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:
The morphism f is étale at x.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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