---- > [!theorem] Theorem. ([[when do Čech and sheaf cohomology agree?]]) > Let $X$ be a [[locally Noetherian scheme|Noetherian scheme]] with an [[affine scheme|open affine]] [[cover]] $\mathcal{U}=\{ U_{i} \}_{i \in I}$, $I$ [[well-ordered set|well-ordered]], with the property that $U_{i_{1},\dots,i_{n}}=U_{i_{1} } \cap \dots \cap U_{i _{n}}$ also [[affine scheme|affine]]. Then if $\mathcal{F}$ is a [[quasicoherent sheaf|quasicoherent]] [[sheaf]] on $X$, we have $ H^{p}(X, \mathcal{F}) \cong \check H^{p}(\mathscr{U}, \mathcal{F}),$ i.e., [[Čech sheaf cohomology|Čech cohomology]] and [[sheaf cohomology]] agree for all $p$. ^theorem > [!basicexample] > If $X$ is [[scheme over a field|defined over]] an [[affine scheme]] $S$ (e.g., $S=\text{Spec }k$) with $X \to S$ [[separated scheme morphism|separated]], then *any* affine open cover of $X$ satisfies the hypothesis. > If $X \to S$ is not separated, things can fail... e.g. $\mathbb{A}^{2}$ with doubled origin is [[scheme gluing|by construction]] covered by two affines $U_{0}\cong \mathbb{A}^{2}$, $U_{1}\cong \mathbb{A}^{2}$, but $U_{0}\cap U_{1}=\mathbb{A}^{2}-\{ 0 \}$ is note affine (as ES4 shows — using [[singular cohomology|cohomology]]). ^basic-example > [!proof]- Proof. ([[when do Čech and sheaf cohomology agree?]]) > Omitted in our course. ---- #### ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```