Documentation

Mathlib.AlgebraicGeometry.Spec

Spec as a functor to locally ringed spaces. #

We define the functor Spec from commutative rings to locally ringed spaces.

Implementation notes #

We define Spec in three consecutive steps, each with more structure than the last:

  1. Spec.toTop, valued in the category of topological spaces,
  2. Spec.toSheafedSpace, valued in the category of sheafed spaces and
  3. Spec.toLocallyRingedSpace, valued in the category of locally ringed spaces.

Additionally, we provide Spec.toPresheafedSpace as a composition of Spec.toSheafedSpace with a forgetful functor.

The adjunction Γ ⊣ Spec is constructed in Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean.

The spectrum of a commutative ring, as a topological space.

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The induced map of a ring homomorphism on the ring spectra, as a morphism of topological spaces.

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The spectrum, as a contravariant functor from commutative rings to topological spaces.

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The spectrum of a commutative ring, as a SheafedSpace.

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The induced map of a ring homomorphism on the ring spectra, as a morphism of sheafed spaces.

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Spec, as a contravariant functor from commutative rings to sheafed spaces.

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The spectrum of a commutative ring, as a LocallyRingedSpace.

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The induced map of a ring homomorphism on the prime spectra, as a morphism of locally ringed spaces.

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Spec, as a contravariant functor from commutative rings to locally ringed spaces.

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The counit (SpecΓIdentity.inv.op) of the adjunction Γ ⊣ Spec is an isomorphism.

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The stalk map of Spec M⁻¹R ⟶ Spec R is an iso for each p : Spec M⁻¹R.

For an algebra f : R →+* S, this is the ring homomorphism S →+* (f∗ 𝒪ₛ)ₚ for a p : Spec R. This is shown to be the localization at p in isLocalizedModule_toPushforwardStalkAlgHom.

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This is the AlgHom version of toPushforwardStalk, which is the map S ⟶ (f∗ 𝒪ₛ)ₚ for some algebra R ⟶ S and some p : Spec R.

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