eq_zero_of_injective — Mathlib

English

Another formulation of Ext vanishing under injectivity: certain Ext classes are forced to be zero when injective coefficients appear.

Русский

Другая формулировка исчезновения Ext под инъективностью: некоторые классы Ext вынужденно равны нулю при инъективных коэффициентах.

LaTeX

$$Eq e 0 for injective conditions$$

Lean4

theorem eq_zero_of_injective [HasExt.{w} C] {X I : C} {n : ℕ} [Injective I] (e : Ext X I (n + 1)) : e = 0 :=
  by
  let K := (CochainComplex.singleFunctor C 0).obj X
  have := K.isStrictlyGE_of_ge (-n) 0 (by cutsat)
  letI := HasDerivedCategory.standard C
  apply homEquiv.injective
  simp only [← cancel_mono (((singleFunctors C).shiftIso (n + 1) (-(n + 1)) 0 (by cutsat)).hom.app _), zero_hom,
    Limits.zero_comp]
  exact to_singleFunctor_obj_eq_zero_of_injective (K := K) (n := -n) _ (by cutsat)

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