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The Stacks project

Lemma 29.36.15. Let f : X \to S be a morphism of schemes. Let x \in X. Set s = f(x). Assume f is locally of finite presentation. The following are equivalent:

  1. The morphism f is étale at x.

  2. The local ring map \mathcal{O}_{S, s} \to \mathcal{O}_{X, x} is flat and X_ s \to \mathop{\mathrm{Spec}}(\kappa (s)) is étale at x.

  3. The local ring map \mathcal{O}_{S, s} \to \mathcal{O}_{X, x} is flat and X_ s \to \mathop{\mathrm{Spec}}(\kappa (s)) is unramified at x.

  4. The local ring map \mathcal{O}_{S, s} \to \mathcal{O}_{X, x} is flat and the \mathcal{O}_{X, x}-module \Omega _{X/S, x} is zero.

  5. The local ring map \mathcal{O}_{S, s} \to \mathcal{O}_{X, x} is flat and the \kappa (x)-vector space

    \Omega _{X_ s/s, x} \otimes _{\mathcal{O}_{X_ s, x}} \kappa (x) = \Omega _{X/S, x} \otimes _{\mathcal{O}_{X, x}} \kappa (x)

    is zero.

  6. The local ring map \mathcal{O}_{S, s} \to \mathcal{O}_{X, x} is flat, we have \mathfrak m_ s\mathcal{O}_{X, x} = \mathfrak m_ x and the field extension \kappa (x)/\kappa (s) is finite separable.

  7. There exist affine opens U \subset X, and V \subset S such that x \in U, f(U) \subset V and the induced morphism f|_ U : U \to V is standard smooth of relative dimension 0.

  8. There exist affine opens \mathop{\mathrm{Spec}}(A) = U \subset X and \mathop{\mathrm{Spec}}(R) = V \subset S with x \in U corresponding to \mathfrak q \subset A, and f(U) \subset V such that there exists a presentation

    A = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ n)

    with

    g = \det \left( \begin{matrix} \partial f_1/\partial x_1 & \partial f_2/\partial x_1 & \ldots & \partial f_ n/\partial x_1 \\ \partial f_1/\partial x_2 & \partial f_2/\partial x_2 & \ldots & \partial f_ n/\partial x_2 \\ \ldots & \ldots & \ldots & \ldots \\ \partial f_1/\partial x_ n & \partial f_2/\partial x_ n & \ldots & \partial f_ n/\partial x_ n \end{matrix} \right)

    mapping to an element of A not in \mathfrak q.

  9. There exist affine opens U \subset X, and V \subset S such that x \in U, f(U) \subset V and the induced morphism f|_ U : U \to V is standard étale.

  10. There exist affine opens \mathop{\mathrm{Spec}}(A) = U \subset X and \mathop{\mathrm{Spec}}(R) = V \subset S with x \in U corresponding to \mathfrak q \subset A, and f(U) \subset V such that there exists a presentation

    A = R[x]_ Q/(P) = R[x, 1/Q]/(P)

    with P, Q \in R[x], P monic and P' = \text{d}P/\text{d}x mapping to an element of A not in \mathfrak q.

Proof. Use Lemma 29.36.14 and the definitions to see that (1) implies all of the other conditions. For each of the conditions (2) – (10) combine Lemmas 29.34.14 and 29.35.14 to see that (1) holds by showing f is both smooth and unramified at x and applying Lemma 29.36.5. Some details omitted. \square


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