Let \(X\) be an affine variety over the algebraically closed ground field \(k\), and let \[ f\in\mco(X) \] be nonconstant. Show that every irreducible component \(Z\) of \(f^{-1}(0)\) has \[ \dim Z=\dim X-1. \]
Solution
Put \[ A=\mco(X). \] Since \(X\) is an affine variety, \(A\) is a finitely generated integral \(k\)-algebra.
1. If \(f^{-1}(0)\) is nonempty, then \(f\) is a nonzero nonunit of \(A\).
Because \(f\) is nonconstant, it is not the zero element of \(A\). Since \(A\) is a domain, \(f\) is therefore not a zero divisor.
If \(f\) were a unit, then it would vanish nowhere on \(X\), so \[ f^{-1}(0)=\varnothing. \] Thus whenever the zero fibre has an irreducible component, \(f\) is also a nonunit.
2. Let \(Z\) be an irreducible component of \(f^{-1}(0)\). Then there is aprime ideal \(\mathfrak p\subseteq A\), minimal over \((f)\), such that\[ Z=V(\mathfrak p). \]
The zero fibre is the closed subset \[ f^{-1}(0)=V(f)\subseteq X. \] The irreducible components of a closed subset of an affine variety correspond to the primes minimal over its defining ideal. Therefore each irreducible component \(Z\) is \(V(\mathfrak p)\) for a prime \(\mathfrak p\) minimal over \((f)\).
3. Every prime \(\mathfrak p\) occurring in step 2 has\[ \height\mathfrak p=1. \]
By step 1, \(f\) is neither a unit nor a zero divisor in the Noetherian ring \(A\). Krull’s principal ideal theorem Hauptidealsatz therefore says that every prime minimal over \((f)\) has height one. This applies to the prime \(\mathfrak p\) from step 2.
4. Every irreducible component \(Z\) of \(f^{-1}(0)\) has\[ \boxed{ \dim Z=\dim X-1. } \]
For the prime \(\mathfrak p\) corresponding to \(Z\), one has \[ \dim Z=\dim(A/\mathfrak p). \] The affine height-dimension formula P-AGH2320DIMENSION gives \[ \dim(A/\mathfrak p)+\height\mathfrak p=\dim A. \] By step 3, \[ \height\mathfrak p=1. \] Hence \[ \dim Z = \dim A-1 = \dim X-1. \]
5. Q.E.D.
If \(f^{-1}(0)=\varnothing\), the assertion is vacuous. Otherwise steps <1>2–<1>4 prove the stated dimension for every irreducible component.