M)=(0)$, hence $\dim M=\dim R$.
In particular, a CM torsion-free module is MCM.
