Alphabeta MathHow statement and proof provenance work
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Noetherian topological spaces via ACC on opens or DCC on closed subsets
Definition
Let be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).
The space is Noetherian when every ascending chain of open subsets stabilizes.
Equivalently, is Noetherian when every descending chain of closed subsets stabilizes.
Depends on
Used by
- Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
- Algebraic cycles and the cycle group of a scheme of finite type over a field Definition
- Chain dimension and the empty-space convention Definition
- Genus and arithmetic genus of a curve Definition
- A Noetherian space is a finite union of irreducible closed subsets Lemma
- Affine neighbourhood containing component generic points Lemma
- Classical varieties have finite irreducible decompositions Lemma
- Extension-by-zero generators detect sheaf-cohomology vanishing Lemma
- Filtered colimits and sheaf cohomology on Noetherian spaces Lemma
- Filtered colimits of sheaves and sections over compact opens Lemma
- Morphisms from complete connected schemes to affine schemes are constant Lemma
- Noetherian devissage for coherent proper pushforward Lemma
- Noetherian open subsets are quasi-compact Lemma
- Proper closed subsets of a curve are finite Lemma
- Regular hyperplane step for coherent support induction Lemma
- Schematic closure and agreement on a dense open Lemma
- Subspaces of a Noetherian space and its compact open subsets Lemma
- Degree of the coherent Hilbert polynomial Theorem
- Dimension bound for quasi-coherent cohomology on a Noetherian scheme Theorem
- Euler characteristic is a Hilbert polynomial Theorem
- Grothendieck vanishing on a Noetherian space Theorem
- Serre duality for line bundles on a smooth proper curve, and the residue realization Theorem
- The spectrum of a Noetherian ring is a Noetherian topological space Theorem
Dependency tree · two levels
2 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
- J. S. Milne, A Primer of Commutative Algebra, v4.03, §14 (standard reference, not scraped)
- The Stacks Project, Section 5.9: Noetherian topological spaces (standard reference, not scraped)