The Schwarzian derivative of a holomorphic function $f$ of a complex variable $z$ is defined as $\{f, z\} = \frac{f^{\prime\prime\prime}(z)}{f^\prime(z)} - \frac{3}{2}{\Big(\frac{f^{\prime\prime}(z)}{f^\prime(z)}\Big)}^2$. It is invariant under all linear fractional transformations, that is, $\{g, z\}$ for any linear fractional transformation $g(z) = \frac{az + b}{cz + d}$ equals zero.
The Schwarzian derivative has a nice geometric interpretation based on its definition as an operator that is invariant under linear fractional transformations. (Read Bill Thurston's answer to a question posted on MathOverflow long back for a geometric description.)
The way I understand it, when we're looking from a geometric point of view, a linear fractional transformation is defined as a homography on the complex projective line. However, a linear fractional transformation can also be defined as a homography over the projective line over a ring. (See this for an explanation.)
I'm interested in understanding if the definition of a linear fractional transformation as a homography over the projective line over a ring leads to (or can be used to obtain) an alternative description of the Schwarzian derivative. (Say, a relatively more algebraic description compared to the geometric one obtained from the standard definition.)
