essSurj_of_hasLeftResolutions — Mathlib

English

essSurj_of_hasLeftResolutions: the composite (Φ.functor ⋙ L₂) is essentially surjective when Φ has left resolutions.

Русский

essSurj через левые разрешения: композиция (Φ.functor ⋙ L₂) полно по образу при наличии левых разрешений.

LaTeX

$$$\text{EssSurj}(\Φ.functor \circ L_2)$$$

Lean4

theorem essSurj_of_hasLeftResolutions [Φ.HasLeftResolutions] : (Φ.functor ⋙ L₂).EssSurj where
  mem_essImage
    X₂ := by
    have := Localization.essSurj L₂ W₂
    have L : Φ.LeftResolution (L₂.objPreimage X₂) := Classical.arbitrary _
    exact ⟨L.X₁, ⟨Localization.isoOfHom L₂ W₂ _ L.hw ≪≫ L₂.objObjPreimageIso X₂⟩⟩

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