## III.9: Flat Morphisms

### III.9.1.  #to-work

A flat morphism $f: X \rightarrow Y$ of finite type of noetherian schemes is open, i.e, for every open subset $U \subseteq X, f(U)$ is open in Y.[^hint.3.9.1]

[^hint.3.9.1]: 
Hint: Show that $f(U)$ is constructible and stable under generization (II, Ex. 3.18) and (II, Ex. 3.19).


### III.9.2.  #to-work

Do the calculation of (9.8.4) for the curve of (I, Ex. 3.14). Show that you get an embedded point at the cusp of the plane cubic curve.

### III.9.3.  #to-work

Some examples of flatness and nonflatness.

a. If $f: X \rightarrow Y$ is a finite surjective morphism of nonsingular varieties over an algebraically closed field $k$, then $f$ is flat.

b. Let $X$ be a union of two planes meeting at a point, each of which maps isomorphically to a plane $Y$. Show that $f$ is not flat. For example, let $Y=$ $\operatorname{Spec} k[x, y]$ and 
\[
X=\operatorname{Spec} k[x, y, z, w] /(z, w) \cap(x+z, y+w)
.\]

c. Again let $Y=\operatorname{Spec} k[x, y]$, but take $X=\operatorname{Spec} k[x, y, z, w] /\left(z^2, z w, w^2, x z-y w\right)$. Show that $X_{\text {red }} \cong Y, X$ has no embedded points, but that $f$ is not flat.

### III.9.4. Open Nature of Flatness. #to-work

Let $f: X \rightarrow Y$ be a morphism of finite type of noetherian schemes. Then $\{x \in X \mid f$ is flat at $x\}$ is an open subset of $X$ (possibly empty).[^note.3.9.4.groth]

[^note.3.9.4.groth]: 
See Grothendieck EGA $IV_3,11.1.1$.

### III.9.5.  Very Flat Families. #to-work

For any closed subscheme $X \subseteq \mathbf{P}^n$, we denote by $C(X) \subseteq \mathbf{P}^{n+1}$ the projective cone over $X$ (I, Ex. 2.10). If $I \subseteq k\left[x_0, \ldots, x_n\right]$ is the (largest) homogeneous ideal of $X$, then $C(X)$ is defined by the ideal generated by $I$ in $k\left[x_0, \ldots, x_{n+1}\right]$.

a. Give an example to show that if $\left\{X_t\right\}$ is a flat family of closed subschemes of $\mathbf{P}^n$, then $\left\{C\left(X_t\right)\right\}$ need not be a flat family in $\mathbf{P}^{n+1}$.

b. To remedy this situation, we make the following definition. Let $X \subseteq \mathbf{P}_T^n$ be a closed subscheme, where $T$ is a noetherian integral scheme. For each $t \in T$, let $I_t \subseteq S_t=k(t)\left[x_0, \ldots, x_n\right]$ be the homogeneous ideal of $X_t$ in $\mathbf{P}_{k(t)}^n$. We say that the family $\left\{X_t\right\}$ is **very flat** if for all $d \geqslant 0$,
\[
\operatorname{dim}_{k(t)}\left(S_t / I_t\right)_d
\]
is independent of $t$. Here $(\quad)_d$ means the homogeneous part of degree $d$.


c. If $\left\{X_t\right\}$ is a very flat family in $\mathbf{P}^n$, show that it is flat. Show also that $\left\{C\left(X_t\right)\right\}$ is a very flat family in $\mathbf{P}^{n+1}$, and hence flat.

d. If $\left\{X_{(t)}\right\}$ is an algebraic family of projectively normal varieties in $\mathbf{P}_k^n$, parametrized by a nonsingular curve $T$ over an algebraically closed field $k$, then $\left\{X_{(t)}\right\}$ is a very flat family of schemes.

### III.9.6.  #to-work

Let $Y \subseteq \mathbf{P}^n$ be a nonsingular variety of dimension $\geqslant 2$ over an algebraically closed field $k$. Suppose $\mathbf{P}^{n-1}$ is a hyperplane in $\mathbf{P}^n$ which does not contain $Y$, and such that the scheme $Y^{\prime}=Y \cap \mathbf{P}^{n-1}$ is also nonsingular. Prove that $Y$ is a complete intersection in $\mathbf{P}^n$ if and only if $Y^{\prime}$ is a complete intersection in $\mathbf{P}^{n-1}$.[^hint.3.9.6]

[^hint.3.9.6]: 
Hint: See (II, Ex. 8.4) and use (9.12) applied to the affine cones over $Y$ and $Y^{\prime}$.


