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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. ¹

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.

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

• 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 \begin{align*} X=\operatorname{Spec} k[x, y, z, w] /(z, w) \cap(x+z, y+w) .\end{align*}

• 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 \(\{x \in X \mathrel{\Big|}f\) is flat at \(x\}\) is an open subset of \(X\) (possibly empty). ²

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 \(k\left[x_0, \ldots, x_{n+1}\right]\).

• 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 \(\mathbf{P}^{n+1}\).

• 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 \(d \geqslant 0\), \begin{align*} \operatorname{dim}_{k(t)}\left(S_t / I_t\right)_d \end{align*} is independent of \(t\). Here \((\quad)_d\) means the homogeneous part of degree \(d\).

• 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.

• 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 \(\mathbf{P}^{n-1}\). ³

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 \(Y^{\prime} \subseteq X \underset{\scriptscriptstyle {k} }{\times} D\), 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 \begin{align*} \mathcal{N}_{Y / X}= \mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_Y} \left(\mathcal{I}_Y / \mathcal{I}_Y^2, \mathcal{O}_Y\right) . \end{align*}

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: \begin{align*} 0 \rightarrow J \rightarrow P \rightarrow A \rightarrow 0 . \end{align*} Consider the exact sequence of (II, 8.4A) \begin{align*} J / J^2 \rightarrow \Omega_{P / k} \otimes_P A \rightarrow \Omega_{A / k} \rightarrow 0 . \end{align*} Apply the functor \(\operatorname{Hom}_A(\cdot, A)\), and let \(T^1(A)\) be the cokernel: \begin{align*} \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 . \end{align*} 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.

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

• 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 \(f^{-1}(U) \cong U \times \mathbf{P}^1\).

• * 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 ⁴

\begin{align*} 0 \leqslant p_a(Y) \leqslant \frac{1}{2}(d-1)(d-2) . \end{align*}