We work with abelian sheaves on a topological space $X$. (You can assume that $X$ is "nice", locally compact, etc ...)

First, if $F$ and $G$ are two abelian sheaves, we have a natural map $$\mathcal{H}om(F,G)\otimes F \to G.$$ Indeed $$\text{Hom}(\mathcal{H}om(F,G)\otimes F,G) \simeq \text{Hom}(\mathcal{H}om(F,G),\mathcal{H}om(F,G))$$ so the identity on the RHS defines the desired morphism on the LHS. Now take $H$ an abelian sheaf and tensors the previous map by $H$. We get $$\mathcal{H}om(F,G)\otimes F \otimes H \to G \otimes H.$$ Hence we get the map $$\mathcal{H}om(F,G)\otimes H \to \mathcal{H}om(F,G\otimes H).$$ Is this last map an epimorphism ? If not, could it be one in the derived category ?


Consider the one point space, so that abelian sheaves are just abelian groups.

$$ \hom(\mathbb{Q}, \mathbb{Z}) \otimes \mathbb{Q} \to \hom(\mathbb{Q}, \mathbb{Z} \otimes \mathbb{Q}) $$

is not an epimorphism.


Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.