## III.10: Smooth Morphisms 

### III.10.1. Smooth $\neq$ Regular. #to-work

Over a nonperfect field, smooth and regular are not equivalent. For example, let $k_0$ be a field of characteristic $p>0$, let $k=k_0(t)$, and let $X \subseteq \mathbf{A}_k^2$ be the curve defined by $y^2=x^p-t$. Show that every local ring of $X$ is a regular local ring, but $X$ is not smooth over $k$.

### III.10.2.  #to-work

Let $f: X \rightarrow Y$ be a proper, flat morphism of varieties over $k$. Suppose for some point $y \in Y$ that the fibre $X_y$ is smooth over $k(y)$. Then show that there is an open neighborhood $U$ of $y$ in $Y$ such that $f: f^{-1}(U) \rightarrow U$ is smooth.

### III.10.3. Tale Morphisms. #to-work

A morphism $f: X \rightarrow Y$ of schemes of finite type over $k$ is **étale**if it is smooth of relative dimension 0 . It is **unramified** if for every $x \in X$, letting $y=f(x)$, we have $\mathrm{m}_y \cdot \mathcal{O}_x=\mathfrak{m}_x$, and $k(x)$ is a separable algebraic extension of $k(y)$. 

Show that the following conditions are equivalent:

(i) $f$ is étale;
(ii) $f$ is flat, and $\Omega_{X / Y}=0$;
(iii) $f$ is flat and unramified.

### III.10.4.  #to-work

Show that a morphism $f: X \rightarrow Y$ of schemes of finite type over $k$ is étale if and only if the following condition is satisfied: for each $x \in X$, let $y=f(x)$. Let $\hat{\mathcal{O}}_x$ and $\hat{\mathcal{O}}_y$ be the completions of the local rings at $x$ and $y$. Choose fields of representatives (II, 8.25A) $k(x) \subseteq \hat{\mathcal{O}}_x$ and $k(y) \subseteq \hat{\mathcal{O}}_y$ so that $k(y) \subseteq k(x)$ via the natural map $\hat{\mathcal{O}}_y \rightarrow \hat{\mathcal{O}}_x$. 

Then our condition is that for every $x \in X, k(x)$ is a separable algebraic extension of $k(y)$, and the natural map
is an isomorphism.
\[
\hat{\mathcal{O}}_y \otimes_{k(y)} k(x) \rightarrow \hat{\mathcal{O}}_x
\]

### III.10.5.  Étale Neighborhoods. #to-work

If $x$ is a point of a scheme $X$, we define an **étale neighborhood** of $x$ to be an étale morphism $f: U \rightarrow X$, together with a point $x^{\prime} \in U$ such that $f\left(x^{\prime}\right)=x$. 

As an example of the use of étale neighborhoods, prove the following: if $\mathcal{F}$ is a coherent sheaf on $X$, and if every point of $X$ has an étale neighborhood $f: U \rightarrow X$ for which $f^* \mathcal{F}$ is a free $\mathcal{O}_U$-module, then $\mathcal{F}$ is locally free on $X$.


### III.10.6.  #to-work

Let $Y$ be the plane nodal cubic curve $y^2=x^2(x+1)$. Show that $Y$ has a finite étale covering $X$ of degree 2, where $X$ is a union of two irreducible components, each one isomorphic to the normalization of $Y$ (Fig. 12).

![](Hartshorne_Problems/3_Hartshorne/figures/2022-10-23_00-23-42.png)


### III.10.7. (Serre). A linear system with moving singularities.  #to-work

Let $k$ be an algebraically closed field of characteristic 2. Let $P_1, \ldots, P_7 \in \mathbf{P}_k^2$ be the seven points of the projective plane over the prime field $\mathbf{F}_2 \subseteq k$. 
Let $D$ be the linear system of all cubic curves in $X$ passing through $P_1, \ldots, P_7$.

a. $D$ is a linear system of dimension 2 with base points $P_1, \ldots, P_7$, which determines an inseparable morphism of degree 2 from $X-\left\{P_i\right\}$ to $\mathbf{P}^2$.

b. Every curve $C \in D$ is singular. 

    More precisely, either $C$ consists of 3 lines all passing through one of the $P_i$, or $C$ is an irreducible cuspidal cubic with cusp $P \neq$ any $P_i$. 

    Furthermore, the correspondence $C \mapsto$ the singular point of $C$ is a $1-1$ correspondence between $D$ and $\mathbf{P}^2$. Thus the singular points of elements of $D$ move all over.

### III.10.8. A linear system with moving singularities contained in the base locus (any characteristic). #to-work

In affine 3 -space with coordinates $x, y, z$, let $C$ be the conic $(x-1)^2+$ $y^2=1$ in the $x y$-plane, and let $P$ be the point $(0,0, t)$ on the $z$-axis. Let $Y_t$ be the closure in $\mathbf{P}^3$ of the cone over $C$ with vertex $P$. 

Show that as $t$ varies, the surfaces $\left\{Y_t\right\}$ form a linear system of dimension 1, with a moving singularity at $P$. The base locus of this linear system is the conic $C$ plus the $z$-axis.

### III.10.9.  #to-work

Let $f: X \rightarrow Y$ be a morphism of varieties over $k$. Assume that $Y$ is regular, $X$ is Cohen-Macaulay, and that every fibre of $f$ has dimension equal to $\operatorname{dim} X-\operatorname{dim} Y$. Then $f$ is flat.[^hint.3.10.9]

[^hint.3.10.9]: 
Hint: Imitate the proof of (10.4), using (II, 8.21A).