diff --git a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean index b81f47d19763c..b3aa7de64c852 100644 --- a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean +++ b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean @@ -187,10 +187,11 @@ lemma smul_section_apply (r : R) (U : Opens (PrimeSpectrum.Top R)) lemma smul_stalk_no_nonzero_divisor {x : PrimeSpectrum R} (r : x.asIdeal.primeCompl) (st : (tildeInModuleCat M).stalk x) (hst : r.1 • st = 0) : st = 0 := by - refine Limits.Concrete.colimit_no_zero_smul_divisor (hx := hst) - ⟨op ⟨PrimeSpectrum.basicOpen r.1, r.2⟩, fun U i s hs ↦ Subtype.eq <| funext fun pt ↦ ?_⟩ - exact LocalizedModule.eq_zero_of_smul_eq_zero (hx := congr_fun (Subtype.ext_iff.1 hs) pt) <| - i.unop pt |>.2 + refine Limits.Concrete.colimit_no_zero_smul_divisor + _ _ _ ⟨op ⟨PrimeSpectrum.basicOpen r.1, r.2⟩, fun U i s hs ↦ Subtype.eq <| funext fun pt ↦ ?_⟩ + _ hst + apply LocalizedModule.eq_zero_of_smul_eq_zero _ (i.unop pt).2 _ + (congr_fun (Subtype.ext_iff.1 hs) pt) /-- If `U` is an open subset of `Spec R`, this is the morphism of `R`-modules from `M` to