The Stacks project

Lemma 75.20.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $x \in |X|$. Assume that $Y$ is locally Noetherian and $f$ locally of finite type. The following are equivalent:

  1. $f$ is smooth at $x$,

  2. for every solid commutative diagram

    \[ \xymatrix{ X \ar[d]_ f & \mathop{\mathrm{Spec}}(B) \ar[d]^ i \ar[l]^-\alpha \\ Y & \mathop{\mathrm{Spec}}(B') \ar[l]_-{\beta } \ar@{-->}[lu] } \]

    where $B' \to B$ is a surjection of local rings with $\mathop{\mathrm{Ker}}(B' \to B)$ of square zero, and $\alpha $ mapping the closed point of $\mathop{\mathrm{Spec}}(B)$ to $x$ there exists a dotted arrow making the diagram commute, and

  3. same as in (2) but with $B' \to B$ ranging over small extensions (see Algebra, Definition 10.141.1).

Proof. Condition (1) means there is an open subspace $X' \subset X$ such that $X' \to Y$ is smooth. Hence (1) implies conditions (2) and (3) by Lemma 75.19.6. Condition (2) implies condition (3) trivially. Assume (3). Choose a commutative diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] \\ Y & V \ar[l] } \]

with $U$ and $V$ affine, horizontal arrows étale and such that there is a point $u \in U$ mapping to $x$. Next, consider a diagram

\[ \xymatrix{ X \ar[d] & U \ar[l] \ar[d] & \mathop{\mathrm{Spec}}(B) \ar[d]^ i \ar[l]^-\alpha \\ Y & V \ar[l] & \mathop{\mathrm{Spec}}(B') \ar[l]_-{\beta } } \]

as in (3) but for $u \in U \to V$. Let $\gamma : \mathop{\mathrm{Spec}}(B') \to X$ be the arrow we get from our assumption that (3) holds for $X$. Because $U \to X$ is étale and hence formally étale (Lemma 75.16.8) the morphism $\gamma $ has a unique lift to $U$ compatible with $\alpha $. Then because $V \to Y$ is étale hence formally étale this lift is compatible with $\beta $. Hence (3) holds for $u \in U \to V$ and we conclude that $U \to V$ is smooth at $u$ by More on Morphisms, Lemma 37.12.1. This proves that $X \to Y$ is smooth at $x$, thereby finishing the proof. $\square$


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