feat: generalize finrank_add_finrank_quotient to PIDs
#9151
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erdOne
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Dec 19, 2023
erdOne
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erdOne
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finrank_add_finrank_quotient to PIDsfinrank_add_finrank_quotient to PIDs
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Dec 19, 2023
| @@ -644,6 +644,16 @@ theorem equiv [Finite R M] (e : M ≃ₗ[R] N) : Finite R N := | |||
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| instance ulift [Finite R M] : Finite R (ULift M) := equiv ULift.moduleEquiv.symm | |||
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| theorem iff_FG {N : Submodule R M} : Module.Finite R N ↔ N.FG := Module.finite_def.trans (fg_top _) | |||
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| theorem iff_FG {N : Submodule R M} : Module.Finite R N ↔ N.FG := Module.finite_def.trans (fg_top _) | |
| theorem iff_fg {N : Submodule R M} : Module.Finite R N ↔ N.FG := Module.finite_def.trans (fg_top _) |
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Dec 22, 2023
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| instance [IsDomain R] : NoZeroSMulDivisors R (M ⧸ Submodule.torsion R M) := by | ||
| constructor | ||
| intros c x hcx | ||
| rw [or_iff_not_imp_left] | ||
| intro hc | ||
| obtain ⟨x, rfl⟩ := Submodule.mkQ_surjective _ x | ||
| rw [← LinearMap.map_smul, Submodule.mkQ_apply, Submodule.Quotient.mk_eq_zero] at hcx | ||
| obtain ⟨n, hn⟩ := hcx | ||
| simp only [Submodule.mkQ_apply, Submodule.Quotient.mk_eq_zero, Submonoid.mk_smul, exists_prop] | ||
| refine ⟨n * ⟨c, mem_nonZeroDivisors_of_ne_zero hc⟩, ?_⟩ | ||
| simpa [Submonoid.smul_def, smul_smul] using hn |
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Isn't this exactly Submodule.QuotientTorsion.noZeroSMulDivisors?
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| instance [IsDomain R] [IsPrincipalIdealRing R] : Module.Free R (M ⧸ Submodule.torsion R M) := | ||
| Module.free_of_finite_type_torsion_free' |
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Dec 23, 2023
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| lemma FiniteDimensional.finrank_add_finrank_quotient_le (N : Submodule R M) : | ||
| finrank R N + finrank R (M ⧸ N) ≤ finrank R M := by |
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Dec 26, 2023
… results about `Module.rank` in #9151 (#9253) + Proves that `Sup` (ciSup) commutes with cardinal addition (`ciSup_add_ciSup`) and multiplication. Generalize results in Cardinal/Basic introduced in #8842 to achieve this. + Use `ciSup_add_ciSup` to prove that the rank of a module is always at least the rank of a submodule plus the rank of the quotient by the submodule. Deduce that the rank of a product module is at least the sum of the ranks of the two factors. + Show that quotienting by a torsion submodule preserves the rank. + Golf `rank_zero_iff_forall_zero` using a recently added lemma. Co-authored-by: Andrew Yang <the.erd.one@gmail.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>
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Closing in favor of #9298 |
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