A perfect complex over a local Cohen--Macaulay ring whose canonical dual is concentrated in a single degree

Question

Let $R$ be a complete local Cohen--Macaulay ring with dualizing module $\omega$. Let $M$ be a perfect complex over $R$. If the homology of $\mathbf R\text{Hom}_R(M,\omega)$ is concentrated in a single degree, then is the homology of $M$ also concentrated in a single degree ? What if I also assume $R$ is Gorenstein?

Answer 1

Not even for a Gorenstein ring.

Let $R$ be $\mathbb{Z}_p$, the ring of $p$-adic integers. It is Gorenstein, and therefore its own dualizing module.

Let $M$ be the direct sum of the two complexes $$\cdots\to0\to\mathbb{Z}_p\xrightarrow{p}\mathbb{Z}_p\to0\to\cdots$$ and $$\cdots\to0\to\mathbb{Z}_p\xrightarrow{}0\to0\to\cdots$$ with the first nonzero terms in degree zero.

Then $M$ has nonzero homology in two different degrees, but $\mathbf R\text{Hom}_R(M,R)$ has homology, isomorphic to $\mathbb{Z}_p/(p)\oplus\mathbb{Z}_p$, only in degree zero.