Let $F$, $F'$ and $F''$ be coherent sheaves on $\mathbb{P}^{3}$. Consider the following exact sequence $$0 \longrightarrow F' \longrightarrow F \longrightarrow F'' \longrightarrow 0 $$
Suppose that $F''$ and $F$ are respectively, $(m-1)$ and $m$ Castelnuovo-Mumford regular.
So $F'$ is m-regular? True or false?
My attempt: Taking the exact long sequence in cohomology and writing the part that interests us, we have $$H^{i-1}\bigl(F''(m-1 - (i-1)) \bigr) \longrightarrow H^{i}\bigl(F'(m-i) \bigr) \longrightarrow H^{i}\bigl(F(m-i) \bigr).$$
For $i \geq 2$, the extremal terms are zero, then $H^{i}\bigl(F'(m-i) \bigr) = 0$.
For $i = 1$, to complete the proof, I need to show that $H^{0}\bigl(F''(m-1 ) \bigr) = 0.$
By Serre's duality, we have $$H^{0}\bigl(F''(m-1 ) \bigr) \simeq H^{3}\bigl(F''^{\vee}(-m + 1) \otimes \omega_{\mathbb{P}^{3}}\bigr) \simeq H^{3}\bigl(F''^{\vee}(-m - 3)\bigr).$$
Now, by Hartshorne (Algebraic Geometry, III. Theorem 7.1 and Proposition 6.7), we have $$H^{3}\bigl(F''^{\vee}(-m - 3)\bigr) \simeq \text{Ext}^{0}(F''^{\vee}(-m - 3), \omega_{\mathbb{P}^{3}}\bigr) \simeq \text{Ext}^{0}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr).$$
Also $$\text{Ext}^{0}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr) = \text{Hom}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr) = \Gamma\bigl(\mathbb{P}^{3}, \text{Hom}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr).$$
If I didn't make any mistakes in the steps above, I have to show that $$\text{Hom}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr) = \Gamma\bigl(\mathbb{P}^{3}, \text{Hom}\bigl(F''^{\vee}, \mathcal{O}_{\mathbb{P}^{3}}(m-1)\bigr)\bigr) = 0. \tag{*}$$
Is that correct? How can we show $(*)$?
Thank you very much.
This assertion is not correct. First, some quick calculations verify that $\mathcal{O}$ is $0$-regular on $\Bbb P^n$ for any $n$, and $\mathcal{O}(d)$ is $(-d)$-regular. Consider the well-known (twist of the) Euler exact sequence on $\Bbb P^1_k$:
$$ 0 \to \mathcal{O}(-1)\to \mathcal{O}^{\oplus 2}\to \mathcal{O}(1) \to 0$$
The center sheaf is $0$-regular, the right sheaf is $(-1)$-regular, and the left sheaf is $1$-regular. Write $i:\Bbb P^1\to \Bbb P^3$ for the standard closed immersion of $\Bbb P^1=V(x_2,x_3)$ into $\Bbb P^3$. Then applying $i_*$ to the above exact sequence gives coherent sheaves on $\Bbb P^3$ with the same cohomology after any twist, which demonstrates the original assertion in your post is not correct.