V \\ {\cal
      H}'\otimes {{\cal H}'_{\text{aux}}} @<< G^+_{{\cal
        H}\,\otimes\,{{\cal H}_{\text{aux}}}} <
    {\cal H}\otimes {{\cal H}_{\text{aux}}}\\
\end{CD}$$ Diagram (\[glob1\]) represents an exact relation which implies all the necessary constraints for the RG procedure as can be verified from equation (\[inveq\]). We therefore choose relation (\[glob1\]) as the basic relation in defining local RGTs.\
Decomposing the global RGT (\[glob1\]) into local RGTs given by (\[comm4\]) demands for a decomposition of $\cal O$ into commuting blocks. From previous considerations we conclude that this is impossible for quantum chains due to the correlation blocks occurring in a decomposition of a quantum physical system. Therefore the idea is to use the auxiliary space to decompose the chain into commuting blocks by storing the information about the correlations of adjacent system blocks within the auxiliary space. By the decompositions discussed so far we then decompose a chain into system blocks and try to find an auxiliary space ${\left({\cal H}_{\text{aux}}\right)}_{I}$ for each system block which takes over the role of the correlation blocks within the RGT as visualized in figure \[auxpic\].

This statement can be made more precise.

A local RGT is said to be [*perfect*]{} if there exists a local operator\
$$\begin{aligned}
{\cal O}_{{\cal H}'_{I}\otimes
   {\left({\cal 