Chevalley’s theorem in algebraic geometry

E559865

theorem in algebraic geometry

Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.

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Statements (43)

Predicate	Object
instanceOf	theorem in algebraic geometry ⓘ
appearsIn	Grothendieck’s Éléments de géométrie algébrique (EGA) NERFINISHED ⓘ Hartshorne’s Algebraic Geometry NERFINISHED ⓘ standard textbooks on scheme theory ⓘ
appliesTo	morphisms of finite type of schemes ⓘ morphisms of finite type of varieties ⓘ
assumes	morphism is of finite type to guarantee constructibility of the image ⓘ
clarifies	how algebraic maps behave with respect to the Zariski topology ⓘ
conclusion	image of a finite type morphism is constructible in the Zariski topology ⓘ
contrastsWith	continuous maps of general topological spaces, whose images need not be constructible ⓘ
domain	scheme theory ⓘ theory of varieties ⓘ
field	algebraic geometry ⓘ
generalizes	classical results on images of polynomial maps between affine varieties ⓘ
hasVariant	Chevalley’s theorem for constructible images in the setting of varieties over an algebraically closed field ⓘ Chevalley’s theorem for morphisms of finite type of Noetherian schemes NERFINISHED ⓘ
historicalContext	proved in the mid-20th century in the development of modern algebraic geometry ⓘ
holdsUnderAssumption	schemes are Noetherian in many standard formulations ⓘ source scheme is of finite type over the target scheme ⓘ
implies	constructible sets are stable under taking images by finite type morphisms ⓘ images of finite type morphisms need not be open or closed but are always constructible ⓘ
influenced	model-theoretic treatments of algebraically closed fields via constructible sets ⓘ
isFundamentalFor	Grothendieck’s foundations of scheme-theoretic algebraic geometry ⓘ the study of images and fibers of morphisms in modern algebraic geometry ⓘ
isKeyToolFor	Noetherian induction arguments on schemes ⓘ dimension theory of schemes ⓘ elimination of quantifiers in algebraically closed fields (via constructibility) ⓘ generic properties of fibers of morphisms ⓘ studying behavior of geometric properties under morphisms ⓘ
namedAfter	Claude Chevalley NERFINISHED ⓘ
oftenFormulatedAs	if f : X → Y is a morphism of finite type of schemes, then f(X) is a constructible subset of Y ⓘ
relatedTo	Noetherian topological spaces NERFINISHED ⓘ upper semicontinuity phenomena in algebraic geometry ⓘ
reliesOn	Noetherian property of the underlying topological spaces in common formulations ⓘ
states	the image of a morphism of finite type between schemes is a constructible set in the target scheme ⓘ
typeOfResult	topological property of images of morphisms ⓘ
usedInProofOf	constructibility of various stratifications of schemes ⓘ generic flatness results ⓘ results on openness of loci defined by fiber conditions ⓘ
usesConcept	Zariski topology NERFINISHED ⓘ constructible set ⓘ morphism of finite type ⓘ scheme ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Claude Chevalley → knownFor → Chevalley’s theorem in algebraic geometry ⓘ