If \(x\) is a point of a scheme \(X\), we define an étale neighborhood of \(x\) to be an étale morphism \(f: U \to X\), together with a point \(x' \in U\) such that \(f(x') = x\).
As an example of the use of étale neighborhoods, prove the following: if \(\mcf\) is a coherent sheaf on \(X\), and if every point of \(X\) has an étale neighborhood \(f: U \to X\) for which \(f^* \mcf\) is a free \(\mco_U\dash\)module, then \(\mcf\) is locally free on \(X\).
Solution
Fix a point \(x\in X\). Choose an étale neighborhood \[ f:U\longrightarrow X, \qquad x'\in U, \qquad f(x')=x, \] such that \(f^*\mcf\) is free.
Put \[ A=\OO_{X,x}, \qquad B=\OO_{U,x'}, \qquad M=\mcf_x. \]
1. The local homomorphism \(A\to B\) is faithfully flat and\[ \mathfrak m_A B=\mathfrak m_B. \]
An étale morphism is flat and unramified by Exercise III.10.3. Hence \(A\to B\) is flat and Hartshorne’s unramified condition gives \[ \mathfrak m_A B=\mathfrak m_B. \]
A flat local homomorphism of local rings is faithfully flat. Indeed, if \(N\) is a nonzero \(A\)-module and \(0\ne n\in N\), then the cyclic submodule \(An\) is isomorphic to \(A/I\) for the proper ideal \(I=\operatorname{Ann}(n)\). Flatness preserves the injection \(An\hookrightarrow N\), while \[ (A/I)\tensor_A B=B/IB\ne0 \] because \(I\subseteq\mathfrak m_A\) and hence \(IB\subseteq\mathfrak m_B\). Thus \(N\tensor_A B\ne0\), so tensoring with \(B\) detects nonzero modules.
2. The stalk \(M=\mcf_x\) is a free \(A\)-module.
Since pullback commutes with stalks, \[ (f^*\mcf)_{x'}\cong M\tensor_A B. \] By hypothesis this is a free \(B\)-module, say of rank \(r\).
Let \[ k=A/\mathfrak m_A, \qquad k'=B/\mathfrak m_B. \] Using step 1, \[ (M/\mathfrak m_A M)\tensor_k k' \cong (M\tensor_A B)/\mathfrak m_B(M\tensor_A B) \cong (k')^{\oplus r}. \] Therefore \[ \dim_k M/\mathfrak m_A M=r. \]
Choose elements \(m_1,\ldots,m_r\in M\) whose classes form a \(k\)-basis of \(M/\mathfrak m_A M\). Because \(M\) is finite over the local ring \(A\), Nakayama’s lemma gives a surjection \[ \varphi:A^{\oplus r}\twoheadrightarrow M, \qquad e_i\longmapsto m_i. \]
Tensor with the flat \(A\)-algebra \(B\). We get a surjection \[ \varphi_B:B^{\oplus r}\twoheadrightarrow M\tensor_A B. \] Modulo \(\mathfrak m_B\), this is an isomorphism between \(r\)-dimensional \(k'\)-vector spaces. Since the target is free of rank \(r\), Nakayama’s lemma applied to a matrix for \(\varphi_B\) shows that \(\varphi_B\) is an isomorphism.
Let \(K=\ker\varphi\). Flatness of \(B/A\) gives \[ K\tensor_A B=\ker\varphi_B=0. \] Faithful flatness from step 1 therefore gives \(K=0\). Hence \[ \boxed{M\cong A^{\oplus r}}. \]
3. The sheaf \(\mcf\) is free on a Zariski neighborhood of \(x\).
2. After shrinkingto an open neighborhood \(V\) of \(x\), each \(m_i\) is represented by a section of\(\mcf|_V\). These sections define a morphism\[ \psi:\OO_V^{\oplus r}\longrightarrow\mcf|_V \]whose stalk at \(x\) is an isomorphism.
Because \(\mcf\) is coherent, the kernel and cokernel of \(\psi\) are coherent. Their stalks at \(x\) vanish, so after shrinking \(V\) once more both sheaves vanish identically. Thus \[ \psi:\OO_V^{\oplus r}\xrightarrow{\sim}\mcf|_V. \] Hence \(\mcf\) is free on a Zariski neighborhood of \(x\).
4. The sheaf \(\mcf\) is locally free on \(X\).
The point \(x\in X\) was arbitrary. Step 3 produces a Zariski neighborhood of every point on which \(\mcf\) is free. Therefore \(\mcf\) is locally free.
5. Q.E.D.
Steps <1>1–<1>2 descend freeness from an étale stalk, and steps <1>3–<1>4 spread the resulting free stalk to ordinary Zariski neighborhoods across all of \(X\).