functionField_isScalarTower — Mathlib

English

For an irreducible X and a point x ∈ X, the stalk O_{X,x} carries a natural algebra structure over X.functionField, given by the specialization map from the germ to the stalk.

Русский

Для непрерывного X и точки x ∈ X, локальный стержень O_{X,x} имеет естественную структуру X.functionField-алгебры, задаваемую специализацией через герм.

LaTeX

$$$\text{Algebra}(X.presheaf.stalk x)\;\text{ over } X.functionField$ with structure map given by $X.presheaf.germ\;((\text{genericPoint spec}).$$$

Lean4

instance functionField_isScalarTower [IrreducibleSpace X] (U : X.Opens) (x : U) [Nonempty U] :
    IsScalarTower Γ(X, U) (X.presheaf.stalk x) X.functionField :=
  by
  apply IsScalarTower.of_algebraMap_eq'
  simp_rw [RingHom.algebraMap_toAlgebra]
  change _ = (X.presheaf.germ U x x.2 ≫ _).hom
  rw [X.presheaf.germ_stalkSpecializes]

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