et $\bfq$ be the homomorphism from $V_{\mathbf{GL_0}}$ to the multiplicative monoid $(\bbA^0,\cdot )$ defined by $(f,z_0)\mapsto \det(f)z_0^{-0}$: $$\xymatrix{ V_{\mathbf{GL_0}} \ar[d]_{\bfq} \\ \bbA^0. }$$ The