Theorems · Definition · commutative algebra

Module.DirectLimit.linearEquiv

{R : Type u_1} →
  [inst : Semiring R] →
    {ι : Type u_2} →
      [inst_1 : Preorder ι] →
        (G : ι → Type u_3) →
          [inst_2 : (i : ι) → AddCommMonoid (G i)] →
            [inst_3 : (i : ι) → Module R (G i)] →
              (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) →
                [inst_4 : DecidableEq ι] →
                  [inst_5 : Nonempty ι] →
                    [inst_6 : IsDirectedOrder ι] →
                      [inst_7 : DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] →
                        Module.DirectLimit G f ≃ₗ[R] DirectLimit G f

The direct limit constructed as a quotient of the direct sum is isomorphic to the direct limit constructed as a quotient of the disjoint union.

Defined in

Mathlib.Algebra.Colimit.Module

Cited by

3 results in Mathlib

Foundations

Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringPreorderAddCommMonoidModuleDecidableEqNonemptyIsDirectedOrderDirectedSystem