Analytically unramified local ring of dimension $1$ always admit canonical module?

Question

Let $(R, \mathfrak m)$ be a reduced local ring of dimension $1$ such that the completion of $R$ is also reduced (such rings are called analytically unramified). Note that $R$ is Cohen-Macaulay since it is reduced and has dimension $1$.

My question is:

Does $R$ necessarily admit a canonical module? I.e., is $R$ a homomorphic image of a local Gorenstein ring?

Answer 1

Yes, $R$ necessarily has a canonical module. This is a special case of the following:

Theorem [Herzog, Satz 6.21]. Let $R$ be a one-dimensional Cohen–Macaulay local ring. Then, the following statements are equivalent:

1. A canonical ideal exists.
2. For every minimal prime ideal $\mathfrak{p}$ of the completion $\hat{R}$ of $R$, the localization $\hat{R}_{\mathfrak{p}}$ is a Gorenstein ring.