Theorems · Definition · commutative algebra

LinearEquiv.restrictScalars

(R : Type u_1) →
  {S : Type u_4} →
    {M : Type u_5} →
      {M₂ : Type u_7} →
        [inst : Semiring R] →
          [inst_1 : Semiring S] →
            [inst_2 : AddCommMonoid M] →
              [inst_3 : AddCommMonoid M₂] →
                [inst_4 : Module R M] →
                  [inst_5 : Module R M₂] →
                    [inst_6 : Module S M] →
                      [inst_7 : Module S M₂] → [LinearMap.CompatibleSMul M M₂ R S] → (M ≃ₗ[S] M₂) → M ≃ₗ[R] M₂

If M and M₂ are both R-semimodules and S-semimodules and R-semimodule structures are defined by an action of R on S (formally, we have two scalar towers), then any S-linear equivalence from M to M₂ is also an R-linear equivalence. See also LinearMap.restrictScalars.

Defined in

Mathlib.Algebra.Module.Equiv.Basic

Cited by

46 results in Mathlib

Foundations

Depth 27 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleModuleModuleLinearMap.CompatibleSMul