$G_{{\cal H}'\otimes{\cal H}'_{\text{aux}}}$ according to (\[grad2\]).\
We are now able to give the precise definition of the local RGT in the form $$\label{comm4}
\begin{CD}
  {\cal H}'_{I}\otimes {\left({\cal H}'_{\text{aux}}\right)}_{I} @> G_{{\cal
      H}'_{I}\,\otimes\,{\left({\cal H}'_{\text{aux}}\right)}_{I}} >>
  {\cal H}_{I}\otimes {\left({\cal H}_{\text{aux}}\right)}_{I}\\ @V {{\cal
      O}_{{\cal H}'_{I}\otimes{\left({\cal H}'_{\text{aux}}\right)}_{I}}
    \left({\bf K}'\right)} VV @VV {{\cal O}_{{\cal H}_{I}\otimes{\left({\cal
            H}_{\text{aux}}\right)}_{I}} \left({\bf K}\right)} V \\ {\cal
      H}'_{I}\otimes {\left({\cal H}'_{\text{aux}}\right)}_{I} @<< G^+_{{\cal
        H}_{I}\,\otimes\,{\left({\cal H}_{\text{aux}}\right)}_{I}} <
    {\cal H}_{I}\otimes {\left({\cal H}_{\text{aux}}\right)}_{I}\\
\end{CD}$$ where we refer to $G^+_{{\cal H}_{I}\,\otimes\,{\left({\cal H}_{\text{aux}}\right)}_{I}}$ and $G_{{\cal H}'_{I}\,\otimes\,{\left({\cal H}'_{\text{aux}}\right)}_{I}}$ as the generators of the transformation. By the explanations of section \[decom\] $$\begin{aligned}
\label{prod1l}
{\cal O}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \,=\,
 {\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \;\,\cdot\!\!\prod_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset I}}
  \!{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,j,\dots\right\}}
    \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,j,\dots\right\}}}
     \left({\bf K}\right)\end{aligned}$$ $$\begin{aligned}
\label{prod2l}
\text{or}\qquad{\cal O}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \,=\,
 {\cal O}^{\text{system}}_{{\cal H}_{I}
  \otimes{\left({\cal H}_{\text{aux}}\right)}_{I}}\left({\bf K}\right)
 \;\,+\!\!\sum_{\substack{\left\{i,i-1,\dots\right\}\\[0.1cm] \subset I}}
  \!{\cal O}^{\text{correlation}}_{{\cal H}_{\left\{i,j,\dots\right\}}
    \otimes{\left({\cal H}_{\text{aux}}\right)}_{\left\{i,j,\dots\right\}}}
  \left({\bf K}\right)\,\end{aligned}$$ and analogously for ${{\cal O}_{{\cal H}'_{I}\otimes{\left({\cal H}'_{\text{aux}}\right)}_{I}}
    \left({\bf K}'\right)}$.

Perfect and exact local RG transformations
==========================================

In this section we study the relationship between (\[comm4\]) and the global RGT\
$$\label{glob1}
\begin{CD}
  {\cal H}'\otimes {{\cal H}'_{\text{aux}}} @> G_{{\cal
      H}'\,\otimes\,{{\cal H}'_{\text{aux}}}} >>
  {\cal H}\otimes {{\cal H}_{\text{aux}}}\\ @V {{\cal
      O}_{{\cal H}'\otimes{{\cal H}'_{\text{aux}}}}
    \left({\bf K}'\right)} VV @VV {{\cal O}_{{\cal H}\otimes{\left({\cal
            H}_{\text{aux}}\right)}} \left({\bf K}\right)} 