Grothendieck’s scheme-theoretic framework

E627992

foundational framework in algebraic geometry reformulation of classical algebraic geometry scheme-theoretic approach

Grothendieck’s scheme-theoretic framework is a foundational reformulation of algebraic geometry that generalizes varieties using schemes, enabling powerful tools like sheaf theory, cohomology, and modern number-theoretic applications.

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Observed surface forms (1)

Surface form	Occurrences
Grothendieck school of algebraic geometry	1

Statements (48)

Predicate	Object
instanceOf	foundational framework in algebraic geometry ⓘ reformulation of classical algebraic geometry ⓘ scheme-theoretic approach ⓘ
allows	definition of schemes over arbitrary commutative rings ⓘ systematic use of local-to-global methods ⓘ uniform treatment of geometry over fields and rings ⓘ
basedOn	concept of schemes ⓘ
coreConcept	Grothendieck topology NERFINISHED ⓘ Yoneda lemma (as a guiding principle) NERFINISHED ⓘ Zariski topology NERFINISHED ⓘ base change ⓘ coherent sheaf ⓘ fiber product of schemes ⓘ flat morphism ⓘ functor of points ⓘ morphism of schemes ⓘ proper morphism ⓘ quasi-coherent sheaf ⓘ representable functor ⓘ scheme ⓘ separated morphism ⓘ sheaf cohomology ⓘ site ⓘ structure sheaf ⓘ étale morphism ⓘ
developedBy	Alexander Grothendieck NERFINISHED ⓘ
emphasizes	functorial viewpoint ⓘ structural and categorical methods ⓘ universal properties ⓘ
enables	Weil conjectures approach via étale cohomology ⓘ formulation of Grothendieck’s version of class field theory ⓘ modern formulation of algebraic number theory ⓘ study of arithmetic schemes ⓘ
field	algebraic geometry ⓘ number theory ⓘ
generalizes	algebraic varieties ⓘ classical affine varieties ⓘ classical projective varieties ⓘ
influenced	modern arithmetic geometry ⓘ modern moduli theory ⓘ theory of stacks ⓘ
introducedInWork	Séminaire de Géométrie Algébrique (SGA) NERFINISHED ⓘ Éléments de géométrie algébrique NERFINISHED ⓘ
replaced	classical variety-based foundations of algebraic geometry ⓘ
usesTool	category theory ⓘ cohomology ⓘ homological algebra ⓘ sheaf theory ⓘ

Referenced by (2)

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

Zariski topology → extendedBy → Grothendieck’s scheme-theoretic framework ⓘ

Éléments de géométrie algébrique → influenced → Grothendieck’s scheme-theoretic framework ⓘ

this entity surface form: Grothendieck school of algebraic geometry