Let $X$ be a smooth projective variety of dimension $n\geq 3$. Let $Z$ be a smooth subvariety of $X$ of codimension at least 3. Let $Y$ be the blow up of $X$ along $Z$ and let $f:Y\longrightarrow X$ be the blow up map. Suppose we have an exact sequence of sheaves:

$0\longrightarrow F\longrightarrow G\longrightarrow H\rightarrow 0$ on $Y$.

Consider the pushforward of this exact sequence:

$0\longrightarrow f_*F\longrightarrow f_*G\longrightarrow f_*H$ on $X$. By definition of pushforward, it is left exact. Is it exact as well?

We just need to check the surjectivity of the last map at stalk level. On $X\setminus Z$ it is surjective. If we take a point in $z\in Z$, how can we check this?

Thank you.


No. Here is an explicit example. Let $X=\mathbf A^3$ and $Z=\{0\}$. The blow-up $Y$ of $X$ at $Z$ is isomorphic to the total space of the line bundle $\mathcal O(-1)$ over $\mathbf P^2$. Take a sheaf on $\mathbf P^2$ having a nonzero $H^1$, for example, the ideal sheaf $\mathcal I$ of the union of two distinct closed points $P$ and $Q$ on $\mathbf P^2$. The short exact sequence $$ 0\rightarrow \mathcal I\rightarrow \mathcal O\rightarrow i_\star k\oplus j_\star k \rightarrow 0, $$ where $i$ and $j$ are the inclusions of $P$ and $Q$ in $\mathbf P^2$ and $k$ is the base field, shows that $$ H^1(\mathbf P^2,\mathcal I)\cong k. $$

Let $p\colon Y\rightarrow \mathbf P^2$ be the projection. Since $p$ is flat, we have an induced short exact sequence $$ 0\rightarrow p^\star\mathcal I\rightarrow \mathcal O\rightarrow p^\star i_\star k\oplus p^\star j_\star k \rightarrow 0 $$ on $Y$. Note that $R^1f_\star\mathcal O=0$ and $R^1f_\star p^\star\mathcal I$ is the sheaf $k$ on $X$ concentrated in $Z=\{0\}$, by the computation above. Therefore, the morphism $$ f_\star\mathcal O\rightarrow f_\star (p^\star i_\star k\oplus p^\star j_\star k) $$ is not surjective.


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