liftEquiv' — Mathlib

English

The map decomposes S/ I as a finite direct sum of cyclic quotients by the Smith-coefficients.

Русский

Отображение разлагает S/ I на конечную прямую сумму циклических фактор-модулей по Smith-coefficients.

LaTeX

$$S/I ≃_F ⨁ i, R/ span({I.smithCoeffs b I hI i})$$

Lean4

/-- The universal property of `ModN G n` in terms of `ZMod n`-modules: `ZMod n`-linear maps from
`ModN G n` are the same as monoid homomorphisms from `G` whose values are `n`-torsion. -/
protected def liftEquiv' [AddCommGroup H] [Module (ZMod n) H] :
    (ModN G n →ₗ[ZMod n] H) ≃ { φ : G →+ H // ∀ g, n • φ g = 0 } :=
  (AddMonoidHom.toZModLinearMapEquiv n).symm.toEquiv.trans ModN.liftEquiv

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