Characterization of Completions of Unique Factorization Domains

DOI10.2307/2154327zbMath0792.13011OpenAlexW4242882045MaRDI QIDQ4695373

Raymond C. Heitmann

Publication date: 25 July 1994

Full work available at URL: https://doi.org/10.2307/2154327

zbMATH Keywords

complete local ringnormal chain conjecturecompletion of a UFD

Mathematics Subject Classification ID

Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) (13H10) Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial) (13F15) Completion of commutative rings (13B35) Complete rings, completion (13J10)

Related Items (45)

CONSTRUCTING CHAINS OF EXCELLENT RINGS WITH LOCAL GENERIC FORMAL FIBERS ⋮ Semilocal generic formal fibers ⋮ On the completeness of factor rings ⋮ A class of local noetherian domains ⋮ Constructing local generic formal fibers ⋮ Two counterexamples in normalization ⋮ Descent and ascent of perinormality in some ring extension settings ⋮ Descent via Koszul extensions ⋮ Noetherian rings between a semilocal domain and its completion ⋮ Geometric divisors in normal local domains ⋮ Cardinalities of prime spectra of precompletions ⋮ Reduction of the small Cohen-Macaulay conjecture to excellent unique factorization domains ⋮ Characterization of completions of excellent domains of characteristic zero. ⋮ On some permanence properties of (derived) splinters ⋮ Every finite poset is isomorphic to a saturated subset of the spectrum of a Noetherian UFD ⋮ Semilocal formal fibers of principal prime ideals ⋮ Are complete intersections complete intersections? ⋮ Controlling the dimensions of formal fibers of a unique factorization domain at the height one prime ideals ⋮ Completions of countable excellent domains and countable noncatenary domains ⋮ Structure of spectra of precompletions ⋮ Characterization of completions of noncatenary local domains and noncatenary local UFDs ⋮ Controlling the generic formal fiber of local domains and their polynomial rings ⋮ Characterization of completions of reduced local rings ⋮ Unique factorization in invariant power series rings ⋮ Controlling formal fibers of principal prime ideals ⋮ Vanishing of cohomology over Cohen-Macaulay rings ⋮ Extension of ideal-theoretic properties of a domain to submodules of its quotient field ⋮ Direct-sum decompositions over local rings ⋮ Completely controlling the dimensions of formal fiber rings at prime ideals of small height ⋮ Generic formal fibers of polynomial rings ⋮ Tight closure in non-excellent rings ⋮ Some results on tight closure and completion ⋮ Almost excellent unique factorization domains ⋮ Local rings of finite Cohen-Macaulay type ⋮ Excellent rings with local generic formal fibers ⋮ Direct Sum Decompositions Over Two-Dimensional Local Domains ⋮ Maximal chains of prime ideals of different lengths in unique factorization domains ⋮ Completions of uncountable local rings with countable spectra ⋮ Indecomposable Gorenstein modules of odd rank ⋮ Dimensions of Formal Fibers of Height One Prime Ideals ⋮ Constructing (non-)Catenarian rings ⋮ Chains of Rings with Local Formal Fibers ⋮ Srinivas' problem for rational double points ⋮ Gluing minimal prime ideals in local rings ⋮ Local rings of bounded Cohen-Macaulay type

Cites Work

• The depth conjecture: A counterexample
• Chain conjectures in ring theory. An exposition of conjectures on catenary chains
• A Brief History and Survey of the Catenary Chain Conjectures
• On Quasi-Unmixed Local Domains, the Altitude Formula, and the Chain Condition for Prime Ideals, (I)
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