Vanishing of Quasi-Coherent Sheaf Cohomology of Affine Scheme

Theorem

Let $X = \Spec A$ be the spectrum of a 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

This theorem requires a proof. In particular: Proof following EGA III (1.3.1) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. 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 {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

• 1961: Alexander Grothendieck: Éléments de Géométrie Algébrique: Volume $\text { III }$ $(1.3.1)$
• 1977: Robin Hartshorne: Algebraic Geometry Remark $\text{III}.3.5.1$