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89 lines
2.8 KiB
Text
89 lines
2.8 KiB
Text
/-
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We don't want to reinvent the wheel, but finding
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things in Mathlib can be pretty annoying. This is
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a temporary file intended to be a dumping ground for
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useful lemmas and definitions
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-/
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import Mathlib.RingTheory.Ideal.Basic
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import Mathlib.RingTheory.Noetherian
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import Mathlib.RingTheory.Artinian
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import Mathlib.RingTheory.FiniteType
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import Mathlib.Order.Height
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import Mathlib.RingTheory.MvPolynomial.Basic
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import Mathlib.RingTheory.Ideal.Over
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Algebra.Homology.ShortExact.Abelian
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variable {R M : Type _} [CommRing R] [AddCommGroup M] [Module R M]
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--ideals are defined
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#check Ideal R
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variable (I : Ideal R)
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--as are prime and maximal (they are defined as typeclasses)
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#check (I.IsPrime)
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#check (I.IsMaximal)
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--a module being Noetherian is also a class
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#check IsNoetherian M
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#check IsNoetherian I
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--a ring is Noetherian if it is Noetherian as a module over itself
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#check IsNoetherianRing R
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--ditto for Artinian
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#check IsArtinian M
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#check IsArtinianRing R
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--I can't find the theorem that an Artinian ring is noetherian. That could be a good
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--thing to add at some point
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--Here's the main defintion that will be helpful
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#check Set.chainHeight
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--this is the polynomial ring R[x]
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#check Polynomial R
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--this is the polynomial ring with variables indexed by ℕ
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#check MvPolynomial ℕ R
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--hopefully there's good communication between them
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--There's a preliminary version of the going up theorem
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#check Ideal.exists_ideal_over_prime_of_isIntegral
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--Theorems relating primes of a ring to primes of its localization
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#check PrimeSpectrum.localization_comap_injective
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#check PrimeSpectrum.localization_comap_range
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--Theorems relating primes of a ring to primes of a quotient
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#check PrimeSpectrum.range_comap_of_surjective
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--There's a lot of theorems about finite-type algebras
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#check Algebra.FiniteType.polynomial
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#check Algebra.FiniteType.mvPolynomial
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#check Algebra.FiniteType.of_surjective
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-- There is a notion of short exact sequences but the number of theorems are lacking
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-- For example, I couldn't find anything saying that for a ses 0 -> A -> B -> C -> 0
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-- of R-modules, A and C being FG implies that B is FG
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open CategoryTheory CategoryTheory.Limits CategoryTheory.Preadditive
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variable {𝒜 : Type _} [Category 𝒜] [Abelian 𝒜]
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variable {A B C : 𝒜} {f : A ⟶ B} {g : B ⟶ C} {h : LeftSplit f g} {h' : RightSplit f g}
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#check ShortExact
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#check ShortExact f g
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-- There are some notion of splitting as well
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#check Splitting
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#check LeftSplit
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#check LeftSplit f g
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-- And there is a theorem that left split implies splitting
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#check LeftSplit.splitting
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#check LeftSplit.splitting h
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-- Similar things are there for RightSplit as well
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#check RightSplit.splitting
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#check RightSplit.splitting h'
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-- There's also a theorem about ismorphisms between short exact sequences
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#check isIso_of_shortExact_of_isIso_of_isIso
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