disambiguating completion and localization

Inversion: adjoin a formal inverse. \begin{align*}{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] } = \left\{{{a\over b}\in {\mathbf{Q}}{~\mathrel{\Big\vert}~}b=p^k}\right\} .\end{align*} Localization at \(p\): invert all primes \(\ell\neq p\). Geometrically: invert all functions which are no in the ideal \(\left\langle{p}\right\rangle\), so function that do not vanish at \(p\), by adjoining inverses to all elements in \({\mathbf{Z}}\setminus\left\langle{p}\right\rangle\) \begin{align*}{\mathbf{Z}}_p \coloneqq { L_p {\mathbf{Z}}}= {\mathbf{Z}} \left[ { \scriptstyle { { ({\left\langle{p}\right\rangle}^c) }^{-1}} } \right] = \left\{{{a\over b}\in {\mathbf{Q}}{~\mathrel{\Big\vert}~}b\not\in\left\langle{p}\right\rangle}\right\} = \left\{{x\in {\mathbf{Q}}{~\mathrel{\Big\vert}~}v_p(x)\geq 0}\right\} = {\mathbf{Z}} { \left[ \scriptstyle {\left\{{\ell^{-1}{~\mathrel{\Big\vert}~}\ell\neq p }\right\}} \right] } \end{align*} These are functions that are defined locally on a neighborhood of the point \(p\).

Completion (adically) at \(p\): compatible systems of lifts: \begin{align*} { { {\mathbf{Z}}_{\widehat{p}} }}= \lim\qty{ \cdots \leftarrow^{\operatorname{mod}p^3} {\mathbf{Z}}/p^2 {\mathbf{Z}}\leftarrow^{\operatorname{mod}p^2} {\mathbf{Z}}/p{\mathbf{Z}}} \subseteq \prod_{n\geq 0} {\mathbf{Z}}/p^n {\mathbf{Z}} \end{align*} These are functions defined on a formal neighborhood of the point \(p\), and the projections \({\mathbf{Z}}\to { {\mathbf{Z}}_{\widehat{p}} }\to {\mathbf{Z}}/p{\mathbf{Z}}\) correspond to \(\left\{{p}\right\} \hookrightarrow\operatorname{Spf}{ {\mathbf{Z}}_{\widehat{p}} }\hookrightarrow\operatorname{Spec}{\mathbf{Z}}\).

Completion (profinite): a limit over all finite quotients (normal subgroups with finite index) over the subgroup lattice: \begin{align*} { \widehat{{\mathbf{Z}}} }= \lim\left\{{{\mathbf{Z}}/n{\mathbf{Z}}\to {\mathbf{Z}}/m{\mathbf{Z}}{~\mathrel{\Big\vert}~}m\mathrel{\Big|}n }\right\} \cong \prod_{p} { {\mathbf{Z}}_{\widehat{p}} }= {\mathbf{A}}_{\mathbf{Z}}^{<\infty} \end{align*} where \({\mathbf{A}}_Z = {\mathbf{R}}\times \prod_p { {\mathbf{Z}}_{\widehat{p}} }\).

Prufer groups, which are the Pontrayagin duals of \({ {\mathbf{Z}}_{\widehat{p}} }\): \begin{align*}{\mathbf{Z}}/p^\infty \coloneqq\colim({\mathbf{Z}}/p \xrightarrow{p} {\mathbf{Z}}/p^2 \xrightarrow{p} \cdots) \cong \left\langle{\zeta_{p}, \zeta_{p^2}, \cdots}\right\rangle = {\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }/{\mathbf{Z}}= \left\{{(x_i) {~\mathrel{\Big\vert}~}x_1^p = 1, x_2^p = x_1, x_3^p = x_2,\cdots}\right\}.\end{align*}

Ideal localization: \(L_I {\mathbf{Z}}\) is localization at the monoid \({\mathbf{Z}}\setminus I\) and allows inverting all functions that do not vanish on \(V(I) \subseteq \operatorname{Spec}{\mathbf{Z}}\).

Prime localization: \(L_{p^c} {\mathbf{Z}}\) is localization at \(\left\{{1, p, p^2,\cdots}\right\}\) allows inverting all functions that vanish on \(\left\langle{p}\right\rangle \in \operatorname{Spec}{\mathbf{Z}}\), so \(\operatorname{Spec}L_{p^c} {\mathbf{Z}}= (\operatorname{Spec}{\mathbf{Z}}) \setminus\left\langle{p}\right\rangle\) is a punctured \(\operatorname{Spec}{\mathbf{Z}}\).

Questions

What does \(\operatorname{Spec}{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] } \to \operatorname{Spec}{\mathbf{Z}}\) represent?
What does \(\operatorname{Spec}{ {\mathbf{Z}}_{\widehat{p}} }\to \operatorname{Spec}{\mathbf{Z}}\) represent?
What does \(\operatorname{Spec}{ L_p {\mathbf{Z}}}\to \operatorname{Spec}{\mathbf{Z}}\) represent?
What does \(\operatorname{Spec}{ \widehat{{\mathbf{Z}}} }\to \operatorname{Spec}{\mathbf{Z}}\) represent?