Dimension of subsets of nonsigular variety Could  you help me to prove this question? Do you have any idea?
Let $X$ be a nonsingular  variety and $Y \subseteq X$ is  close and nonsingular . Then for any $x \in Y$   which $\dim Y_x=\dim X_x -n$ there is an affine open neighborhood $ U$, $ x\in U\subseteq X$ such that $Y\cap U=V(f_1,\ldots,f_n)$ where   $ (f_1,\ldots,f_n) $ is a regular sequence of $\mathcal{O}(U)  $.

Because $X$ and $Y$ are non-sigular, the conormal bundle to $Y$ in $X$
is a locally free sheaf on $Y$ of rank $n$.  Choose an affine n.h. $U = $ 
Spec $A$ of $x$ in $X$ such that over the intersection $U \cap Y$ the conormal
bundle becomes free.  If we let $I$ be the ideal in $A$ cutting out $Y$
(so $U \cap Y =$ Spec $A/I$), then the conormal bundle is the sheaf associated to the module $I/I^2$.   Thus $I/I^2$ is free of rank $n$ over $A/I$,
and so is generated by $n$ elements.  
A Nakayama-type argument will now show that $I$ is generated by $n$ elements,
say $f_1,\ldots,f_n$.  The fact that these form a regular sequence is related 
to the fact that they give a free basis for $I/I^2$.  
