Unique factorisation domain

Depends on

• Divisibility and associates in an integral domain
• Irreducible and prime elements of an integral domain

Used by

• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• Divisors on a smooth proper curve Definition
• Locally factorial scheme Definition
• The Alexander polynomial from the zeroth elementary ideal Definition
• An ordinary node has two smooth branches Example
• Principal divisors on the projective line have degree zero Example
• The rank-one 2 by 2 determinantal cone Example
• Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes Lemma
• Finite local length exactly when no common local branch Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Gauss lemma over a UFD Lemma
• Irreducible holomorphic germs are prime Lemma
• Krull dimension of the holomorphic germ ring Lemma
• Reduced preparation has nonzero discriminant Lemma
• Square-free reduction of a holomorphic equation Lemma
• The Laurent polynomial ring is Noetherian and a unique factorisation domain Lemma
• The Picard group and intersection form of a product of projective lines Lemma
• The Picard group of a point blowup of the projective plane Lemma
• The resultant detects finitely many common projective points Lemma
• Total fractions split over the branches of a reduced hypersurface Lemma
• Cartier and Weil divisors agree on a smooth curve Theorem
• Every principal ideal domain is a unique factorisation domain Theorem
• Finite unique irreducible components of a hypersurface germ Theorem
• For every field F, F[x] is a unique factorisation domain Theorem
• Symmetry, additivity and local nature of intersection multiplicity Theorem
• The ring of holomorphic germs is a UFD Theorem
• Under AC, Cartier and Weil divisors agree on a locally factorial Noetherian integral scheme Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Sharifi, Abstract Algebra, Advanced Ring Theory (standard reference, not scraped)