Alphabeta MathHow statement and proof provenance work
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The reduction of a scheme
Definition
For a scheme , let be the ideal sheaf whose germs are the nilpotent elements of the local rings of ; equivalently, a section lies in when it is locally nilpotent on . The reduction is the closed subscheme with the same underlying topological space and structure sheaf . On it is .
Depends on
Used by
- Agreement on a schematically dense open Corollary
- Blowing up a nonzero ideal on an integral scheme is birational Corollary
- Global functions on geometrically connected and geometrically reduced proper schemes Corollary
- The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
- Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
- Contact order of two regular components at a point Definition
- Geometric properties of fibres Definition
- Integral schemes Definition
- Irreducible components as schemes Definition
- Locally factorial scheme Definition
- Normalization defect delta of a reduced curve Definition
- The scheme-theoretic tangent cone at a point Definition
- Weil divisor normal noetherian scheme Definition
- A proper singular curve with weight-zero cohomology Example
- Reduction of the dual-number point Example
- The cusp retains a doubled tangent line Example
- The twists on the projective line have degree n Example
- A smooth group of multiplicative type is the only closed subscheme containing all its finite subgroups Lemma
- A tangent direction is realized by a local smooth curve Lemma
- Integrality and reducedness of blowups from the Rees charts Lemma
- Noetherian devissage for coherent proper pushforward Lemma
- Normalization is unchanged under finite birational maps of reduced curves Lemma
- Rational points of smooth finite-type schemes over a separably closed field are schematically dense Lemma
- The intersection matrix of a point blowup of a regular surface Lemma
- The normalization defect is an Euler characteristic and a weighted sum of local lengths Lemma
- The product of two projective lines is an integral smooth projective surface Lemma
- A support does not determine its scheme structure Remark
- Bertini smoothness away from the base locus Theorem
- Dense regular loci on every component Theorem
- Fixed-point schemes and centralizers of linearly reductive actions Theorem
- Normalization of a reduced curve is finite Theorem
- Representability and smoothness of concentrator subschemes Theorem
- Resolution of reduced plane curves by point blowups and the delta recurrence Theorem
- Smooth projective proper hyperresolutions Theorem
- Strict transforms of closed subschemes are blowups of the subscheme Theorem
- Universal property of scheme reduction Theorem
Dependency tree · two levels
7 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, Schemes, Lemma 12.4 (standard reference, not scraped)