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
• Euler characteristic in a proper flat family is locally constant Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• Vanishing of the deformation tangent space forces rigidity of deformation classes Corollary
• A morphism of ringed spaces need not be a morphism of locally ringed spaces Counterexample
• A locally ringed space Definition
• Degree of a nonconstant morphism of curves Definition
• embedding dimension and regular local ring Definition
• Equicharacteristic local rings and coefficient fields Definition
• Henselian pairs and Henselian local rings Definition
• Intersection multiplicity of closed subschemes at a point Definition
• Local intersection multiplicity of two plane curves Definition
• Locally factorial scheme Definition
• Minimal complexes and the perverse truncations in the real type-A Soergel category Definition
• Minimal Free Resolution Over A Local Ring Definition
• Order codimension one rational function Definition
• Order of an ideal sheaf at a point Definition
• Rational equivalence and the Chow group of cycles Definition
• Relative dimension of a smooth morphism at a point Definition
• Sheaf total quotient rings Definition
• Simple normal crossings divisors and simultaneous normal crossings position Definition
• Square-zero extensions, small extensions and first-order thickenings Definition
• Strict normal crossings divisor on a regular surface Definition
• Systems of parameters and parameter ideals Definition
• The local ring A₀ of rational functions regular at q=0 Definition
• The ring of holomorphic germs at 0 and its maximal ideal Definition
• Uniformising parameters at smooth points of a plane curve Definition
• Weil divisor normal noetherian scheme Definition
• Continuous real-valued functions make a space into a locally ringed space Example
• A divisor supported in a special fibre has positive conormal degree on some component Lemma
• A flat local map is faithfully flat Lemma
• A function cutting the components of a special fibre Lemma
• 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
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Cycles of coherent sheaves and of closed subschemes, with flat pullback Lemma
• Dense relative-dimension strata in flat finitely presented fibres Lemma

…and 44 more results.

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)