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54 lines
2.1 KiB
Markdown
54 lines
2.1 KiB
Markdown
# Commutative algebra in Lean
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Welcome to the repository for adding definitions and theorems related to Krull dimension and Hilbert polynomials to mathlib.
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We start the commutative algebra project with a list of important definitions and theorems and go from there.
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Feel free to add, modify, and expand this file. Below are starting points for the project:
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- Definitions of an ideal, prime ideal, and maximal ideal:
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```lean
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def Mathlib.RingTheory.Ideal.Basic.Ideal (R : Type u) [Semiring R] := Submodule R R
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class Mathlib.RingTheory.Ideal.Basic.IsPrime (I : Ideal α) : Prop
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class IsMaximal (I : Ideal α) : Prop
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```
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- Definition of a Spec of a ring: `Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic.PrimeSpectrum`
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- Definition of a Noetherian and Artinian rings:
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```lean
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class Mathlib.RingTheory.Noetherian.IsNoetherian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop
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class Mathlib.RingTheory.Artinian.IsArtinian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop
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```
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- Definition of a polynomial ring: `Mathlib.RingTheory.Polynomial.Basic`
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- Definitions of a local ring and quotient ring: `Mathlib.RingTheory.Ideal.Quotient.?`
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```lean
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class Mathlib.RingTheory.Ideal.LocalRing.LocalRing (R : Type u) [Semiring R] extends Nontrivial R : Prop
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```
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- Definition of the chain of prime ideals and the length of these chains
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- Definition of the Krull dimension (supremum of the lengh of chain of prime ideal): `Mathlib.Order.KrullDimension.krullDim`
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- Krull dimension of a module
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- Definition of the height of prime ideal (dimension of A_p): `Mathlib.Order.KrullDimension.height`
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Give Examples of each of the above cases for a particular instances of ring
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Theorem 0: Hilbert Basis Theorem:
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```lean
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theorem Mathlib.RingTheory.Polynomial.Basic.Polynomial.isNoetherianRing [inst : IsNoetherianRing R] : IsNoetherianRing R[X]
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```
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Theorem 1: If A is a nonzero ring, then dim A[t] >= dim A +1
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Theorem 2: If A is a nonzero noetherian ring, then dim A[t] = dim A + 1
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Theorem 3: If A is nonzero ring then dim A_p + dim A/p <= dim A
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Lemma 0: A ring is artinian iff it is noetherian of dimension 0.
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Definition of a graded module
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