The purpose of this note is to demonstrate for my Math 831 class that the coordinate ring of the real \(n\)-sphere of radius one is a UFD, for \(n\geq 2\). In other words, if \(n\geq 2\), then the ring
is a UFD. Recall from our lectures that when \(n = 1\), the coordinate ring of the real 1-sphere is not a UFD, though the coordinate ring of the complex 1-sphere is a UFD. The proof we give that the coordinate ring of the real \(n\)-sphere is a UFD is the one appearing in the celebrated paper of R. Swan, Vector bundles and projective modules. The proof below uses the facts that \(X_1^2+\cdots +X_{n}^2\) and \(X_1^2+\cdots + X_{n}^2-1\) are irreducible polynomials over \(\mathbb{R}\) when \(n\geq 2\), facts that can be proven (relatively) easily by induction, using techniques from the homework. Changing notation, we have:
Proposition. For \(n\geq 3\), set \(R := \mathbb{R}[X_1, X_2, \ldots, X_n]/(X_1^2+X_2^2+\cdots + X_n^2-1)\). Then \(R\) is a unique factorization domain. In particular, the coordinate ring of the real 2-sphere is a UFD.
Proof. Throughout, if \(T\) is an indeterminate in a polynomial ring \(S\) over a commutative ring, we use \(t\) to denote its image in any homomorphic image of \(S\). Let \(W_1, \ldots, W_n, T\) be indeterminates over \(\mathbb{R}\). We first note that \(B := \mathbb{R}[W_1T,\ldots, W_nT, T]\) is isomorphic to \(A := \mathbb{R}[X_1, \ldots, X_n, T]\). For this, it suffices to show that \(W_1T, \ldots, W_nT, T\) are algebraically independent over \(\mathbb{R}\), for then we can obtain an isomorphism \(\psi : A \to B\) by sending \(X_i\) to \(W_iT\) and \(T\) to \(T\). However, the quotient field \(K\) of \(B\) is just \(\mathbb{R}(W_1, \ldots, W_n, T)\), so the transcendence degree of \(K\) over \(\mathbb{R}\) equals \(n+1\). Since \(K\) is generated as a field over \(\mathbb{R}\) by \(n+1\) elements, namely \(W_1T, \ldots, W_nT, T\), those elements must be algebraically independent over \(\mathbb{R}\). Thus, \(A\) is isomorphic to \(B\), as rings. The isomorphism \(\psi\) extends to an isomorphism from \(A[T^{-1}]\) to \(B[T^{-1}]\) which we call \(\psi_0\).
Set \(F:= X_1^2+\cdots X_n^2-1 \in A\) and \(G := (W_1T)^2+\cdots + (W_nT)^2-1 \in B\). Then \(\psi_0\) takes the principal ideal \((F)\) in \(A[T^{-1}]\) to the principal ideal \((G)\) in \(B[T^{-1}]\). Therefore, \(A[T^{-1}]/(F)\) is isomorphic to \(B[T^{-1}]/(G)\). We claim \(B[T^{-1}]/(G)\) is a UFD. If so, then \(A[T^{-1}]/(F) = R[T, T^{-1}]\) is a UFD. Since \(T\) is a prime element in \(R[T]\), by Nagata's Lemma, \(R[T]\) is a UFD, and thus, \(R\) is a UFD.
To see that \(C := B[T^{-1}]/(G)\) is a UFD, we set
Note that that \(C = D[t^{-1}]\), since \(t\) is a unit in \(C\). Since \(D/tD = \mathbb{R}[W_1, \ldots, W_n]/(W_1^2+\cdots W_n^2)\), \(t\) is a prime element in \(D\). Thus, if \(D\) is a UFD, then \(C\) is a UFD. Finally, to show that \(D\) is a UFD, in \(\mathbb{R}[W_1,\ldots, W_n, T]\) we set \(U := T-W_n\) and \(V = T+W_n\). Then
Now, \(D/uD\) is isomorphic to
Since \(n-1 \geq 2\), \(W_1^2+\cdots +W_{n-1}^2\) is irreducible over \(\mathbb{R}\), so that \(D/uD\) is an integral domain, and thus, \(u\) is a prime element. Since \(D[u^{-1}] =\mathbb{R}[W_1, \ldots, W_n, U, U^{-1}]\), \(D[u^{-1}]\) is a UFD, and so by Nagata's Lemma, \(D\) is a UFD, which completes the proof.
Remark. When \(n = 3\), over \(\mathbb{C}\) we may write \(f := X_1^2+X_2^2+X_3^2-1 = (X_1+iX_2)(X_1-iX_2)+ (X_3-1)(X_3+1)\) so that in the ring \(\mathbb{C}[X_1,X_2,X_3]/(f)\), we can write the image of \((X_1+iX_2)(X_1-iX_2)\) as a product of irreducible elements in two different ways, showing that the coordinate ring of the complex 2-sphere is not a UFD. However, if \(n\geq 4\), the same proof as above will show that the coordinate ring of the complex \((n-1)\)-sphere is a UFD, since in the crucial step in the last paragraph of the proof, where
\(W_1^2+\cdots +W_{n-1}^2\) is irreducible over \(\mathbb{C}\) if \(n\geq 4\), but if \(n = 3\), then \(W_1^2+W_2^2 = (W_1+iW_2)(W_1-iW_2)\), so the proof breaks down in this case, as we have seen previously.