Suppose we have a topological space $X$ and an open subset $U \subseteq X$ with inclusion $j \colon U \hookrightarrow X$. If we have a sheaf $\mathcal F \in \text{Sh}(X)$, then we know that in general the canonical sheaf morphism $$ \mathcal F \to j_*\mathcal F|_U $$ is not surjective. It is, though, for flasque sheaves.

I was wondering if the surjectivity still holds if $\mathcal F$ is only soft, but $X$ is $T_1$, i.e. every single-point-set is closed. I think the answer is yes, because as $\{x\}$ is closed for every $x \in X$, the softness of $\mathcal F$ implies the surjectivity of the canonical map $$ \mathcal F(X) \to \mathcal F_x $$ which also implies the surjectivity of $$ \mathcal F_x \to (j_*\mathcal F|_U)_x. $$

Is this correct or am I missing something? Thanks in advance for any help / comments!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.