feat(RingTheory): local isomorphisms

[chrisflav]

We add the class of algebras that are locally (on the geometric source) standard open immersions. This will be used to define ind-Zariski algebras, which are an important tool to study ind-étale algebras.

From Proetale.

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• depends on: chore(RingTheory/Localization): mirror tensor product compatibilities and golf #38177

[github-actions]

PR summary 718d4e20c0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.LocalIso (new file)	1934

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Declarations diff

+ Algebra.IsLocalIso
+ IsLocalization.Away.tensorProductEquivTMulRight
+ IsLocalization.Away.tensorProductEquivTMulRight_tmul
+ IsLocalization.tensorProductEquivOfMapIncludeRight
+ IsLocalization.tensorProductEquivOfMapIncludeRight_tmul
+ IsLocalization.tensorProduct_tensorProduct_right
+ IsStableUnderBaseChange.tensorProduct
+ Locally.span_eq_top
+ iff_span_isStandardOpenImmersion_eq_top
+ instance (priority := 100) [IsStandardOpenImmersion R S] : IsLocalIso R S
+ instance : IsStandardOpenImmersion R R
+ instance [Algebra R T] [IsLocalIso R S] : IsLocalIso T (T ⊗[R] S) := by
+ isScalarTower_localizationAlgebra
+ locally_iff_span_eq_top
+ of_bijective
+ of_isPushout
+ of_span_eq_top
+ of_span_range_eq_top
+ pi_of_finite
+ span_isStandardOpenImmersion_eq_top
++ iff_of_algEquiv
++ of_algEquiv
++ trans
- IsStandardOpenImmersion.trans
- instance : IsScalarTower R (Localization M) (Localization (Algebra.algebraMapSubmonoid S M))

You can run this locally as follows

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

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

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

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Increase in tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type
6511	1	backward.isDefEq.respectTransparency

Current commit 673b89f9ff
Reference commit 718d4e20c0

You can run this locally as

./scripts/reporting/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).

[mathlib-dependent-issues]

This PR/issue depends on:

• chore(RingTheory/Localization): mirror tensor product compatibilities and golf #38177
  By Dependent Issues (🤖). Happy coding!