Let $\mathscr{R}$ and $\mathscr{T}$ be two sheaves of rings over a topological space $X$. If we take $\mathscr{S}$ to be a sheaf of abelian groups over $X$, we say that $\mathscr{S}$ is a coherent sheaf of $\mathscr{R}$-modules, or $\mathscr{T}$-modules, if for all $w \in X$, and integer $m \geq 0$, we have the existence of an open neighbourhood $U$ such that over $U$, $\mathscr{R}$ (resp. $\mathscr{T})$, admits a chain of syzygies of length $m$.
Can someone provide an elementary example of a sheaf $\mathscr{S}$ that is coherent with respect to one sheaf of rings $\mathscr{R}$ but not coherent with respect to $\mathscr{T}$?
Please note that I am aware of the way many algebraic geometers define a coherent sheaf. That is, in Hartshorne, we have the following definition: Let $(X, \mathscr{O}_X)$ be a scheme. A sheaf of $\mathscr{O}_X$-modules $\mathscr{F}$ is quasi-coherent if $X$ can be covered by open affine subsets $U_i = \text{Spec} A_i$, such that for each $i$ there is an $A_i$-module $M_i$ with $\mathscr{F} \vert_{U_i} \cong \widetilde{M}_i$. We say that $\mathscr{F}$ is coherent if furthermore each $M_i$ can be taken to be a finitely generated $A_i$-module.
I would rather however stick to the chain of syzygies definition.
References for what I am reading are: Gunning and Rossi's Analytic Functions of Several Complex Variables and Kaup and Kaup's Holomorphic Functions of Several Variables.