Given a complete local (Noetherian) ring $T$, we find necessary and sufficient conditions on $T$ such that there exists a local domain $A$ with $|A| < |T|$ 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 $|\mbox{ Spec}(A)| < |\mbox{ Spec}(T)|$. Finally, we use "partial completions" to create local rings $A$ with $\widehat{ A} = T$ such that $\mbox{ Spec}(A)$ has varying cardinality in different varieties.
Categories: math.AC