I'm trying to compute $H^i_{\operatorname{fppf}}(\mathbb{P}^1(k),\alpha_p)$, where $\mathbb{P}^1(k)$ is a projective line over the field $k$ of characteristic $p>0$. I start with with following exact sequence in fppf topology.
\begin{equation} 0\to \alpha_p \to \mathbb G_a \stackrel{(-)^p}\to \mathbb G_a \to 0\tag{1} \end{equation}
which gives the long exact sequence \begin{equation} 0 \to \Gamma(\mathbb{P}^1(k),\alpha_p) \to \Gamma(\mathbb{P}^1(k),\mathbb G_a ) \to \Gamma(\mathbb{P}^1(k),\mathbb G_a ) \to H^1_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p) \to H^1_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) \to H^1_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) \to H^2_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p) \to H^2_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) \end{equation}
Now,$ H^i_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) = H^i_{\operatorname{zar}}(\mathbb{P}^1, \mathbb G_a)$. So, $ H^2_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) = 0$ and $ H^1_{\operatorname{fppf}}(\mathbb{P}^1, \mathbb G_a) = 0$.
For the above I'm using the following theorem
https://stacks.math.columbia.edu/tag/03P2
So, we get $H^2_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p) = H^1_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p) =H^0_{\operatorname{fppf}}(\mathbb{P}^1,\alpha_p) = 0 $.
Is the above computation correct?