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Is there a unified explanation to the following phenomena?

1) $\mathbb{R} [X, Y] / (X^2 + Y^2 - 1)$ is not a UFD.

2) $\mathbb{C} [X, Y] / (X^2 + Y^2 - 1)$ is a UFD.

3) $\mathbb{R} [X, Y, Z] / (X^2 + Y^2 + Z^2 - 1)$ is a UFD.

4) $\mathbb{C} [X, Y, Z] / (X^2 + Y^2 + Z^2 - 1)$ is not a UFD.

I guess that

1) holds as we can regard $\mathbb{R} [X, Y] / (X^2 + Y^2 - 1)\cong \mathbb{R} [\sin \theta, \cos \theta]$ by setting $X = \sin \theta$ and $Y = \cos \theta$,

2) holds as we can regard $\mathbb{C} [X, Y] / (X^2 + Y^2 - 1)\cong \mathbb{C} [t, t^{-1}]$ by setting $t = X + iY$.

But my guess does not look promising to show that 3) and 4) hold.

user26857
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eltonjohn
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  • this will help http://math.stackexchange.com/questions/244460/ring-of-trigonometric-functions-with-real-coefficients . – Charlie Mar 12 '16 at 14:19
  • A Noetherian ring is a UFD iff it's ideal class group vanishes, and for normal things this is isomorphic to the Picard group of line bundles, so (1-2) says that all real alg l.b.'s become trivial when we tensor with $\mathbb{C}$, and (3-4) says there a nontrivial one over $\mathbb{C}$ that doesn't exist over $\mathbb{R}$. I'd be excited if someone: 1. answered this in terms of a base-change formula for one of these groups, or 2. proved (3) using some sort of topological argument. (Maybe it's connected to the hairy ball theorem, which I have seen used to prove algebraic facts about that ring.) – Xander Flood Mar 12 '16 at 16:59
  • http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/stablyfree.pdf The tangent bundle of S^2 is the standard example of a stably free module which is not free, and the standard proof uses the hairy ball theorem. I have no great reason to think this is related, but it's interesting. Also, affine complex varieties are Stein manifolds, so Catran's Theorem B guarantees that we have a UFD provided there are topological nontrivial line bundles. This means (2) and (4) have a purely topological argument, but I don't think that reasoning works over $\mathbb{R}$. – Xander Flood Mar 12 '16 at 17:06

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