Cardinalities of Prime Spectra of Precompletions

Given a complete local (Noetherian) ring $T$, we find necessary and sufficient conditions on $T$ such that there exists a local domain $A$ with $\mathrm{\mid }A\mathrm{\mid }<\mathrm{\mid }T\mathrm{\mid }$ and $\widehat{A}=T$, where $\widehat{A}$ denotes the completion of $A$ with respect to its maximal ideal. We then find necessary and sufficient conditions on $T$ such that there exists a domain $A$ with $\widehat{A}=T$ and $\mathrm{\mid }Spec(A)\mathrm{\mid }<\mathrm{\mid }Spec(T)\mathrm{\mid }$. Finally, we use “partial completions” to create local rings $A$ with $\widehat{A}=T$ such that $Spec(A)$ has varying cardinality in different varieties.