## II.8: Differentials


### II.8.1  #to-work

Here we will strengthen the results of the text to include information about the sheaf of differentials at a not necessarily closed point of a scheme $X$.


a. Generalize (8.7) as follows. Let $B$ be a local ring containing a field $k$, and assume that the residue field $k(B)=B / \mathfrak{m}$ of $B$ is a separably generated extension of $k$. Then the exact sequence of (8.4A),
\[
0 \rightarrow \mathrm{m} / \mathrm{m}^2 \stackrel{\delta}{\rightarrow} \Omega_{B / k} \otimes k(B) \rightarrow \Omega_{k(B) / k} \rightarrow 0
\]
is exact on the left also.[^hint.2.8.1]



b. Generalize (8.8) as follows. With $B, k$ as above, assume furthermore that $k$ is perfect, and that $B$ is a localization of an algebra of finite type over $k$. Then show that $B$ is a regular local ring if and only if $\Omega_{B / k}$ is free of rank $=\operatorname{dim} B+$ tr.d. $k(B) / k$.

c. Strengthen (8.15) as follows. Let $X$ be an irreducible scheme of finite type over a perfect field $k$, and let $\operatorname{dim} X=n$. For any point $x \in X$, not necessarily closed, show that the local ring $\mathcal{O}_{x, X}$ is a regular local ring if and only if the stalk $\left(\Omega_{X / k}\right)_x$ of the sheaf of differentials at $x$ is free of rank $n$.

d. Strengthen (8.16) as follows. If $X$ is a variety over an algebraically closed field $k$, then $U=\left\{x \in X \mid \mathcal{O}_x\right.$ is a regular local ring $\}$ is an open dense subset of $X$.

[^hint.2.8.1]: 
Hint: In copying the proof of (8.7), first pass to $B / \mathrm{m}^2$, which is a complete local ring, and then use (8.25A) to choose a field of representatives for $B / \mathrm{m}^2$.

### II.8.2. #to-work

Let $X$ be a variety of dimension $n$ over $k$. Let $\mathcal{E}$ be a locally free sheaf of rank $>n$ on $X$, and let $V \subseteq \Gamma(X, \mathcal{E})$ be a vector space of global sections which generate $\mathcal{E}$. Then show that there is an element $s \in V$, such that for each $x \in X$, we have $s_x \notin \mathfrak{m}_x \mathcal{E}_x$. Conclude that there is a morphism $\mathcal{O}_X \rightarrow \mathcal{E}$ giving rise to an exact sequence
\[
0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime} \rightarrow 0
\]
where $\mathcal{E}'$ is also locally free.[^hint.2.8.2]

[^hint.2.8.2]: 
Hint: Use a method similar to the proof of Bertini's theorem (8.18).]

### II.8.3. Product Schemes. #to-work


a. Let $X$ and $Y$ be schemes over another scheme $S$. Use (8.10) and (8.11) to show that 
\[
\Omega_{X \fiberprod{S} Y/S}  \cong p_1^* \Omega_{X / S} \oplus p_2^* \Omega_{Y / S}
.\]

b. If $X$ and $Y$ are nonsingular varieties over a field $k$, show that 
\[
\omega_{X \times Y} \cong p_1^* \omega_X \otimes
p_2^* \omega_Y
.\]

c. Let $Y$ be a nonsingular plane cubic curve, and let $X$ be the surface $Y \times Y$. Show that $p_g(X)=1$ but $p_a(X)=-1$ (I, Ex. 7.2). This shows that the arithmetic genus and the geometric genus of a nonsingular projective variety may be different.

