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36 lines
1.2 KiB
Markdown
36 lines
1.2 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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```
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- Definition of a Spec of a ring
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- Definition of a Noetherian and Artinian rings
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- Definitions of a local ring and quotient ring
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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)
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- Definition of the height of prime ideal (dimension of A_p)
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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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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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Definition of a graded module
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