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 étaleif 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:
- \(f\) is étale;
- \(f\) is flat, and \(\Omega_{X / Y}=0\);
- \(f\) 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 \(\widehat{\mathcal{O}}_x\) and \(\widehat{\mathcal{O}}_y\) be the completions of the local rings at \(x\) and \(y\). Choose fields of representatives (II, 8.25A) \(k(x) \subseteq \widehat{\mathcal{O}}_x\) and \(k(y) \subseteq \widehat{\mathcal{O}}_y\) so that \(k(y) \subseteq k(x)\) via the natural map \(\widehat{\mathcal{O}}_y \rightarrow \widehat{\mathcal{O}}_x\).
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. \begin{align*} \widehat{\mathcal{O}}_y \otimes_{k(y)} k(x) \rightarrow \widehat{\mathcal{O}}_x \end{align*}
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).
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 \(P_1, \ldots, P_7\).
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\(D\) 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 \(\mathbf{P}^2\).
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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. 1