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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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