---- > [!theorem] Theorem. ([[sheaf cohomology is well-behaved with respect to the dimension of Noetherian spaces]]) > If $X$ is a [[Noetherian topological space|Noetherian]] [[topological space]] of [[dimension of a topological space|dimension]] $n$, and $\mathcal{F}$ is a [[sheaf]] of [[abelian group|abelian groups]] on $X$, then $H^{i}(X; \mathcal{F})=0$ for all $i>n$. ^theorem > [!note] Remark. > This theorem is not useful in ordinary algebraic topology because e.g. a [[manifold]] with its standard topology will not be [[Noetherian topological space|Noetherian]] (and moreover, the notion of dimension used in algebraic geometry is not really relevant). ^note > [!proof]- Proof. ([[sheaf cohomology is well-behaved with respect to the dimension of Noetherian spaces]]) > Not 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 > ```