The sheaf of an injective module over a Noetherian ring is flasque #
Let R be a Noetherian ring and I an injective R-module. This file proves that the
quasi-coherent sheaf I^~ on Spec R is flasque: every section of I^~ over an open subset
extends to a global section (Hartshorne, Algebraic Geometry, Proposition III.3.4).
Flasque sheaves have no higher cohomology, and every R-module embeds into an injective one, so
this is the input for Serre's vanishing theorem Hⁱ(Spec R, M^~) = 0 for i > 0 on a Noetherian
affine scheme.
Implementation notes #
Every open subset of Spec R is a finite union of basic open subsets D(g), and the proof
inducts on the number of them. Suppose that every section over W = D(g₁) ∪ ⋯ ∪ D(gₙ) extends,
and let s be a section over D(f) ∪ W. Subtracting an extension of s|_W reduces to the case
that s vanishes on W. Then σ = s|_{D(f)} is an element of I_f that vanishes on each
D(f gᵢ), so it is killed by a power of every gᵢ: it lies in the 𝔞-primary component
Γ_𝔞(I_f) for 𝔞 = (g₁, …, gₙ). By Ideal.primaryComponent_map_surjective it is the
image of some u ∈ Γ_𝔞(I), which is Hartshorne's Lemma III.3.2 and Proposition III.3.3 combined.
The global section u restricts to σ on D(f) and to zero on each D(gᵢ), so it extends s.
This induction on basic open subsets replaces the Noetherian induction on the support of I^~ in
Hartshorne's proof.
Main declarations #
TauCeti.AlgebraicGeometry.isFlasque_tilde_of_injective:I^~is flasque.
References #
- R. Hartshorne, Algebraic Geometry, Chapter III, Proposition 3.4.
The sheaf of an injective module over a Noetherian ring is flasque (Hartshorne,
Algebraic Geometry, Proposition III.3.4): if R is Noetherian and M is an injective
R-module, then every section of M^~ over an open subset of Spec R extends to a larger open
subset.