A local ring is a nonzero commutative ring with a unique maximal ideal

Depends on

• Commutative ring
• Prime ideals and maximal ideals in a commutative ring
• The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring
• $R/M$ is a field if and only if $M$ is a maximal ideal

Used by

• A Noetherian local domain has dimension zero exactly when it is a field Corollary
• Assuming the Axiom of Choice, minimal generators over a local ring are exactly residue-field bases Corollary
• A morphism of ringed spaces need not be a morphism of locally ringed spaces Counterexample
• A locally ringed space Definition
• embedding dimension and regular local ring Definition
• Equicharacteristic local rings and coefficient fields Definition
• Henselian pairs and Henselian local rings Definition
• Minimal Free Resolution Over A Local Ring Definition
• Systems of parameters and parameter ideals Definition
• The ring of holomorphic germs at 0 and its maximal ideal Definition
• Continuous real-valued functions make a space into a locally ringed space Example
• A local domain has a dominating valuation overring Lemma
• A local morphism of stalks induces a residue-field map Lemma
• Choose a parameter that misses the top-dimensional minimal components Lemma
• Finite-type field extensions with zero Ω Lemma
• Formal power-series substitution converges in a complete local algebra Lemma
• Invertible Jacobian minor gives regular parameters in a polynomial fibre Lemma
• Local flatness criterion by regular parameters Lemma
• Local Koszul Acyclicity Inductive Converse Lemma
• Local Koszul H One Detects First Regularity Failure Lemma
• Regularity ascends and descends along a flat local homomorphism Lemma
• Standard smooth algebras are finitely presented and flat Lemma
• The module-relative Hilbert–Samuel polynomial exists without a dimension theorem Lemma
• Unramified residue extensions are finite separable Lemma
• A germ is a unit exactly when its value at 0 is nonzero, so O_m,0 is local Proposition
• An Artinian local ring has nilpotent maximal ideal, and its finite modules have finite length Theorem
• Assuming the Axiom of Choice, a nonzero commutative ring is local exactly when its nonunits form an ideal, exactly when one of x and 1-x is a unit for every x Theorem
• Completion of a Noetherian local ring is local with the same residue field Theorem
• Equivalent characterizations of a DVR Theorem
• Holomorphic germs at a point form a local ring Theorem
• Indecomposable type-A diagrammatic Soergel objects are indexed by permutations and shifts Theorem
• Rₚ is local with unique maximal ideal pRₚ Theorem
• Valuative uniqueness detects separatedness Theorem

Dependency tree · two levels

18 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

• The Stacks Project, Section 10.18: Local rings (standard reference, not scraped)