### II.8.4. Complete Intersections in $\mathbf{P}^n$. #to-work

A closed subscheme $Y$ of $\mathbf{P}_k^n$ is called a **(strict, global) complete intersection** if the homogeneous ideal $I$ of $Y$ in $S=k\left[x_0, \ldots, x_n\right]$ can be generated by $r=\operatorname{codim}\left(Y, \mathbf{P}^n\right)$ elements (I, Ex. 2.17).

a. Let $Y$ be a closed subscheme of codimension $r$ in $\mathbf{P}^n$. Then $Y$ is a complete intersection if and only if there are hypersurfaces (i.e., locally principal subschemes of codimension 1) $H_1, \ldots, H_r$, such that $Y=H_1 \cap \ldots \cap H_r$ as schemes, i.e., $\mathcal{I}_Y=\mathcal{I}_{H_1}+\ldots+\mathcal{I}_{H_r}$.[^hint.2.8.4.a]

b. If $Y$ is a complete intersection of dimension $\geqslant 1$ in $\mathbf{P}^n$, and if $Y$ is normal, then $Y$ is projectively normal (Ex. 5.14).[^hint.2.8.4.b]

c. With the same hypotheses as (b), conclude that for all $l \geqslant 0$, the natural map $\Gamma\left(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_Y(l)\right)$ is surjective. In particular, taking $l=0$, show that $Y$ is connected.

d. Now suppose given integers $d_1, \ldots, d_r \geqslant 1$, with $r<n$. Use Bertini's theorem (8.18) to show that there exist nonsingular hypersurfaces $H_1, \ldots, H_r$ in $\mathbf{P}^n$, with deg $H_i=d_i$, such that the scheme $Y=H_1 \cap \ldots \cap H_r$ is irreducible and nonsingular of codimension $r$ in $\mathbf{P}^n$.

e. If $Y$ is a nonsingular complete intersection as in (d), show that 
\[
\omega_Y \cong
\mathcal{O}_Y\left(\sum d_i-n-1\right)
.\]

f. If $Y$ is a nonsingular hypersurface of degree $d$ in $\mathbf{P}^n$, use (c) and (e) above to show that $p_g(Y)={d-1\choose n}$. 
Thus $p_g(Y)=p_a(Y)$ (I, Ex. 7.2). In particular, if $Y$ is a nonsingular plane curve of degree $d$, then 
\[
p_g(Y)=\frac{1}{2}(d-1)(d-2)
.\]

g. If $Y$ is a nonsingular curve in $\mathbf{P}^3$, which is a complete intersection of nonsingular surfaces of degrees $d, e$, then 
\[
p_g(Y)=\frac{1}{2} d e(d+e-4)+1
.\]
Again the geometric genus is the same as the arithmetic genus (I, Ex. 7.2).

[^hint.2.8.4.a]: 
Hint: Use the fact that the uniqueness theorem holds in $S$ (Matsumura $[2, p. 107]$).

[^hint.2.8.4.b]: 
Hint: Apply (8.23) to the affine cone over $Y$.

### II.8.5. Blowing up a Nonsingular Subvariety. #to-work

As in (8.24), let $X$ be a nonsingular variety, let $Y$ be a nonsingular subvariety of codimension $r \geqslant 2$, let $\pi: \widetilde{X} \rightarrow X$ be the blowing-up of $X$ along $Y$, and let $Y^{\prime}=\pi^{-1}(Y)$.

a. Show that the maps $\pi^*:\Pic X \rightarrow\Pic \tilde{X}$, and $\mathbf{Z} \rightarrow \Pic X$ defined by $n \mapsto$ class of $n Y^{\prime}$, give rise to an isomorphism $\Pic \tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}$.

b. Show that[^hint.2.8.5.b]
\[
\omega_{\tilde{x}} \cong f^* \omega_X \otimes \mathcal{L}\left((r-1) Y^{\prime}\right)
.\]

[^hint.2.8.5.b]: Hint: By (a) we can write in any case 
\[
\omega_{\tilde{X}} \cong f^* \mathcal{M} \otimes \mathcal{L}\left(q Y^{\prime}\right)
\]
for some invertible sheaf $\mathcal{M}$ on $X$, and some integer q. By restricting to $\tilde{X}-Y^{\prime} \cong X-Y$, show that $\mathcal{M} \cong \omega_X$. To determine $q$, proceed as follows. 

