TITLE: Question on a theorem of Eisenbud's and Harris' "The geometry of schemes"
QUESTION [6 upvotes]: My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\mathbb{N}$.

Given a closed subscheme $X$ of $\mathbb{P}^n$ with Hilbert Polynomial $P$. The tangent space $T_{[X]}$ to the Hilbert scheme $\mathcal{H}_{P}$ at the point corresponding to $X$ is the space of global sections of the normal sheaf $\mathcal{N}_{\mathbb{P}^n/X}$.

I don't know what "is" precisely means here. $T_{[X]}$ has a natural $K([X])$-vector space structure and I would like to have an isomorphism of $K([X])$-vector spaces but my problem begins much earlier in the definition of $\mathcal{H}om$-sheaves:
Let $J$ be the ideal sheaf of $X$. The presheaf
$$
U \mapsto \mathcal{H}om_{\mathcal{O}_X|_U}(J/J^2|_U,\mathcal{O}_X|_U)
$$
is already a sheaf of $\mathcal{O}_X$-modules (this does not depend on the special arguments here). This sheaf $\mathcal{N}_{\mathbb{P}^n/X}$ is called the normal sheaf of $X$ in $\mathbb{P}^n$. The global sections
$$
\mathcal{N}_{\mathbb{P}^n/X}(X)=\mathcal{H}om_{\mathcal{O}_X}(J/J^2|_X,\mathcal{O}_X)
$$
of this normal sheaf is an $\mathcal{O}_X(X)$-module. Is there an isomorphism of $\mathcal{N}_{\mathbb{P}^n/X}(X)$ to some $hom$ of rings or modules where I can work with? What is a possible $K([X])$-vector space structure on it?

REPLY [4 votes]: 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 
$U\cap X$ (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.