Integral closure in an extension ring and integrally closed domains

Depends on

• Integral ring maps and integral extensions
• The field of fractions $\operatorname{Frac}(D)=(D\setminus\{0\})^{-1}D$ of an integral domain
• Zero divisor, and integral domain: a commutative ring with $1 \ne 0$ and no zero divisors

Used by

• Proper birational normal curves agree off finitely many points Corollary
• Rings of integers are Dedekind domains Corollary
• The normalization of an irreducible affine variety is finite Corollary
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Normalization of a non-normal surface is not a point blowup Counterexample
• Quasi-finite does not imply finite Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Dedekind domains Definition
• Delta invariant of a curve singularity Definition
• Integral elements subalgebra of an arbitrary ring map Definition
• normal noetherian ring Definition
• Normal points and normal varieties Definition
• Ring of integers Definition
• The normalization of an irreducible affine variety Definition
• Total quotient ring and normalisation of a reduced plane curve germ Definition
• Divisor of a rational function on the projective line Example
• Normalization of a nodal affine plane curve Example
• Normalization of the cusp semigroup ring Example
• Normalization of the t³,t⁴,t⁵ monomial curve Example
• Normalizing the cuspidal plane curve Example
• Normalizing the nodal plane curve Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• A finite birational morphism onto a normal variety is an isomorphism Lemma
• Finite algebras over a strongly transcendental variable are nowhere quasi-finite Lemma
• Finite normalization commutes with principal localization Lemma
• Finite-variable polynomial algebras over fields are integrally closed Lemma
• Integral basis and discriminant of a coprime-discriminant compositum Lemma
• Integral closure in a purely inseparable rational envelope is finite Lemma
• Integral closure is unchanged across an integral intermediate domain Lemma
• Minimal polynomials of integral elements over an integrally closed domain have coefficients in the domain Lemma
• Normalization is unchanged under finite birational maps of reduced curves Lemma
• Polynomial rings over normal domains are normal Lemma
• Power extension over a normal affine domain Lemma
• The conductor is an ideal of both rings Lemma
• A domain is integrally closed if and only if its prime localisations are, equivalently if and only if its maximal localisations are Theorem
• A finite-type domain over a field has finite normalization Theorem
• Equivalent characterizations of a DVR Theorem
• Finite generation of invariants and the affine categorical quotient Theorem
• Finite separable integral closures over normal Noetherian domains are module-finite Theorem

…and 10 more results.

Dependency tree · two levels

8 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

• Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Definition (10.30) (standard reference, not scraped)
• J. S. Milne, A Primer of Commutative Algebra, v4.03, Definitions 6.6 and 6.9 (standard reference, not scraped)