Is the affine line isomorphic to the punctured affine line?

This is the story of Johan Commelin and myself working through the first sections of Hartshorne almost 10 years ago (nothing creates a bond like reading Hartshorne together…). This post is about problem I.1.1(b), which is essentially the following:

Exercise. Let k be a field. Show that k[x] and k[x,x^{-1}] are not isomorphic.

In my next post, I will explain why I’m coming back to exactly this problem. There are many ways to solve it, for example:

Solution 1. The k-algebra k[x] represents the forgetful functor \mathbf{Alg}_k \to \mathbf{Set}, whereas k[x,x^{-1}] represents the unit group functor R \mapsto R^\times. These functors are not isomorphic, for example because the inclusion k \to k[x] induces an isomorphism on unit groups, but not on additive groups. \qed

A less fancy way to say the same thing is that all k-algebra maps k[x,x^{-1}] \to k[x] factor through k, while the same evidently does not hold for k-algebra maps k[x] \to k[x].

However, we didn’t like this because it only shows that k[x] and k[x,x^{-1}] are not isomorphic as k-algebras (rather than as rings). Literal as we were (because we’re undergraduates? Lenstra’s influence?), we thought that this does not answer the question. After finishing all unstarred problems from section I.1 and a few days of being unhappy about this particular problem, we finally came up with:

Solution 2. The set k[x]^\times \cup \{0\} is closed under addition, whereas k[x,x^{-1}]^\times \cup \{0\} is not. \qed

This shows more generally that k[x] and \ell[x,x^{-1}] are never isomorphic as rings for any fields k and \ell.

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