Smooth morphisms and relative dimension

(Smooth)

$f : X \to Y$ is smooth if it is flat, locally of finite presentation, and has geometrically regular fibres. It is smooth of relative dimension $n$ if in addition every irreducible component of every fibre has dimension $n$.

(Criteria for smoothness of relative dimension $n$)

For a morphism $f \colon X \to Y$ the following are equivalent.

$f$ is smooth of relative dimension $n$.
$f$ is flat and locally of finite presentation, every fibre has pure dimension $n$, and $\Omega_{X/Y}$ is locally free of rank $n$.
$f$ is flat and locally of finite presentation, and every fibre $X_y$ is smooth of pure dimension $n$ over $\kappa(y)$.
$f$ is flat and locally of finite presentation, and every geometric fibre is a regular scheme of pure dimension $n$.
Locally on $X$ and $Y$, $f$ is induced by $B \to B[x_1, \ldots, x_{n+r}]/(g_1, \ldots, g_r)$ with $\det(\partial g_i / \partial x_j)_{1 \leq i, j \leq r}$ invertible.

The fibre-dimension hypothesis in condition 2 cannot be dropped. Over $k = \FF_p$, the morphism $\Spec k[x]/(x^p) \to \Spec k$ is flat and of finite presentation, and $\Omega = (k[x]/(x^p))\, dx$ is free of rank $1$ because $d(x^p) = p x^{p-1} dx = 0$; but the fibre has dimension $0$, not $1$, and it is not reduced, so the morphism is not smooth.

Over a non-perfect field a fibre can be regular and not geometrically regular: $\Spec k[x]/(x^p - t)\to\Spec k$ over $k = \FF_p(t)$ is flat and of finite presentation, its source is a regular point, and it becomes non-reduced after base change to $\kbar$; so this morphism is not smooth.

For a smooth morphism of relative dimension $n$, $\Omega_{X/Y}$ is the relative cotangent bundle and $\bigwedge^n\Omega_{X/Y}$ is the relative canonical sheaf. Smooth morphisms are stable under base change and composition. Smooth morphisms of relative dimension $0$ are étale. A smooth morphism of relative dimension $n$ locally factors as an étale map to $\AA^n_Y$ followed by the projection.

What is a smooth morphism?

What is a smooth morphism of relative dimension n?

What properties follow from smoothness?

Dependencies

D-MORFTLocally of finite type, finite type, and finite presentationuses
D-MORETALEÉtale morphismsuses

Backlinks

D-MORNORMREGNormal and regular morphismsrelated-to
D-ELLSCHElliptic curves over a schemeuses
T-MORGENGeneric smoothnessuses
T-MORSMREGSmooth over an algebraically closed field equals regularuses
FE-MORFROBThe Frobenius morphism is nowhere smoothuses
PR-MORJACThe jacobian criterion for smoothnessuses

What links to this

Smooth, unramified, and étale