### III.9.7.  #to-work

Let $Y \subseteq X$ be a closed subscheme, where $X$ is a scheme of finite type over a field $k$. Let $D=k[t] / t^2$ be the ring of dual numbers, and define an infinitesimal deformation of $Y$ as a closed subscheme of $X$, to be a closed subscheme $Y^{\prime} \subseteq X \fiberprod{k} D$, which is flat over $D$, and whose closed fibre is $Y$. 

Show that these $Y^{\prime}$ are classified by $H^0\left(Y, \mathcal{N}_{Y / X}\right)$, where
\[
\mathcal{N}_{Y / X}=
\sheafhom_{\OO_Y} \left(\mathcal{I}_Y / \mathcal{I}_Y^2, \mathcal{O}_Y\right) .
\]

### III.9.8.  \* #to-work

Let $A$ be a finitely generated $k$-algebra. Write $A$ as a quotient of a polynomial ring $P$ over $k$, and let $J$ be the kernel:
\[
0 \rightarrow J \rightarrow P \rightarrow A \rightarrow 0 .
\]
Consider the exact sequence of (II, 8.4A)
\[
J / J^2 \rightarrow \Omega_{P / k} \otimes_P A \rightarrow \Omega_{A / k} \rightarrow 0 .
\]
Apply the functor $\operatorname{Hom}_A(\cdot, A)$, and let $T^1(A)$ be the cokernel:
\[
\operatorname{Hom}_A\left(\Omega_{P / k} \otimes A, A\right) \rightarrow \operatorname{Hom}_A\left(J / J^2, A\right) \rightarrow T^1(A) \rightarrow 0 .
\]
Now use the construction of (II, Ex. 8.6) to show that $T^1(A)$ classifies infinitesimal deformations of $A$, i.e., algebras $A^{\prime}$ flat over $D=k[t] / t^2$, with $A^{\prime} \otimes_D k \cong A$. It follows that $T^1(A)$ is independent of the given representation of $A$ as a quotient of a polynomial ring $P$. 

### III.9.9.  #to-work

A $k$-algebra $A$ is said to be **rigid** if it has no infinitesimal deformations, or equivalently, by (Ex. 9.8) if $T^1(A)=0$. Let $A=k[x, y, z, w] /(x, y) \cap(z, w)$, and show that $A$ is rigid. This corresponds to two planes in $\mathbf{A}^4$ which meet at a point.

### III.9.10.  #to-work

A scheme $X_0$ over a field $k$ is rigid if it has no infinitesimal deformations.

a. Show that $\mathbf{P}_k^1$ is rigid, using (9.13.2).

b. One might think that if $X_0$ is rigid over $k$, then every global deformation of $X_0$ is locally trivial. Show that this is not so, by constructing a proper, flat morphism $f: X \rightarrow \mathbf{A}^2$ over $k$ algebraically closed, such that $X_0 \cong \mathbf{P}_k^1$, but there is no open neighborhood $U$ of 0 in $\mathbf{A}^2$ for which $f^{-1}(U) \cong U \times \mathbf{P}^1$.

c. \* Show, however, that one can trivialize a global deformation of $\mathbf{P}^1$ after a flat base extension, in the following sense: let $f: X \rightarrow T$ be a flat projective morphism, where $T$ is a nonsingular curve over $k$ algebraically closed. Assume there is a closed point $t \in T$ such that $X_t \cong \mathbf{P}_k^1$. Then there exists a nonsingular curve $T^{\prime}$, and a flat morphism $g: T^{\prime} \rightarrow T$, whose image contains $t$, such that if $X^{\prime}=X \times_T T^{\prime}$ is the base extension, then the new family $f^{\prime}: X^{\prime} \rightarrow T^{\prime}$ is isomorphic to $\mathbf{P}_{T^{\prime}}^1 \rightarrow T^{\prime}$.


### III.9.11.  #to-work

Let $Y$ be a nonsingular curve of degree $d$ in $\mathbf{P}_k^n$, over an algebraically closed field $k$. Show that[^hint.3.9.11]

\[
0 \leqslant p_a(Y) \leqslant \frac{1}{2}(d-1)(d-2) .
\]

[^hint.3.9.11]: 
Hint: Compare $Y$ to a suitable projection of $Y$ into $\mathbf{P}^2$, as in (9.8.3) and (9.8.4).