Let \(Y\) be a scheme of finite type over an algebraically closed field \(k\). Show that the function \[ \varphi(y) = \dim_k (\mfm_y / \mfm_y^2) \] is upper semicontinuous on the set of closed points of \(Y\).
Solution
1. It is enough to prove the assertion on an affine neighbourhood\[ U=\Spec A \]of an arbitrary closed point \(y\in Y\).
Upper semicontinuity is local on the source. Since \(Y\) is of finite type over \(k\), every point has an affine neighbourhood of finite type over \(k\). Thus it suffices to prove the required upper semicontinuity on the closed points of each such affine open.
2. Write\[ A=k[x_1,\ldots,x_N]/I, \qquad I=(f_1,\ldots,f_r). \]For every closed point \(z\in U\), let\[ J(z)= \left( \frac{\partial f_i}{\partial x_j}(z) \right)_{\substack{1\le i\le r\\1\le j\le N}}. \]Then\[ \boxed{ \varphi(z)=N-\operatorname{rank}J(z). } \]
Because \(k\) is algebraically closed and \(A\) is of finite type over \(k\), the residue field of every closed point \(z\) is \(k\). Let \[ \mathfrak n_z=(x_1-a_1,\ldots,x_N-a_N) \subseteq k[x_1,\ldots,x_N] \] be the maximal ideal corresponding to \(z=(a_1,\ldots,a_N)\), and let \(\mfm_z\) be the maximal ideal of \(A_z\).
The conormal sequence for \[ k[x_1,\ldots,x_N]\longrightarrow A \] and base change to \(k(z)=k\) gives a right-exact sequence \[ I/I^2\otimes_A k \longrightarrow k^N \longrightarrow \Omega_{A/k}\otimes_A k \longrightarrow0. \] The class of \(f_i\) maps to \[ df_i(z) = \sum_{j=1}^N \frac{\partial f_i}{\partial x_j}(z)\,dx_j. \] Since the \(f_i\) generate \(I\), the image of the first arrow is the row space of \(J(z)\). Hence \[ \dim_k\bigl(\Omega_{A/k}\otimes_A k\bigr) = N-\operatorname{rank}J(z). \]
At a \(k\)-rational point, the cotangent-space identification gives \[ \mfm_z/\mfm_z^2 \cong \Omega_{A/k}\otimes_A k. \] Therefore \[ \varphi(z) = \dim_k(\mfm_z/\mfm_z^2) = N-\operatorname{rank}J(z), \] as claimed.
3. For every integer \(q\), the subset of closed points\[ \{z\in U:\varphi(z)\ge q\} \]is closed in the subspace of closed points of \(U\).
By step 2, \[ \varphi(z)\ge q \quad\Longleftrightarrow\quad \operatorname{rank}J(z)\le N-q. \] For \[ 0\le N-q<\min\{r,N\}, \] the latter condition says precisely that every \((N-q+1)\times(N-q+1)\) minor of the matrix \[ J= \left( \frac{\partial f_i}{\partial x_j} \right) \] vanishes at \(z\). These minors are regular functions on \(U\), so their common zero locus is closed. If \(N-q<0\), the set is empty; if \(N-q\ge\min\{r,N\}\), it is all of \(U\). Thus the assertion holds for every \(q\).
4. The function\[ \varphi(y)=\dim_k(\mfm_y/\mfm_y^2) \]is upper semicontinuous on the closed points of \(Y\).
1.Since the property is local, it holds on all of \(Y\).
5. Q.E.D.
Step 4 is the required upper semicontinuity.