Serre's vanishing theorem on a Noetherian affine scheme #
Let R be a Noetherian ring. This file proves that a quasi-coherent sheaf of modules on Spec R
has no cohomology in positive degrees: Hⁿ(Spec R, M~) = 0 for every R-module M and every
n > 0. Together with the Mayer–Vietoris sequence this is how the cohomology of a quasi-coherent
sheaf on a Noetherian scheme is computed from an affine open cover.
The proof is by dimension shifting. Embed M into an injective R-module I, with cokernel Q.
Since M ↦ M~ is exact, 0 ⟶ M~ ⟶ I~ ⟶ Q~ ⟶ 0 is a short exact sequence of
𝒪_{Spec R}-modules, and I~ is flasque because R is Noetherian. Hence the long exact sequence
gives H¹(M~) = coker(Γ(I~) ⟶ Γ(Q~)) = coker(I ⟶ Q) = 0 and Hⁿ⁺²(M~) ≅ Hⁿ⁺¹(Q~), and induction
on n concludes, since Q is again an arbitrary R-module.
Main declarations #
AlgebraicGeometry.Scheme.Modules.subsingleton_cohomology_succ_of_isQuasicoherent: a quasi-coherent sheaf of modules on the spectrum of a Noetherian ring has vanishing cohomology in every positive degree.
Implementation notes #
The Noetherian hypothesis enters only through the flasqueness of I~. The vanishing holds on
every affine scheme, but the proof of that generality goes through Čech cohomology rather than
injective modules.
References #
- R. Hartshorne, Algebraic Geometry, Chapter III, Theorem 3.5 and Proposition 3.4.
Serre's vanishing theorem (Hartshorne, Algebraic Geometry, Theorem III.3.5): on the spectrum of a Noetherian ring, a quasi-coherent sheaf of modules has vanishing cohomology in every positive degree.