----- > [!proposition] Proposition. ([[exactness can be tested on stalks]]) > A cochain complex of [[sheaf|sheaves]] over a [[topological space]] $X$ $\cdots \to \mathcal{F}^{i-1} \to \mathcal{F}^{i} \xrightarrow{d} \mathcal{F}^{i+1} \to \cdots$ > is [[exact sequence|exact]] at $i$ if and only if for every $p \in X$, the [[chain complex of modules|cochain complex of abelian groups]] $\cdots \to \mathcal{F}^{i-1}_{p} \to \mathcal{F}^{i}_{p} \xrightarrow{d_{p}} \mathcal{F}^{i+1}_{p} \to \cdots$ > is [[exact sequence|exact]] at $i$. > [!proof]- Proof. ([[exactness can be tested on stalks]]) > > $\to$. Suppose $\mathcal{F}^{\bullet}$ is [[exact sequence|exact]] at $i$. The definition of exactness of $\mathcal{F}^{\bullet}$ at $i$ is $\im d^{i}=\ker d^{i+1}$. Since [[(pre)sheaf kernel|taking kernels and stalks commute]] and [[sheaf image|likewise for images]], we have for all $p$: $(\im d^{i})_{p}=\im d^{i}_{p} \text{ and } (\ker d^{i+1})_{p}=\ker d^{i+1}_{p}. \ \ (*)$ > Since the two left-hand sides are equal, the two right-hand sides are equal as well. > > $\leftarrow$. Suppose that, for all $p$, $\im d_{p}^{i}=\ker d_{p}^{i+1}$. Then (by $(*)$) the [[(pre)sheaf stalk|stalks]] of $\im d^{i}$ and $\ker d^{i+1}$ are equal at all $p$. Now the result follows from [[equality of subsheaf and sheaf can be tested on stalks]], at least assuming that we are given a chain complex. ----- #### ---- #### 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 > ```