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