    - First show that $\omega_{Y^{\prime}} \cong f^* \omega_X \otimes \mathcal{O}_{Y^{\prime}}(-q-1)$. 
    - Then take a closed point $y \in Y$ and let $Z$ be the fibre of $Y^{\prime}$ over $y$. 
    - Then show that $\omega_Z \cong$ $\mathcal{O}_Z(-q-1)$. But since $Z \cong \mathbf{P}^{r-1}$, we have $\omega_Z \cong \mathcal{O}_Z(-r)$, so $q=r-1$.


### II.8.6. The Infinitesimal Lifting Property. #to-work

The following result is very important in studying deformations of nonsingular varieties. Let $k$ be an algebraically closed field, let $A$ be a finitely generated $k$-algebra such that $\operatorname{Spec} A$ is a nonsingular variety over $k$. Let 
\[
0 \rightarrow I \rightarrow B^{\prime} \rightarrow B \rightarrow 0
\]
be an exact sequence, where $B^{\prime}$ is a $k$-algebra, and $I$ is an ideal with $I^2=0$. Finally suppose given a $k$-algebra homomorphism $f: A \rightarrow B$. Then there exists a $k$-algebra homomorphism $g: A \rightarrow B^{\prime}$ making a commutative diagram

\begin{tikzcd}
	&& {} \\
	\\
	&& I \\
	&& {B'} \\
	A && B
	\arrow["f", from=5-1, to=5-3]
	\arrow["g", dashed, from=5-1, to=4-3]
	\arrow[hook, from=3-3, to=4-3]
	\arrow[two heads, from=4-3, to=5-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCw0LCJBIl0sWzIsNCwiQiJdLFsyLDIsIkkiXSxbMiwwXSxbMiwzLCJCJyJdLFswLDEsImYiXSxbMCw0LCJnIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNCwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XV0=)

We call this result the **infinitesimal lifting property for $A$**. We prove this result in several steps.

a. First suppose that $g: A \rightarrow B^{\prime}$ is a given homomorphism lifting $f$. If $g^{\prime}: A \rightarrow B^{\prime}$ is another such homomorphism, show that $\theta=g-g^{\prime}$ is a $k$-derivation of $A$ into $I$, which we can consider as an element of $\operatorname{Hom}_A\left(\Omega_{A / k}, I\right)$. Note that since $I^2=0, I$ has a natural structure of $B$-module and hence also of $A$-module. Conversely, for any $\theta \in \operatorname{Hom}_A\left(\Omega_{A / k}, I\right), g^{\prime}=g+\theta$ is another homomorphism lifting $f$. (For this step, you do not need the hypothesis about Spec $A$ being nonsingular.)

b. Now let $P=k\left[x_1, \ldots, x_n\right]$ be a polynomial ring over $k$ of which $A$ is a quotient, and let $J$ be the kernel. Show that there does exist a homomorphism $h: P \rightarrow B^{\prime}$ making a commutative diagram,
\begin{tikzcd}
	J && I \\
	P && B \\
	A && {B'}
	\arrow["f"', from=3-1, to=3-3]
	\arrow["h"', from=2-1, to=2-3]
	\arrow[hook, from=1-1, to=2-1]
	\arrow[hook, from=1-3, to=2-3]
	\arrow[two heads, from=2-1, to=3-1]
	\arrow[two heads, from=2-3, to=3-3]
\end{tikzcd}
> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJKIl0sWzAsMSwiUCJdLFswLDIsIkEiXSxbMiwwLCJJIl0sWzIsMSwiQiJdLFsyLDIsIkInIl0sWzIsNSwiZiIsMl0sWzEsNCwiaCIsMl0sWzAsMSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMyw0LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsxLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFs0LDUsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==)

    and show that $h$ induces an $A$-linear map $\bar h: J / J^2 \rightarrow I$.

c. Now use the hypothesis Spec $A$ nonsingular and (8.17) to obtain an exact sequence
\[
0 \rightarrow J / J^2 \rightarrow \Omega_{P / k} \otimes A \rightarrow \Omega_{A / k} \rightarrow 0 .
\]
Show furthermore that applying the functor $\operatorname{Hom}_A(\cdot, I)$ gives an exact sequence
\[
0 \rightarrow \operatorname{Hom}_A\left(\Omega_{A / k}, I\right) \rightarrow \operatorname{Hom}_P\left(\Omega_{P / k}, I\right) \rightarrow \operatorname{Hom}_A\left(J / J^2, I\right) \rightarrow 0 .
\]
Let $\theta \in \operatorname{Hom}_P\left(\Omega_{P / k}, I\right)$ be an element whose image gives $\bar h\in \operatorname{Hom}_A\left(J / J^2, I\right)$. Consider $\theta$ as a derivation of $P$ to $B^{\prime}$. Then let $h^{\prime}=h-\theta$, and show that $h^{\prime}$ is a homomorphism of $P \rightarrow B^{\prime}$ such that $h^{\prime}(J)=0$. Thus $h^{\prime}$ induces the desired homomorphism $g: A \rightarrow B^{\prime}$.

### II.8.7. #to-work

As an application of the infinitesimal lifting property, we consider the following general problem. Let $X$ be a scheme of finite type over $k$, and let $\mathcal{F}$ be a coherent sheaf on $X$. We seek to classify schemes $X^{\prime}$ over $k$, which have a sheaf of ideals $\mathcal{I}$ such that $\mathcal{I}^2=0$ and $\left(X^{\prime}, \mathcal{O}_X / \mathcal{I}\right) \cong\left(X, \mathcal{O}_X\right)$, and such that $\mathcal{I}$ with its resulting structure of $\mathcal{O}_X$-module is isomorphic to the given sheaf $\mathcal{F}$. Such a pair $X^{\prime}, \mathcal{I}$ we call an **infinitesimal extension of the scheme $X$ by the sheaf $\mathcal{F}$**. 

One such extension, the trivial one, is obtained as follows. Take $\mathcal{O}_{X^{\prime}}=\mathcal{O}_X \oplus \mathcal{F}$ as sheaves of abelian groups, and define multiplication by 
\[
(a \oplus f) \cdot\left(a^{\prime} \oplus f^{\prime}\right)=a a^{\prime} \oplus \left(a f^{\prime}+a^{\prime} f\right)
.\]
Then the topological space $X$ with the sheaf of rings $\mathcal{O}_{X^{\prime}}$ is an infinitesimal extension of $X$ by $\mathcal{F}$.

The general problem of classifying extensions of $X$ by $\mathcal{F}$ can be quite complicated. So for now, just prove the following special case: if $X$ is affine and nonsingular, then any extension of $X$ by a coherent sheaf $\mathcal{F}$ is isomorphic to the trivial one. See (III, Ex. 4.10) for another case.

### II.8.8. Plurigenus and (some) Hodge numbers are birational invariants. #to-work

Let $X$ be a projective nonsingular variety over $k$. For any $n>0$ we define the **$n$th plurigenus of $X$** to be 
\[
P_n=\operatorname{dim}_k \Gamma\left(X, \omega_X^{\otimes n}\right)
.\]
Thus in particular $P_1=p_g$. Also, for any $q, 0 \leqslant q \leqslant \operatorname{dim} X$ we define an integer 
\[
h^{q, 0}=\operatorname{dim}_k \Gamma\left(X, \Omega_{X / k}^q\right)
\quad\text{where}\quad
\Omega_{X / k}^q=\bigwedge^q \Omega_{X / k}
.\]
is the sheaf of regular $q$-forms on $X$. In particular, for $q=\operatorname{dim} X$, we recover the geometric genus again. The integers $h^{q, 0}$ are called **Hodge numbers**.

Using the method of $(8.19)$, show that $P_n$ and $h^{q, 0}$ are birational invariants of $X$, i.e., if $X$ and $X^{\prime}$ are birationally equivalent nonsingular projective varieties, then $P_n(X)=P_n\left(X^{\prime}\right)$ and $h^{q, 0}(X)=h^{q, 0}\left(X^{\prime}\right)$.