Let T be a complete local (Noetherian) ring, and let A be a local subring of T such that the completion of A with respect to its maximal ideal is T . We investigate the possible structures of the partially ordered set Spec(A). Specifically, we explore the minimal prime ideals of A and their formal fibers, the maximal chains of prime ideals in A, and the number of prime ideals in A containing combinations of minimal prime ideals of A.

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

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.

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