Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Irreducible topological spaces and irreducible subsets in the subspace topology
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 irreducible when and whenever with closed, one has or .
If , then is an irreducible subset of when the subspace is irreducible, where is the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
Depends on
Used by
- A plane intersection with no common component is nonempty and zero-dimensional Corollary
- Under AC, proper integral finite-type schemes over fields with multiple points are not affine Counterexample
- Chain dimension and the empty-space convention Definition
- Classical algebraic prevarieties, regular maps, and varieties Definition
- Generic points of irreducible closed subsets Definition
- Integral affine schemes Definition
- Integral schemes Definition
- Irreducible components as schemes Definition
- Irreducible components of a topological space Definition
- Plane projective curves and their components Definition
- The twists on the projective line have degree n Example
- A Noetherian space is a finite union of irreducible closed subsets Lemma
- Complex algebraic boundary points admit normalized curve arcs Lemma
- Constant sheaves on irreducible spaces are flasque and acyclic Lemma
- Existence and basic properties of irreducible components Lemma
- Irreducibility via nonempty open subsets, connectedness and open subspaces Lemma
- Nonzero maps from an invertible sheaf to a locally free sheaf are injective Lemma
- Proper closed subsets of a curve are finite Lemma
- Regular hyperplane step for coherent support induction Lemma
- Torsion-free coherent modules on a smooth curve are locally free Lemma
- Upper semicontinuity of proper fibre dimension Lemma
- A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point Theorem
- Grothendieck vanishing on a Noetherian space Theorem
- Irreducible components of the spectrum correspond to minimal prime ideals Theorem
- Jacobian criterion and openness of the regular locus over a perfect field Theorem
- Nonconstant morphisms of proper curves are finite and surjective Theorem
Dependency tree · two levels
6 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, Definition 14.5 (standard reference, not scraped)
- The Stacks Project, Definition 5.8.1 (standard reference, not scraped)