Theorems · Definition · commutative algebra

IsLocalizedModule.lift

{R : Type u_1} →
  [inst : CommSemiring R] →
    (S : Submonoid R) →
      {M : Type u_2} →
        {M' : Type u_3} →
          {M'' : Type u_4} →
            [inst_1 : AddCommMonoid M] →
              [inst_2 : AddCommMonoid M'] →
                [inst_3 : AddCommMonoid M''] →
                  [inst_4 : Module R M] →
                    [inst_5 : Module R M'] →
                      [inst_6 : Module R M''] →
                        (f : M →ₗ[R] M') →
                          [IsLocalizedModule S f] →
                            (M →ₗ[R] M'') → (∀ (x : ↥S), IsUnit ((algebraMap R (Module.End R M'')) ↑x)) → M' →ₗ[R] M''

If M' is a localized module and g is a linear map M → M'' such that all scalar multiplication by s : S is invertible, then there is a linear map M' → M''.

Defined in

Mathlib.Algebra.Module.LocalizedModule.Basic

Cited by

10 results in Mathlib

Foundations

Depth 50 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModuleIsLocalizedModule