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 $n\in\mathbb{N}$.



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> 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 $\mathcal{N}\_{\mathbb{P}^n/X}$.
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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?



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 
$U\cap X$ (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.


