An interesting precursor concerning units in real´´ quadratic function fields is a paper by Abel Sur l'integration de la formule differentielle $\frac{\rho \,dx}{\sqrt{R(x)}}$, $R$ et $\rho$ étant des fonctions entières.´´ There he proves (over the complex numbers) that there is a unit precisely when the appropriate continued fraction expansion is (eventually) periodic. The only thing one would need to add in the finite coefficient field case is that because the appropriate polynomials appearing in the continued fraction expansion are of bounded degree they will eventually repeat.

My guess is that they are referring to what is usually called Borel-Moore homology (or homology with closed support). That fits at least with their given example. (A quick glance at Wikipedia indicates that the entry on Borel-Moore homology gives a correct description of it.)

One comment about a CDE square vs a triangle. The whole theory only sees simple modules so one may start by dividing out by the radical of $R_K$ (and its intersection with $R$) making $R_K$ semi-simple.

There are two things here. To define $C$ one needs for $A$ to be local Henselian only (a minor point). To have $D$ defined it is necessary to have that the kernel of $K_0(R_A) \to K_0(R_K)$ ($K_0$ being the Grothendieck group of f.g. modules) map to zero under the reduction map $K_0(R_A) \to K_0(R_k)$ (and for that map to be defined one needs $R$ to be regular). The traditional case is that $A$ is a DVR, I am unsure about the right generality. One could however assume only that $(A,(q))$ is a Henselian couple ($(q)$ need not be maximal) and $k=A/(q)$ and $K=A[q^{-1}]$.