which is the infimum of Gorenstein colengths in birational maps.

For a local ring $R$, we define $$bg(R)\coloneqq\inf\left\{\ell_S(R/S) \middle|
\begin{array}{l}
 S\text{ is Gorenstein and }S\subseteq R\text{ is a module-finite}\\
\text{residually rational birational map of local rings}
\end{array}
\right\}.$$

We will state the main results of this paper by using this invariant. The first one is the following theorem, which gives a one-dimensional analogue of Theorem \[thHVAAM\].

\[thA\] Let $(R,\m)$ be a one-dimensional Cohen-Macaulay local ring having a canonical module $\omega$. Consider the following conditions.

-   $bg(R)\leq 1$.

-   Either $R$ is Gorenstein or there exists a Gorenstein local ring $(S,\mathfrak{n})$ of dimension one such that $R\cong \End_S(\mathfrak{n})$.

-   Either $R$ is Gorenstein or $\m\cong \m^\dag$.

-   There is a short exact sequence $0\to \omega \to \m \to k \to 0$.

-   There is an ideal $I$ of $R$ such that $I\cong \omega$ (i.e. $I$ is a canonical ideal of $R$) and $l(R/I)\leq 2$.

Then the implications $(1)\implies (2) \implies (3) \iff (4) \iff (5)$ hold. The direction $(5) \implies (1)$ also holds if $R$ contains an infinite field $k$ as a subalgebra which is isomorphic to $R/\m$ via the projection $R\to R/\m$, i.e. $R$ has an infinite coefficient field $k\subseteq 