Structure of spectra of precompletions

@article{Barrett2019StructureOS,
  title={Structure of spectra of precompletions},
  author={Erica Barrett and Emil Graf and Susan Loepp and Kimball Strong and Sharon Zhang},
  journal={Rocky Mountain Journal of Mathematics},
  year={2019}
}
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. 
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2 Citations
Background Citations
1
Methods Citations
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2 Citations

Completions of Uncountable Local Rings with Countable Spectra

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that…

Cardinalities of prime spectra of precompletions

Given a complete local (Noetherian) ring, necessary and sufficient conditions are found on the inline-formula content-type to allow for a local domain to exist such that there exists a localdomain of T.

References

SHOWING 1-7 OF 7 REFERENCES

Characterization of completions of unique factorization domains

It is shown that a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or it has depth at least two and no element of its…

Characterization of completions of noncatenary local domains and noncatenary local UFDs

Completions of reduced local rings with prescribed minimal prime ideals

Semi-local formal fibers of minimal prime ideals of excellent reduced local rings

Completions of Local Rings with an Isolated Singularity

Abstract It is shown that all equicharacteristic complete local rings can be realized as completions of local rings whose proper localizations are all regular. The only restriction in the mixed…

Formal fibers with countably many maximal elements

A method for constructing bad noetherian local rings