A morphism \(f: X \to Y\) 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 \(\mfm_y \cdot \mco_x = \mfm_x\), 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.
Solution
Because \(X\) and \(Y\) are of finite type over \(k\), the morphism \(f\) is locally of finite presentation.
1. For a morphism locally of finite type,\[ f\text{ is unramified} \quad\Longleftrightarrow\quad \Omega_{X/Y}=0. \]
Fix \(x\in X\), put \(y=f(x)\), and write \[ A=\OO_{Y,y}, \qquad B=\OO_{X,x}, \qquad \kappa=\kappa(y), \qquad K=\kappa(x). \] The local ring of the fibre \(X_y\) at \(x\) is \[ R=(B/\mfm_yB)_{\mathfrak m_x/\mathfrak m_yB}, \] and formation of relative differentials commutes with base change: \[ \Omega_{R/\kappa} \cong (\Omega_{X/Y})_x\tensor_B R. \]
Suppose first that \(f\) is unramified at \(x\). Then \[ \mfm_yB=\mfm_x, \] so \(R=K\). The extension \(K/\kappa\) is finite separable by definition. Therefore \[ \Omega_{R/\kappa}=\Omega_{K/\kappa}=0. \] Nakayama’s lemma applied to the finite \(B\)-module \((\Omega_{X/Y})_x\) now gives \[ (\Omega_{X/Y})_x=0. \]
Conversely, suppose \((\Omega_{X/Y})_x=0\). Then \[ \Omega_{R/\kappa}=0. \] For a local algebra essentially of finite type over a field, vanishing of the module of differentials is the zero-dimensional unramified criterion: the local ring is its residue field and that residue field is finite separable over the base field. Thus \[ R=K, \qquad K/\kappa\text{ finite separable}. \] The equality \(R=K\) means that the maximal ideal of \(R\) is zero, i.e. \[ \mfm_x/\mfm_yB=0. \] Hence \[ \mfm_yB=\mfm_x, \] which is exactly Hartshorne’s local definition of unramifiedness at \(x\).
Since this holds at every point, \[ \boxed{f\text{ unramified}\iff\Omega_{X/Y}=0}. \] This is the differential characterization recorded in D-MORUNR.
2. Conditions (ii) and (iii) are equivalent.
Both conditions require \(f\) to be flat. By step 1, their remaining conditions \[ \Omega_{X/Y}=0 \qquad\text{and}\qquad f\text{ unramified} \] are equivalent. Hence \[ \boxed{\text{(ii)}\iff\text{(iii)}}. \]
3. Condition (i) implies condition (ii).
If \(f\) is étale in the sense of the problem, it is smooth of relative dimension zero. A smooth morphism is flat, and for a smooth morphism of relative dimension \(r\) the sheaf of relative differentials is locally free of rank \(r\) by the differential criterion for smoothness. Here \(r=0\), so \[ \Omega_{X/Y}=0. \] Thus (ii) holds.
4. If condition (iii) holds, then every fibre of \(f\) is geometrically regular of dimension zero.
Assume \(f\) is flat and unramified. Fix \(x\in X\) and put \(y=f(x)\). As in step 1, the local ring of the fibre at \(x\) is \[ \OO_{X_y,x} =(\OO_{X,x}/\mfm_y\OO_{X,x})_{\mathfrak m_x/\mathfrak m_y\OO_{X,x}}. \] Unramifiedness gives \[ \mfm_y\OO_{X,x}=\mfm_x, \] so \[ \OO_{X_y,x}=\kappa(x). \] Moreover \(\kappa(x)/\kappa(y)\) is finite separable.
After any field extension \(L/\kappa(y)\), the algebra \[ \kappa(x)\tensor_{\kappa(y)}L \] is finite étale over \(L\) after passage to the corresponding field factors; in particular all its local rings are fields. Thus the fibre remains regular after every field extension. Hence \(X_y\) is geometrically regular of pure dimension zero at \(x\). Since \(x\) was arbitrary, every fibre is geometrically regular of dimension zero.
5. Condition (iii) implies condition (i).
Under (iii), \(f\) is flat by hypothesis and locally of finite presentation because it is a morphism between schemes of finite type over \(k\). Step 4 shows that all fibres are geometrically regular of pure dimension zero. The fibre criterion for smoothness D-MORSM therefore says that \(f\) is smooth of relative dimension zero. By definition, \(f\) is étale. Thus \[ \boxed{\text{(iii)}\Longrightarrow\text{(i)}}. \]
6. The three conditions are equivalent.
Step 3 gives \[ \text{(i)}\Longrightarrow\text{(ii)}, \] step 2 gives \[ \text{(ii)}\Longleftrightarrow\text{(iii)}, \] and step 5 gives \[ \text{(iii)}\Longrightarrow\text{(i)}. \] Therefore \[ \boxed{\text{(i)}\iff\text{(ii)}\iff\text{(iii)}}. \]
7. Q.E.D.
Steps <1>1–<1>2 identify vanishing relative differentials with Hartshorne’s unramified condition, and steps <1>3–<1>5 identify flat unramified morphisms with smooth morphisms of relative dimension zero.