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Let $Y$ be a scheme of finite type over an algebraically closed field $k$. Show that the function $\phi(y) = dim_k(m_y/{m_y}^2)$ is upper semicontinuous on the set of closed point of $Y$ (i.e. for any point $y$, there exists an open neighborhood $U$, such that for any $x \in U, \phi(y) \geq \phi(x)$ ).

I have two thoughts about this problem:

1) If $y$ is a smooth point, then using the property that singular set is closed, one can show semicontinuity. So the difficulty comes from the singular point.

2) I would like to using semicontinuity theorem of cohomology of fibers, but I don't know how to construct the corresponding coherent sheaf.

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up vote 3 down vote accepted

Since the question is local you may assume that $Y=Spec(A)\subset \mathbb A^N_k $, where $A=k[X_1,...,X_N]/(f_1,...,f_m)$.
In other words $Y$ is the fiber of $0$ for the morphism $f=(f_1,...,f_m):\mathbb A^N_k \to \mathbb A^m_k$ .
As in good old calculus we have a Jacobian matrix with value for each closed $x\in Y$ : $$J(x)= (Jac (f))(x) =(\frac{\partial f_i }{\partial x_j}(x)) \quad (i=1,...,m \;; j=1,...,N) $$
The number you are interested in is exactly the nullity of that matrix:
$$ \phi(x)=dim(T_x(Y))=dim_k( ker\:J(x))=m-rank (J(x)) $$
The conclusion follows : if $\phi(y)=d$, then some $(m-d)\times (m-d)$ minor of $J(y)$ is $\neq 0$.
It will remain $\neq0$ for all $x$ in a neighbourhood of $y$ and thus in that neighbourhood we will
have $rank (J(x))\geq m-d$ so that $\phi(x)=m-rank (J(x))\leq d$ as desired.

Note that neither (non)-singularity nor cohomology are evoked.

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As Georges shows, this is classical algebraic geometry. I wonder whether a more general statement is true: if $X$ is a locally noetherian scheme, then is the function $x \mapsto \dim_{k(x)} \mathfrak{m}_x / \mathfrak{m}_x^2$ upper-semicontinuous? – Andrea Apr 23 '12 at 18:25
@Andrea: consider the affine line over $\mathbb C$. – user18119 Apr 24 '12 at 1:12
@Georges Elencwajg Thank you so much for the proof!! – Li Zhan Apr 24 '12 at 2:10
@Andrea QiL is right, since $Spec(\mathbb{C}[x])$ is smooth, $\phi(p)=dim(\mathbb{C}[x]_p)=1\quad $ except $p=0$, at that case, $\phi(0)=0$. But thank you all the same for making the clarification for the result. – Li Zhan Apr 24 '12 at 2:14
I thank QiL for his usual clarity. My question was really stupid. – Andrea Apr 24 '12 at 16:08

I think you can apply semicontinuity to sheaf of differentials.

But I don't see why affine line over C is a counterexample to andreas's question.

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