Let $f:X \to Y$ be a projective morphism of complex Noetherian schemes. Assume $Y$ is smooth and for all $y \in Y$, $f^{-1}(y)$ is of pure dimension $1$. Let $\mathcal{F}\_1, \mathcal{F}\_2$ and $\mathcal{F}\_3$ be coherent sheaves on $X$ flat over $Y$ satisfying the following short exact sequence:


$$0 \to \mathcal{F}\_1 \to \mathcal{F}\_2 \to \mathcal{F}\_3 \to 0.$$


Assume $R^0f\_\*(\mathcal{F}\_1)$ and $R^1f\_\*(\mathcal{F}\_3)$ are zero. We then have the associated long exact sequence of right derived functor:


$$0 \to R^0f\_\*(\mathcal{F}\_2) \to R^0f\_\*(\mathcal{F}\_3) \to R^1f\_\*(\mathcal{F}\_1) \to R^1f\_\*(\mathcal{F}\_2) \to 0.$$ 


Assume $R^0f\_\*(\mathcal{F}\_3)$ and $R^1f\_\*(\mathcal{F}\_1)$ are locally free $\mathcal{O}\_Y$-modules. Then,


$1)$ Is it possible that one of $R^0f\_\*(\mathcal{F}\_2)$ or $R^1f\_\*(\mathcal{F}\_2)$ is locally free $\mathcal{O}\_Y$-module?


$2)$ For any $y \in Y$ closed, denote by $k(y)$ the constant sheaf $k(y)$ on the point $y$ (as mentioned in Hartshorne's Algebraic geometry Corollary, III.$9.4$).
Is it then true that $- \otimes k(y)$ (as mentioned in loc. cit. Theorem III.$12.11$) is exact when applied to the above long exact sequence. In particular, is the following sequence exact?


$$0 \to R^0f\_\*(\mathcal{F}\_2)\otimes k(y) \to R^0f\_\*(\mathcal{F}\_3)\otimes k(y) \to R^1f\_\*(\mathcal{F}\_1)\otimes k(y) \to R^1f\_\*(\mathcal{F}\_2)\otimes k(y)$$ 



A functorial property of higher right derived functors
Take the universal extension of $O$ by $O(-2)$ on $P^1$. By definition it is a vector bundle $E$ on $X = P^1\times A^1$ which is an extension
$$
0 \to p\_1^\*O(-2) \to E \to O \to 0.
$$
Let $Y = A^1$ and $f = p\_2$. Then after taking the pushforward you will get
$$
0 \to R^0f\_\*(E) \to O \stackrel{t}\to O \to R^1f\_\*(E) \to 0,
$$
where $t$ is the coordinate on $A^1$. This shows that $R^0f\_\*(E) = 0$ while $R^1f\_\*(E)$ is the structure sheaf of the point $0$. Of course, after tensoring with the structure sheaf of $0$ the sequence is not exact anymore.


