The title made me think of Hartshorne and Serre sending letters to one another on 6-codimensional paper!

Sebastian, I didn't vote your question down, but it could use some more explanation. How exactly is your class of subfactors defined? How do you "easily" define your equivalence $\tilde_1$? What properties do you want from your equivalence? Do you have examples? I feel that this question is too open-ended to get a good answer, as phrased.

I see, thanks very much for the explanation. That makes total sense.

(The elements $d,e$ in my previous comment should be monomials, of course.)

Vladimir, thanks for your answer. The potential problem I saw was in your "clearly". Saying that $a f_\sigma - f_\tau b$ can be reduced to zero means it is in the span of terms of the form $d(W_\mu - f_\mu)e$, where $d W_\mu e \leq a W_\sigma$. I had thought that perhaps coefficients could arise in that sum that had poles at $0$. It feels like this shouldn't be possible but I don't have a rigorous argument why not.

Thanks for your answer, Mariano. What do you mean by "right members"? And how can we be sure that no coefficients arise outside of $R$? That was precisely the question that I was asking myself when thinking about this.