mirror of
https://github.com/GTBarkley/comm_alg.git
synced 2024-12-26 23:48:36 -06:00
83 lines
2.6 KiB
Text
83 lines
2.6 KiB
Text
/-
|
||
We don't want to reinvent the wheel, but finding
|
||
things in Mathlib can be pretty annoying. This is
|
||
a temporary file intended to be a dumping ground for
|
||
useful lemmas and definitions
|
||
-/
|
||
import Mathlib.RingTheory.Ideal.Basic
|
||
import Mathlib.RingTheory.Noetherian
|
||
import Mathlib.RingTheory.Artinian
|
||
import Mathlib.Order.Height
|
||
import Mathlib.RingTheory.MvPolynomial.Basic
|
||
import Mathlib.RingTheory.Ideal.Over
|
||
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
|
||
import Mathlib.Algebra.Homology.ShortExact.Abelian
|
||
|
||
variable {R M : Type _} [CommRing R] [AddCommGroup M] [Module R M]
|
||
|
||
--ideals are defined
|
||
#check Ideal R
|
||
|
||
variable (I : Ideal R)
|
||
|
||
--as are prime and maximal (they are defined as typeclasses)
|
||
#check (I.IsPrime)
|
||
#check (I.IsMaximal)
|
||
|
||
--a module being Noetherian is also a class
|
||
#check IsNoetherian M
|
||
#check IsNoetherian I
|
||
|
||
--a ring is Noetherian if it is Noetherian as a module over itself
|
||
#check IsNoetherianRing R
|
||
|
||
--ditto for Artinian
|
||
#check IsArtinian M
|
||
#check IsArtinianRing R
|
||
|
||
--I can't find the theorem that an Artinian ring is noetherian. That could be a good
|
||
--thing to add at some point
|
||
|
||
|
||
|
||
--Here's the main defintion that will be helpful
|
||
#check Set.chainHeight
|
||
|
||
--this is the polynomial ring R[x]
|
||
#check Polynomial R
|
||
--this is the polynomial ring with variables indexed by ℕ
|
||
#check MvPolynomial ℕ R
|
||
--hopefully there's good communication between them
|
||
|
||
|
||
--There's a preliminary version of the going up theorem
|
||
#check Ideal.exists_ideal_over_prime_of_isIntegral
|
||
|
||
--Theorems relating primes of a ring to primes of its localization
|
||
#check PrimeSpectrum.localization_comap_injective
|
||
#check PrimeSpectrum.localization_comap_range
|
||
--Theorems relating primes of a ring to primes of a quotient
|
||
#check PrimeSpectrum.range_comap_of_surjective
|
||
|
||
-- There is a notion of short exact sequences but the number of theorems are lacking
|
||
-- For example, I couldn't find anything saying that for a ses 0 -> A -> B -> C -> 0
|
||
-- of R-modules, A and C being FG implies that B is FG
|
||
open CategoryTheory CategoryTheory.Limits CategoryTheory.Preadditive
|
||
|
||
variable {𝒜 : Type _} [Category 𝒜] [Abelian 𝒜]
|
||
variable {A B C : 𝒜} {f : A ⟶ B} {g : B ⟶ C} {h : LeftSplit f g} {h' : RightSplit f g}
|
||
|
||
#check ShortExact
|
||
#check ShortExact f g
|
||
-- There are some notion of splitting as well
|
||
#check Splitting
|
||
#check LeftSplit
|
||
#check LeftSplit f g
|
||
-- And there is a theorem that left split implies splitting
|
||
#check LeftSplit.splitting
|
||
#check LeftSplit.splitting h
|
||
-- Similar things are there for RightSplit as well
|
||
#check RightSplit.splitting
|
||
#check RightSplit.splitting h'
|
||
-- There's also a theorem about ismorphisms between short exact sequences
|
||
#check isIso_of_shortExact_of_isIso_of_isIso
|