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Mathlib.RingTheory.Smooth.StandardSmooth

Standard smooth algebras #

A standard smooth algebra is an algebra that admits a SubmersivePresentation. A standard smooth algebra is of relative dimension n if it admits a submersive presentation of dimension n.

While every standard smooth algebra is smooth, the converse does not hold. But if S is R-smooth, then S is R-standard smooth locally on S, i.e. there exists a set { t } of S that generates the unit ideal, such that Sₜ is R-standard smooth for every t (TODO, see below).

Main definitions #

All of these are in the Algebra namespace. Let S be an R-algebra.

TODO #

Notes #

This contribution was created as part of the AIM workshop "Formalizing algebraic geometry" in June 2024.

class Algebra.IsStandardSmooth (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] :

An R-algebra S is called standard smooth, if there exists a submersive presentation.

Instances
    theorem Algebra.SubmersivePresentation.isStandardSmooth (R : Type u) (S : Type v) (ι : Type w) (σ : Type t) [CommRing R] [CommRing S] [Algebra R S] [Finite σ] [Finite ι] (P : SubmersivePresentation R S ι σ) :
    noncomputable def Algebra.IsStandardSmooth.relativeDimension (R : Type u) (S : Type v) [CommRing R] [CommRing S] [Algebra R S] [IsStandardSmooth R S] :

    The relative dimension of a standard smooth R-algebra S is the dimension of an arbitrarily chosen submersive R-presentation of S.

    Note: If S is non-trivial, this number is independent of the choice of the presentation as it is equal to the S-rank of Ω[S/R] (see IsStandardSmoothOfRelativeDimension.rank_kaehlerDifferential).

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    Instances For

      An R-algebra S is called standard smooth of relative dimension n, if there exists a submersive presentation of dimension n.

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