I am in trouble with Exercise 8.4 in Hartshorne's Chapter II; I am doing part (a). It is about (global) complete intersection in $\textbf P^n$. For those without Hartshorne' book at hand, I describe the problem:
$Y$ is a closed subscheme of $\textbf P^n$ of codimension $r
Well, I know that the ring $S=k[x_0,\dots,x_n]$ is Cohen-Macauley, so we have the unmixedness theorem in $S$. That is, whenever an ideal $J\subset S$ of codimension $q$ can be generated by $q$ elements, it is unmixed: it shares its own height with all of its associated primes - the elements of $\textrm{Ass}_S(S/J)$. Thus what I can say (?) is
\begin{equation} r=\textrm{ht}\,(f_1,\dots,f_r)=\textrm{ht}\,I(Y) \end{equation}
and hence $L:=(f_1,\dots,f_r)$ is unmixed. So if $\mathfrak p$ is an associated (necessarily minimal) prime for $L$ then $\textrm{ht}\,\mathfrak p=r$. If I assume, by contradiction, that $I(Y)\neq L$, then what happens? I cannot find a contradiction.
Thank you so much!