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Exercise 110.40.6. Higher dimensions.

  1. Prove that every invertible sheaf on two dimensional affine space is trivial. More precisely, let ${\mathbf A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$ where $k$ is a field. Show that every invertible sheaf on ${\mathbf A}^2_ k$ is trivial. (Hint: One way to do this is to consider the corresponding module $M$, to look at $M \otimes _{k[x, y]} k(x)[y]$, and then use Exercise 110.40.5 (1) to find a generator for this; then you still have to think. Another way to is to use Exercise 110.40.3 and use what we know about ideals of the polynomial ring: primes of height one are generated by an irreducible polynomial; then you still have to think.)

  2. Prove that every invertible sheaf on any open subscheme of two dimensional affine space is trivial. More precisely, let $U \subset {\mathbf A}^2_ k$ be an open subscheme where $k$ is a field. Show that every invertible sheaf on $U$ is trivial. Hint: Show that every invertible sheaf on $U$ extends to one on ${\mathbf A}^2_ k$. Not easy; but you can find it in [H].

  3. Find an example of a nontrivial invertible sheaf on a punctured cone over a field. More precisely, let $k$ be a field and let $C = \mathop{\mathrm{Spec}}(k[x, y, z]/(xy-z^2))$. Let $U = C \setminus \{ (x, y, z) \} $. Find a nontrivial invertible sheaf on $U$. Hint: It may be easier to compute the group of isomorphism classes of invertible sheaves on $U$ than to just find one. Note that $U$ is covered by the opens $\mathop{\mathrm{Spec}}(k[x, y, z, 1/x]/(xy-z^2))$ and $\mathop{\mathrm{Spec}}(k[x, y, z, 1/y]/(xy-z^2))$ which are “easy” to deal with.


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