I am very new to algebraic geometry. At the point of studying pushforward of a sheas I am wondering that does it is an exact function? Many resources (like MSE questions) telling that it is left exact but not right but not providing any concrete examle. But the pushforward under inclusion of a closed set is exact and the same for open set is not true.
Can someone give an explicit example of a this non exactness phenomena on inclusion of open sets?
I was trying with skyscraper Sheaf and constant sheaf, their direct product etc. but I did not get any proper example.
Thanks in advance.
Let $i \colon U \to \mathbb{A}^2$ be the embedding of the complement of the origin. Consider the sequence $$ 0 \to \mathcal{O}_U \stackrel{(-y,x)} \to \mathcal{O}_U \oplus \mathcal{O}_U \stackrel{(x,y)} \to \mathcal{O}_U \to 0. $$ It is exact, because $x$ and $y$ have no common zeros on $U$, hence form a regular sequence. On the other hand, $i_*\mathcal{O}_U \cong \mathcal{O}_{\mathbb{A}^2}$ (by the Hartog's theorem}, hence the pushforward of the sequence looks like $$ 0 \to \mathcal{O}_{\mathbb{A}^2} \stackrel{(-y,x)} \to \mathcal{O}_{\mathbb{A}^2} \oplus \mathcal{O}_{\mathbb{A}^2} \stackrel{(x,y)} \to \mathcal{O}_{\mathbb{A}^2} \to 0, $$ and it is no longer exact because now $x$ and $y$ have common zero at the origin.