I want to solve the following exercise.
Consider the projective line $X = \mathbb{P}^1_{\mathbb{F}_2} = \rm{Proj}_{\mathbb{F}_2}\mathbb{F}_2[t_0,t_1]$ over the field $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Show that $\eta = \frac{t_0}{t_0+t_1} + \frac{t_1^2}{t_0(t_0+t_1)}$ gives rise to a Cech - cocycle with respect to the open covering $\mathcal{U} = \{D_+(t_0),D_+(t_0+t_1)\}$. Relate the resulting invertible sheaf to Serre's twisted sheaves $\mathcal{O}_X(n)$, $n\in\mathbb{Z}$.
Write $A = \mathbb{F}_2[t_0,t_1]$. I am very confused here, since I don't understand the subject well eonough yet. As I see it $\eta$ is only defined on $D_+(t_0(t_0+t_1))$. I can show that $\eta = \frac{t_0+t_1}{t_0}\in A_{(t_0)}$ but I have no clue how $\eta$ is supposed to define a Cech - cocycle.
Thank you in advance for your help.