Lattices with equivalent localizations

Let ρ and σ be representations of a monoid G on finitely generated R-modules V and W, and let A be the localization of R at a submonoid S whose elements act injectively on V and on W. If the scalar extensions A ⊗[R] V and A ⊗[R] W are equivalent representations, then V and W are equivalent up to a scalar of S: there are intertwining maps f : V → W and f' : W → V and an s ∈ S with f' ∘ f = s • id and f ∘ f' = s • id.

The maps are obtained by clearing denominators. The equivalence e restricted to V ⊆ A ⊗[R] V takes values with denominators in S, and because V is finitely generated a single s ∈ S clears all of them (Module.Finite.exists_lift_of_isLocalizedModule_of_injective), so that s • e maps V into W. Intertwining is inherited from e, since W embeds in A ⊗[R] W.

For representations over ℤ, with S the nonzero integers and A = ℚ, this says that two representations on finitely generated torsion-free ℤ-modules with equivalent rationalizations are related by an injective intertwining map with finite cokernel. This is the lattice input to the comparison of the reductions modulo a prime ℓ of two ℤ[G]-lattices with isomorphic rationalizations: the reductions need not be isomorphic, but the finite-index embedding forces their classes in the Grothendieck group of 𝔽_ℓ[G] to agree.

Main results

• Representation.Equiv.exists_intertwiningMap_comp_eq_smul: representations on S-torsion-free finitely generated modules with equivalent localizations are equivalent up to a scalar of S.
• Representation.Equiv.exists_injective_finite_quotient_range: two representations on finitely generated torsion-free ℤ-modules with equivalent rationalizations admit an injective intertwining map from one to the other with finite cokernel.

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), §VII.3, (7.3.3).
• J. S. Milne, Arithmetic Duality Theorems, 2nd ed. (2006), Chapter I, Lemma 2.12.

theorem Representation.Equiv.exists_intertwiningMap_comp_eq_smul {R : Type u_1} [CommSemiring R] {A : Type u_2} [CommSemiring A] [Algebra R A] {G : Type u_3} [Monoid G] {V : Type u_4} {W : Type u_5} [AddCommMonoid V] [Module R V] [AddCommMonoid W] [Module R W] {ρ : Representation R G V} {σ : Representation R G W} (e : (Representation.baseChange A ρ).Equiv (Representation.baseChange A σ)) (S : Submonoid R) [IsLocalization S A] [Module.Finite R V] [Module.Finite R W] (hV : ∀ (s : ↥S), IsSMulRegular V ↑s) (hW : ∀ (s : ↥S), IsSMulRegular W ↑s) : ∃ (f : ρ.IntertwiningMap σ) (f' : σ.IntertwiningMap ρ) (s : ↥S), (∀ (v : V), f' (f v) = ↑s • v) ∧ ∀ (w : W), f (f' w) = ↑s • w

Two representations on finitely generated modules on which S acts injectively, whose scalar extensions to the localization A of R at S are equivalent, are equivalent up to a scalar of S: there are intertwining maps f : V → W and f' : W → V and s ∈ S with f' ∘ f = s • id and f ∘ f' = s • id.

theorem Representation.Equiv.exists_injective_finite_quotient_range {G : Type u_1} [Monoid G] {V : Type u_2} {W : Type u_3} [AddCommGroup V] [Module ℤ V] [Module.Finite ℤ V] [Module.IsTorsionFree ℤ V] [AddCommGroup W] [Module ℤ W] [Module.Finite ℤ W] [Module.IsTorsionFree ℤ W] {ρ : Representation ℤ G V} {σ : Representation ℤ G W} (e : (Representation.baseChange ℚ ρ).Equiv (Representation.baseChange ℚ σ)) : ∃ (f : ρ.IntertwiningMap σ), Function.Injective ⇑f ∧ Finite (W ⧸ f.range)

Two representations of G on finitely generated torsion-free ℤ-modules whose rationalizations ℚ ⊗[ℤ] V and ℚ ⊗[ℤ] W are equivalent admit an injective intertwining map V → W with finite cokernel.