# Vanishing of Quasi-Coherent Sheaf Cohomology of Noetherian Affine Scheme

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## 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 $\text{III}.3.5$