4
$\begingroup$

Let $k$ be a field, $A = k[x_1, x_2, \dots]$ and $\mathfrak{m} \subset A$ be the maximal ideal $(x_1, x_2, \dots)$. Defining $U = \text{Spec}\, A \setminus \mathfrak{m}$, how may one compute the groups $H^i(U, \mathcal{O}_U)$?

I was able to compute the Cech cohomology group for $i=1$ to be zero using the standard cover $D(x_i)$. This is then equal to the sheaf cohomology group because the sheaf is quasi-coherent and finite intersections of this cover is affine, the result applying for any higher cohomology groups as well (look at the Cech-to-derived functor spectral sequence). Is there an easy way to see what the groups should look like for $i > 1$?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.