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In his notes on K-theory, A. Okounkov states the following exercise:

The group $\operatorname{GL}(2)$ acts naturally on $\mathbb{P}^1$ and on line bundles $\mathcal{O}(k)$ over it. Push forward these line bundles under $\pi\colon\mathbb{P}^1 \to \text{ pt}$ using an explicit $T$-invariant ($T$ is supposed to be the maximal torus in $\operatorname{GL}(2)$) Čech covering of $\mathbb{P}^1$.

The pushforward is defined as:

$$\pi_*\mathcal{O}(k) = \sum_i (-1)^i [R^i\pi_*\mathcal{O}(k)].$$

If $\mathbb{P}^1$ is given by the coordinates $x_0,x_1$, the Čech covering $$U_1 = \{y_1 = x_1/x_0 \mid x_0\neq 0\}, U_2 = \{z_0 = x_0/x_1 \mid x_1\neq 0\}$$ is $T$-invariant.

By Theorem II.8.5 in Hartshorne,

$$R^i\pi_*\mathcal{O}(k) = H^i(\mathbb{P}^1,\mathcal{O}(k))^\sim.$$

How do I conclude and where does the action come into play?

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  • $\begingroup$ What do you want to conclude? $\endgroup$
    – Sasha
    Dec 12, 2022 at 12:03
  • $\begingroup$ What the push forward of these line bundles under $\pi\colon\mathbb{P}^1 \to \text{ pt}$ is (using an explicit $T$-invariant Čech covering of $\mathbb{P}^1$). $\endgroup$ Dec 12, 2022 at 12:58
  • $\begingroup$ The pushforward is determined by the two formulas in your question. The \v{C}ech covering is a tool that can be used to compute the cohomology, providing an alternative to the argument from Hartshorne. The action itself plays no role for this computation, but it is useful to observe that the result is also valid in the $\mathrm{GL}(2)$-equivariant category. $\endgroup$
    – Sasha
    Dec 12, 2022 at 13:21
  • $\begingroup$ Thanks. How would one compute $R^i\pi_*\mathcal{O}(k)$ using the covering? $\endgroup$ Dec 12, 2022 at 15:38
  • $\begingroup$ By means of Čech resolution, of course. $\endgroup$
    – Sasha
    Dec 12, 2022 at 17:04

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