feat(Algebra/Module): a finite stably free module M is free if it is locally free of rank 1

[mbkybky]

We show that a finite stably free module M is free if all of its localizations at maximal ideals are free of rank 1

• depends on: feat(Algebra/Module): define stably free modules #36951

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[github-actions]

PR summary 4ec5d95aa6

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Algebra.Module.StablyFree.Basic (new file)	1373
Mathlib.Algebra.Module.StablyFree.FreeOfLocalizedEq (new file)	2218

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Declarations diff

+ IsStablyFree
+ IsStablyFree.equiv
+ IsStablyFree.equiv_iff
+ IsStablyFree.of_free_prod
+ IsStablyFree.out
+ Module.free_of_isStablyFree_of_localized_eq_ring
+ cofactorLinear
+ cofactorToLeft
+ cofactorToLeft_ιMulti_cons
+ instance (priority := low) IsStablyFree.projective [IsStablyFree R M] : Module.Projective R M := by
+ instance [Free R M] : IsStablyFree R M
+ topExteriorLinearEquiv

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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No changes to technical debt.

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:

• feat(Algebra/Module): define stably free modules #36951
  By Dependent Issues (🤖). Happy coding!