Vanishing of Quasi-Coherent Sheaf Cohomology of Noetherian Affine Scheme

This page has been identified as a candidate for refactoring of basic complexity. Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

Theorem

Let $X = \Spec A$ be the spectrum of a noetherian commutative ring $A$.

Let $\FF$ be a quasi-coherent sheaf on $X$.

Then for all $i \in \Z$ with $i > 0$ the $i$-th sheaf cohomology $\map {H^i} {X, \FF} = 0$.

Proof 1

Let $M := \map\Gamma{X, \FF}$ be the $\OO_X$-module of global sections of $\FF$.

By Category of Modules has Enough Injectives the category of $A$-modules has enough injectives.

By Injective Resolution Exists iff Enough Injectives $M$ has an injective resolution $I$.

Let

$\tilde{} : A\text{-}\mathbf{Mod} \to \mathbf{QCoh}(X)$

be the functor from the category of $A$-modules to the category of quasi-coherent $\OO_X$-modules $\mathbf{QCoh}(X)$.

By Equivalence of Categories is Exact and Exact Functor Preserves Exact Sequences $\tilde I$ is an exact cochain complex.

By Equivalence of Modules and Quasi-Coherent Sheaves there is an isomorphism $\FF \cong \tilde M$.

By Sheaf Associated to Injective Module over Noetherian Ring is Flasque, $\tilde I^i$ is flasque for all $i \mathop \in \Z$.

By Flasque Sheaf is Acyclic $\tilde I$ is an acyclic resolution of $\FF$.

By Equivalence of Modules and Quasi-Coherent Sheaves, the cochain complex $\Gamma(X, \tilde I)$ is exact at $i$ for all positive integers $i \in \Z_{\geq 1}$.

By Cochain Complex is Exact iff Cohomology Vanishes $H^i(\Gamma(X, \tilde I)) = 0$ for all positive integers $i \in \Z_{\geq 1}$.

By Right Derived Functor by Acyclic Resolution the cohomology groups $H^i(\Gamma(X, \tilde I))$ are isomorphic to the sheaf cohomology $H^i(X, \FF)$ for all $i \mathop \in \Z$.

Hence $H^i(X, \FF) = 0$ for all positive integers $i \in \Z_{\geq 1}$.

$\blacksquare$

Proof 2

This is a special case of Vanishing of Quasi-Coherent Sheaf Cohomology of Affine Scheme.

$\blacksquare$

Sources

• 1977: Robin Hartshorne: Algebraic Geometry Theorem $\text{III}.3.5$