Hartshorne's Remark 7.8.2

Question

This remark presents another form of proposition 7.3, in terms of linear systems. What troubles me is the second condition of this version. It's formulated as follows.

(2) $\mathfrak{d}$ separates tangent vectors, i.e., given a closed point $P\in X$ and a tangent vector $t\in T_P(X)=(\mathfrak{m}_P/\mathfrak{m}^2_P)'$, there is a $D\in\mathfrak{d}$ such that $P\in\mathrm{Supp}\ D$, but $t\notin T_P(D)$. Here we think of $D$ as a locally principal closed subscheme, in which case the Zariski tangent space $T_P(D)=(\mathfrak{m}_{P,D}/\mathfrak{m}^2_{P,D})'$ is naturally a subspace of $T_P(X)$.

There are many points that I really can't understand. First, some notations in his words are strange for me, e.g. what the prime symbol $'$ means in the tangent space. I have searched through all the pages of this book but I still can't find the related definition about this notation. Then I'm also confused about the associated closed subscheme of $D$. How to describe the $T_P(D)$? What is the $\mathfrak{m}_{P,D}$? Does it, I guess, equal to the maximal ideal of the local ring $\mathscr{O}_{P,X}/\mathscr{I}_{P,D}$, where $\mathscr{I}_{P,D}$ is the ideal sheaf of $D$? And overall, how to prove the equivalence between this and the original form of (7.3) ? How to relate these tangent spaces to the $k$-vector space $\mathfrak{m}_P\mathscr{L}_P/\mathfrak{m}_P^2\mathscr{L}_P$ in (7.3) ?

I hope someone could help explain this remark in detail. And thanks in advance.

Answer 1

(7.3 $\implies$ 7.8.2 ) Suppose we are given a closed point $P \in X$ and a tangent vector $t \in T_P(X)$. Then, there exist $s \in V = \Gamma(X, \mathcal{L})$ by Proposition 7.7 such that $\phi(s_P) \in m_P$ and $t(\phi(\overline{s_P})) \neq 0$, where $\phi : \mathcal{L}_P \to \mathcal{O}_P$ is the obvious isomorphism. Then, let D = $(s)_0$. Then, as $\phi(\overline{s_P}) \not\in ker(t)$, we get that $t \not\in T_{P}(D)$. (Note that $m_{P,D} = m_P/(\phi(s_P))$. We have a surjective map $m_P/m_P^2 \to m_{P,D}/m_{P,D}^2$, which induces an injective map between the duals, $T_{P}(D) \to T_P(X)$. Note that all the maps in $T_P(D)$ have $\phi(\overline{s_P})$ in their kernel).

( 7.8.2 $\implies$ 7.3 ) We first fix a closed point $P \in X$. We know in general that for every subspace W of codimension 1 of $m_P/m_P^2$, there exist an element in the tangent space say $t$, such that $ker(t) = W$. By this obvious fact regarding any tangent space, and the above argument, we get that for every subspace W of $m_P/m_P^2$, there exist an element $s \in V$ such that $\phi(s_P) \in m_P$ and $\phi(\overline{s_P})\not\in W$. Hence, as $m_P/m_P^2$ is a finite dimesnional vector space, we get that $\{s \in V | \phi(s_P) \in m_P\}$ generate $m_P/m_P^2$.

Answer 2

Question:"I hope someone could help explain this remark in detail. And thanks in advance."

Answer: If $I \subseteq A$ is any ideal with $B:=A/I$ and $I \subseteq \mathfrak{p} \subseteq A$ is a prime ideal, you get an ideal $\mathfrak{q}:=\mathfrak{p}/I\mathfrak{p} \subseteq A/I$ and a diagram of maps

$$\require{AMScd} \begin{CD} A @>{p}>> B\\ @VVV @VVV \\ A_{\mathfrak{p}} @>{q}>> B_{\mathfrak{q}} \end{CD} .$$

Let $Y:=Spec(B) \subseteq X:=Spec(A)$. You get an induced surjective map of cotangent spaces

$$q^*: \mathfrak{m}/\mathfrak{m}^2\rightarrow \mathfrak{n}/\mathfrak{n}^2.$$

When you dualize you get an induced injective map of tangent spaces

$$ i: T_{\mathfrak{q}}(Y) \rightarrow T_{\mathfrak{p}}(X).$$

The map $i$ is injective since $q^*$ is surjective. Since tangent and cotangent spaces are defined in terms of local rings and their maximal ideals, it follows that for any closed subscheme $Y \subseteq X$ and any point $y\in Y$, you get an inclusion of tangent spaces

$$T_y(Y) \subseteq T_y(X)$$

Example: If $S:=Supp(D) \subseteq X$ is a closed subscheme and $s\in S$ it follows there is a canonical inclusion of $\kappa(s)$-vector spaces

$$T_s(S) \subseteq T_s(X).$$