feat(Mathlib/Algebra/Module): uniform annihilator for FG modules with subsingleton localization#87
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… FG modules with subsingleton localization Adds `Module.Finite.exists_mem_smul_eq_zero_of_subsingleton_localizedModule`: for a finitely generated `R`-module `M` and a submonoid `T ⊆ R` such that the localized module `T⁻¹ M` is trivial, there exists a single element `t ∈ T` that annihilates all of `M`. This is the uniform version of `LocalizedModule.subsingleton_iff` and is destined for Mathlib. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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…o `IsLocalizedModule` and strengthen to `iff` Addresses review on PR chrisflav#87: - Generalize from `LocalizedModule T M` to any `g : M →ₗ[R] M'` with `[IsLocalizedModule S g]`, matching the pattern of `IsLocalizedModule.subsingleton_iff` / `LocalizedModule.subsingleton_iff`. - Strengthen the existential to an `iff` (`Module.Finite.subsingleton_iff_exists_mem_smul_eq_zero`): under finite generation, a uniform annihilator exists iff the localization is trivial. Keep the forward existential (`Module.Finite.exists_mem_smul_eq_zero_of_subsingleton`) as a thin corollary, with the `_localizedModule` suffix dropped. - Move the file to `Proetale/Mathlib/RingTheory/Finiteness/LocalizedModule.lean` to respect the mirror-tree convention and reflect the `Module.Finite` dependency (the lemma's heavier import is `RingTheory.Finiteness`). - Minor proof polish: `s.mem_attach ⟨x, hx⟩` and `by rw [hs]; exact Submodule.mem_top`. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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Summary
Adds the lemma
Module.Finite.exists_mem_smul_eq_zero_of_subsingleton_localizedModuleto the Mathlib-mirror tree (new file
Proetale/Mathlib/Algebra/Module/LocalizedModule.lean):For a finitely generated
R-moduleMand a submonoidT ⊆ Rsuch that thelocalized module
T⁻¹ Mis trivial, there exists a single elementt ∈ Twhoseaction annihilates all of
M. This is the uniform version ofLocalizedModule.subsingleton_ifffor finitely generated modules.This is upstreamed from the
archonbranch (where it appears as a_root_.helper inside
Proetale/Mathlib/RingTheory/Etale/IndSpreads.lean,named
Module.exists_mem_smul_eq_zero_of_finite_of_subsingleton_localization).Pulled out and renamed to a Mathlib-style location/name so it can be used
independently and is destined for Mathlib proper.
Test plan
lake build Proetale.Mathlib.Algebra.Module.LocalizedModulesucceedslake buildsucceeds (no regressions in dependents)sorryin the new file