Cardinalities of Prime Spectra of Precompletions

Abstract

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 $\hat{A}=T$, where $\hat{A}$ 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 $\hat{A}=T$ and $|\mbox{Spec}(A)| < |\mbox{Spec}(T)|$. Finally, we use "partial completions" to create local rings $A$ with $\hat{A}=T$ such that $\mbox{Spec}(A)$ has varying cardinality in different varieties.

24 pages

References in corpus (1)

• Structure of Spectra of Precompletions