Theorems · Theorem · algebraic geometry

smoothSheafCommRing.isUnit_stalk_iff

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {EM : Type u_1} [inst_1 : NormedAddCommGroup EM]
  [inst_2 : NormedSpace 𝕜 EM] {HM : Type u_2} [inst_3 : TopologicalSpace HM] (IM : ModelWithCorners 𝕜 EM HM)
  {M : Type u} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace HM M] {x : M}
  (f : ↑((smoothSheafCommRing IM (modelWithCornersSelf 𝕜 𝕜) M 𝕜).presheaf.stalk x)),
  IsUnit f ↔ f ∉ RingHom.ker (smoothSheafCommRing.eval IM (modelWithCornersSelf 𝕜 𝕜) M 𝕜 x)

The units of the stalk at x of the sheaf of smooth functions from M to 𝕜, considered as a sheaf of commutative rings, are the functions whose values at x are nonzero.

Defined in

Mathlib.Geometry.Manifold.Sheaf.LocallyRingedSpace

Cited by

1 results in Mathlib

Foundations

Depth 233 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceTopologicalSpaceTopologicalSpaceChartedSpace