Let $R$ be a regular local ring and $M$ a finitely generated reflexive $R$-module. When $R$ has dimension 2, then $M$ is a free $R$-module. This is discussed in [Reflexive modules over a 2-dimensional regular local ring](https://mathoverflow.net/questions/22943/reflexive-modules-over-a-2-dimensional-regular-local-ring).


Suppose $R$ has dimension 3. Are there any results about the structure of $M$ in this case?



Structure of reflexive modules over regular local rings
If $R$ is a Gorenstein (or even Cohen-Macaulay and Gorenstein in codimension $1$) local ring, then $M$ is reflexive if and only if it is a second syzygy (see e.g. [this answer](https://math.stackexchange.com/q/3487112)). As regular rings are Gorenstein, this directly extends the result you quote for regular local rings of dimension $2$. Indeed, a regular local ring of dimension $2$ has global dimension $2$, so second syzygies are free.


