Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 25I / Decomposition of the given vanishing locus / Solution

The algebraic set
$$
V(x^2+y^2-1,x^2-z^2-1)\subseteq\mathbb A_{\mathbb C}^3
$$
has the two irreducible components
$$
V(y-iz,x^2-z^2-1)
\quad\text{and}\quad
V(y+iz,x^2-z^2-1).
$$
Each is irreducible because the linear change $u=x-z$, $v=x+z$ identifies its <coordinate ring> with
$$
\mathbb C[u,v]/(uv-1)\cong\mathbb C[u,u^{-1}],
$$
an <integral domain>.

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 Incoming links: Irreducible components of two intersecting complex hyperbolas