I read somewhere: https://ocw.mit.edu/courses/mathematics/18-726-algebraic-geometry-spring-2009/lecture-notes/MIT18_726s09_lec22_gaga.pdf lemma on page 3.
if $t_1$...$t_n$ are local algebraic coordinates for z in $\mathbb{P_{\mathbb{C}}^r}^{\sim}$ ( projective space viewed as a complex manifold), that there is a "completion morphism" g: $\mathscr{H}_z$ $\rightarrow$ $\mathbb{C}$[[$t_1$...$t_n$]], a faithfully flat morphism. $\mathscr{H}_z$ is $\mathcal{O}_{\mathbb{P_{\mathbb{C}}^r}^{\sim},z}$, the local ring of germs of holomorphic functions in neighborhood of z. (I think...)
How is $\mathbb{C}$[[$t_1$...$t_n$]] the completion of $\mathscr{H}_z$ as well as of $\mathcal{O}_{\mathbb{P_{\mathbb{C}}^r},z}$ (the ring of regular polynomials defined at z??), (the morphism f: $\mathcal{O}_{\mathbb{P_{\mathbb{C}}^r},z}$ $\rightarrow$ $\mathscr{H}_z$ is just inclusion??). Completion with respect to what maximal ideal?? What does a completion morphism mean?