embedding dimension and regular local ring

Depends on

• A local ring is a nonzero commutative ring with a unique maximal ideal
• Left and right Noetherian rings
• Krull dimension of a nonzero ring

Used by

• Regular centers have projective-bundle exceptional divisors Corollary
• A flat family with a nodal special fibre is not smooth at the node Counterexample
• Blowing up a point on a singular surface need not be smooth Counterexample
• Finite normalization alone does not make a curve regular Counterexample
• Frobenius on the affine line is finite flat but not smooth Counterexample
• Normalization and blowup are different operations Counterexample
• Normalization of a non-normal surface is not a point blowup Counterexample
• Regular field factors can have a nonregular tensor product Counterexample
• Smoothness of the source cannot be dropped in the extension of rational maps Counterexample
• The equation must define the intended scheme Counterexample
• The maximal-contact mechanism fails in positive characteristic Counterexample
• The multiplicative group scheme mu p is not a smooth torus Counterexample
• Analytic topology and finite smooth algebraic stratifications of complex varieties Definition
• Contact order of two regular components at a point Definition
• Derivative ideals of an ideal sheaf and of a marked ideal Definition
• Intersection numbers of Cartier divisors on a smooth projective surface Definition
• Marked ideals of maximal order, tangent directions and transversality to the exceptional divisors Definition
• Order of an ideal sheaf at a point Definition
• regular noetherian ring Definition
• Regular points of locally Noetherian schemes Definition
• regular system of parameters Definition
• serre r k and s k conditions Definition
• Simple normal crossings divisors and simultaneous normal crossings position Definition
• Smoothness over a field by geometric regularity Definition
• Strict normal crossings divisor on a regular surface Definition
• Uniformising parameters at smooth points of a plane curve Definition
• A cusp: one blowup, the normalization and the delta drop Example
• A node is resolved by one point blowup Example
• A projective cone with a smooth conic base Example
• cusp local ring not regular Example
• dual numbers not regular Example
• dvrs as regular local rings Example
• embedding dimension versus dimension node Example
• fields as regular local rings Example
• regular flat local map with singular closed fibre Example
• regular local ambient cover minimal dimension Example
• Resolving the quadric cone by one blowup Example
• A birational morphism of regular surfaces factors through the blowup of a point where its inverse is undefined Lemma
• A character of a normal subgroup admits an inverse multiple in a group representation Lemma
• A transverse hyperplane slice is smooth at the chosen point Lemma

…and 41 more results.

Dependency tree · two levels

9 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

• 12.3–12.5, p.115 (standard reference, not scraped)