ac.commutative algebra - Question on a theorem of Eisenbud's and Harris' "The geometry of schemes"

Dear roger123,

This is largely a response to your question aksed as a comment below Charles Siegel's answer, but it won't fit in the comment box. Since $mathcal N_{mathbb P^n/X}$ is a sheaf of modules over the sheaf of rings $mathcal O_{mathbb P^n}$ (the structure sheaf of projective space),
its global sections $mathcal N_{mathbb P^n/X}(mathbb P^n)$ are a module over the ring $mathcal O_{mathbb P^n}(mathbb P^n)$, which in turn are just $k$ (the ground field).
In short, the global sections of $mathcal N_{mathbb P^n/X}$ form a $k$-vector space.

Maybe you are being confused by the fact that $mathcal N_{mathbb P^n/X}$ is a sheaf on $mathbb P^n$ that is supported on $X$, so that people often simultaneously regard it as a sheaf on either $X$ or $mathbb P^n$. This is okay, because if $U$ is any open in $mathbb P^n$ and $mathcal F$ is a sheaf supported on $X$, then the sections over an open subset $U$ of $mathbb P^n$
(when it is regarded as a sheaf on $mathbb P^n$) will coincide with the sections over
$Ucap X$ (when it is regarded as a sheaf on $X$).

In particular, one has the equation $mathcal N_{mathbb P^n/X}(X) =
mathcal N_{mathbb P^n/X}(mathbb P^n)$ (an abuse of notation if taken literally; however
one is supposed to regard $mathcal N_{mathbb P^n/X}$ as a sheaf on $X$ on the left-hand side,
and as a sheaf on $mathbb P^n$ on the right-hand side).

One more thing: If the residue field of the Hilbert scheme at the point $P$ is $k(P)$,
then $P$ is a $k(P)$-valued point of the Hilbert scheme,
and so the corresponding closed subscheme $X$ lies in $mathbb P^n_{k(P)}$. Thus, in the
above discussion, $k$ can (and should) be taken to be $k(P)$. Thus the above discussion explains
why $mathcal N_{mathbb P^n/X}(X)$ is a $k(P)$-vector space, as it should be.