feat(RingTheory/NoetherNormalization): Add Noether normalization lemma

[su00000]

This PR contains a proof of Noether normalization lemma:
Let A be a finitely generated algebra over a field k. Then there exists natural number $r$ and an injective homomorphism from k[X_1, X_2, ..., X_r] to A such that A is integral over k[X_1, X_2, ..., X_r].

Co-authored-by: Riccardo Brasca [email protected]
Co-authored-by: Wan Lin [email protected]
Co-authored-by: Xiaoyang Su [email protected]

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• depends on: [Merged by Bors] - feat(RingTheory/IntegralClosure/IsIntegralClosure/Basic): Add lemmas about MvPolynomial and Polynomial #18244

[github-actions]

PR summary d818c2c745

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.NoetherNormalization (new file)	1395

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Declarations diff

+ T1
+ exists_finite_inj_algHom_of_fg
+ exists_integral_inj_algHom_of_fg
+ exists_integral_inj_algHom_of_quotient
+ map_coe
+ ofDigits_inj_of_len_eq

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(RingTheory/IntegralClosure/IsIntegralClosure/Basic): Add lemmas about MvPolynomial and Polynomial #18244
  By Dependent Issues (🤖). Happy coding!

[riccardobrasca]

Can you please fix the conflict? Thanks!

[riccardobrasca]

Can you add a short description of the strategy of the proof in the docstring at the beginning? It doesn't seem to be the same as in the stack project.

[Ruben-VandeVelde]

Thanks! A few small comments

[riccardobrasca]

I think this is really good and almost ready. In my opinion there are still two things to be done:

• you write a rather detailed proof strategy, but the description is before each theorem. I think it would be better to have it at the beginning of the file, with references to the actual declaration proving what you claim, maybe in a section "# Strategy of the proof" (for example you can say something like so that T maps
  different monomials of f to mvpolynomials with different degrees of X_0, see name_of_the_theorem).
• we try to have descriptive names even for private lemmas. I know, this is annoying but it helps).

The best thing would be to try to extract some results, but I don't know if this is reasonable, they all seem very specific to this proof.