Alphabeta MathHow statement and proof provenance work
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A classical affine variety
Definition
Let be an algebraically closed field. A classical affine variety over is a nonempty affine algebraic set that is irreducible in the Zariski topology, meaning that whenever with and closed in , one has or .
This retains the pre-scheme convention that the empty algebraic set is allowed as an algebraic set but is not called a variety.
Depends on
Used by
- A principal open subset of a classical affine variety Definition
- Regular functions on open subsets of a classical affine variety Definition
- Every nonempty open subset of an affine variety is dense Lemma
- Irreducibility is equivalent to every pair of nonempty open sets meeting Lemma
- A classical affine variety has a domain as its coordinate ring, and conversely Theorem
- The product of affine varieties has coordinate ring k[X] tensorₖ k[Y] Theorem
Dependency tree · one level
1 result within one dependency step 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, Algebraic Geometry, Chapter 3e (standard reference, not scraped)
- Donu Arapura, Notes on Basic Algebraic Geometry, §1.5 (standard reference, not scraped)