Let \(\mfa \subseteq A = k[x_1,\ldots,x_n]\) be an ideal generated by \(r\) elements. Show that every irreducible component of \(Z(\mfa)\) has dimension at least \(n - r\).
Solution
Induct on the generators. Put \(Y_k = V(f_1,\ldots,f_k)\), so that \(Y_k = Y_{k-1} \intersect H_k\) with \(H_k = V(f_k)\) a hypersurface, and \(Y_0 = \AA^n\).
If \(Y_{k-1} \not\subseteq H_k\), the previous exercise applies to each component and drops its dimension by exactly one, giving components of \(Y_k\) of dimension \(n - k\). If \(Y_{k-1} \subseteq H_k\) the intersection is all of \(Y_{k-1}\) and the dimension does not drop at all. Either way each component of \(Y_k\) has dimension at least \(n-k\), and at \(k = r\) this is the claim.
Source Collections
Dependencies
P-AGHHYPSECTIONA hypersurface section drops the dimension by exactly oneuses