Let's say I have $\mathcal{F}= \mathcal{O}(a)\oplus \mathcal{O}(b)$ for concreteness. Then what does a closed element of $\check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))$ look like as an element of $H^1(\mathbb{P}^1, \mathcal{O}(a)\oplus \mathcal{O}(b))$? So I know you would just take the quotient $$ \frac{ker: \check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b)) \to \check{C}^2(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))}{Im: \check{C}^0(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b)) \to \check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))} $$
but one problem I'm having is that I have been taking my cover $\mathscr{U}$ to be the standard one, $U_0$, $U_1$, which works fine for $\check{C}^0$ and $\check{C}^1$ but on $\check{C}^2$, you need to take local sections on the intersection of three patches. But I only have two. Do I just need to take a different open cover? If so, what should it be? Or is there a better work-around?
Let's say I have $\mathcal{F}= \mathcal{O}(a)\oplus \mathcal{O}(b)$ for concreteness. Then what does a closed element of $\check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))$ look like as an element of $H^1(\mathbb{P}^1, \mathcal{O}(a)\oplus \mathcal{O}(b))$? Theoretically, you take the quotient
$$ \frac{ker: \check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b)) \to \check{C}^2(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))}{Im: \check{C}^0(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b)) \to \check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))} $$
but one problem I'm having is that I have been taking my cover $\mathscr{U}$ to be the standard one, $U_0$, $U_1$, which works fine for $\check{C}^0$ and $\check{C}^1$ but on $\check{C}^2$, you need to take local sections on the intersection of three patches. But I only have two. Do I just need to take a different open cover? If so, what should it be? Or is there a better work-around?
Also, you know $H^1(\mathbb{P}^1, \mathcal{O}(a)\oplus \mathcal{O}(b)) \cong H^0(\mathbb{P}^1, \mathscr{O}(-a-2))^* \oplus H^0(\mathbb{P}^1, \mathscr{O}(-b-2))^*$ by Serre duality. This latter piece you can find a nice description of as a vector space so is there a convenient way to write your element of $\check{C}^1(\mathscr{U}, \mathcal{O}(a)\oplus \mathcal{O}(b))$ as an element of $H^0(\mathbb{P}^1, \mathscr{O}(-a-2))^* \oplus H^0(\mathbb{P}^1, \mathscr{O}(-b-2))^*$? Or if there is a better way to write the vector space $H^1((\mathbb{P}^1, \mathcal{O}(a)\oplus \mathcal{O}(b))$ that would work too. I just don't know how to do it without writing global sections.