## 10.142 Overview of results on smooth ring maps

Here is a list of results on smooth ring maps that we proved in the preceding sections. For more precise statements and definitions please consult the references given.

A ring map $R \to S$ is smooth if it is of finite presentation and the naive cotangent complex of $S/R$ is quasi-isomorphic to a finite projective $S$-module in degree $0$, see Definition 10.137.1.

If $S$ is smooth over $R$, then $\Omega _{S/R}$ is a finite projective $S$-module, see discussion following Definition 10.137.1.

The property of being smooth is local on $S$, see Lemma 10.137.13.

The property of being smooth is stable under base change, see Lemma 10.137.4.

The property of being smooth is stable under composition, see Lemma 10.137.14.

A smooth ring map is syntomic, in particular flat, see Lemma 10.137.10.

A finitely presented, flat ring map with smooth fibre rings is smooth, see Lemma 10.137.17.

A finitely presented ring map $R \to S$ is smooth if and only if it is formally smooth, see Proposition 10.138.13.

If $R \to S$ is a finite type ring map with $R$ Noetherian then to check that $R \to S$ is smooth it suffices to check the lifting property of formal smoothness along small extensions of Artinian local rings, see Lemma 10.141.2.

A smooth ring map $R \to S$ is the base change of a smooth ring map $R_0 \to S_0$ with $R_0$ of finite type over $\mathbf{Z}$, see Lemma 10.138.14.

Formation of the set of points where a ring map is smooth commutes with flat base change, see Lemma 10.137.18.

If $S$ is of finite type over an algebraically closed field $k$, and $\mathfrak m \subset S$ a maximal ideal, then the following are equivalent

$S$ is smooth over $k$ in a neighbourhood of $\mathfrak m$,

$S_{\mathfrak m}$ is a regular local ring,

$\dim (S_{\mathfrak m}) = \dim _{\kappa (m)} \Omega _{S/k} \otimes _ S \kappa (\mathfrak m)$.

see Lemma 10.140.2.

If $S$ is of finite type over a field $k$, and $\mathfrak q \subset S$ a prime ideal, then the following are equivalent

$S$ is smooth over $k$ in a neighbourhood of $\mathfrak q$,

$\dim _{\mathfrak q}(S/k) = \dim _{\kappa (\mathfrak q)} \Omega _{S/k} \otimes _ S \kappa (\mathfrak q)$.

see Lemma 10.140.3.

If $S$ is smooth over a field, then all its local rings are regular, see Lemma 10.140.3.

If $S$ is of finite type over a field $k$, $\mathfrak q \subset S$ a prime ideal, the field extension $\kappa (\mathfrak q)/k$ is separable and $S_{\mathfrak q}$ is regular, then $S$ is smooth over $k$ at $\mathfrak q$, see Lemma 10.140.5.

If $S$ is of finite type over a field $k$, if $k$ has characteristic $0$, if $\mathfrak q \subset S$ a prime ideal, and if $\Omega _{S/k, \mathfrak q}$ is free, then $S$ is smooth over $k$ at $\mathfrak q$, see Lemma 10.140.7.

Some of these results were proved using the notion of a standard smooth ring map, see Definition 10.137.6. This is the analogue of what a relative global complete intersection map is for the case of syntomic morphisms. It is also the easiest way to make examples.

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