Cardinalities of prime spectra of precompletions

Abstract

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