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The Stacks project

Exercise 111.60.6. Let $k$ be a field. Let $A = k[x_1, x_2, x_3, \ldots ]$ be the polynomial ring in infinitely many variables. Denote $\mathfrak m$ the maximal ideal of $A$ generated by all the variables. Let $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus \{ \mathfrak m\} $.

  1. Show $H^1(U, \mathcal{O}_ U) = 0$. Hint: Čech cohomology computation.

  2. What is your guess for $H^ i(U, \mathcal{O}_ U)$ for $i \geq 1$?

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View Exercise 111.60.6 as pdf