55 lines
2.1 KiB
Markdown
55 lines
2.1 KiB
Markdown
# 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:
|
||
```lean
|
||
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:
|
||
```lean
|
||
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.?`
|
||
```lean
|
||
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:
|
||
```lean
|
||
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
|