This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(January 2025) (Learn how and when to remove this message)

Inalgebraic geometry, amorphism${\displaystyle f:X\to S}$ betweenschemes is said to besmooth if

• (i) it islocally of finite presentation
• (ii) it isflat, and
• (iii) for everygeometric point${\displaystyle \overline{s}\to S}$ thefiber${\displaystyle X_{\overline{s}}=X{\times }_S\overline{s}}$ is regular.

(iii) means that each geometric fiber off is anonsingular variety (if it isseparated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.

IfS is thespectrum of analgebraically closedfield andf isof finite type, then one recovers the definition of a nonsingular variety.

A singular variety is called smoothable if it can be put in a flat family so that the nearby fibers are all smooth. Such a family is called a smoothning of the variety.

There are many equivalent definitions of a smooth morphism. Let${\displaystyle f:X\to S}$ be locally of finite presentation. Then the following are equivalent.

1. f is smooth.
2. f is formally smooth (see below).
3. f is flat and thesheaf of relative differentials${\displaystyle {\mathrm{\Omega }}_{X/S}}$ is locally free of rank equal to therelative dimension of${\displaystyle X/S}$.
4. For any${\displaystyle x\in X}$, there exists a neighborhood${\displaystyle \mathrm{Spec}B}$ of x and a neighborhood${\displaystyle \mathrm{Spec}A}$ of${\displaystyle f(x)}$ such that${\displaystyle B=A[t_1,\dots ,t_n]/(P_1,\dots ,P_m)}$ and the ideal generated by them-by-m minors of${\displaystyle (\mathrm{\partial }{P}_i/\mathrm{\partial }{t}_j)}$ isB.
5. Locally,f factors into${\displaystyle X\stackrel{g}{\to }{\mathbb{A}}_S^n\to S}$ whereg isétale.

A morphism of finite type is étale if and only if it is smooth andquasi-finite.

A smooth morphism is stable under base change and composition.

A smooth morphism is universallylocally acyclic.

Smooth morphisms are supposed to geometrically correspond to smoothsubmersions indifferential geometry; that is, they are smooth locally trivial fibrations over some base space (byEhresmann's theorem).

Let${\displaystyle f}$ be the morphism of schemes

${\displaystyle {\mathrm{Spec}}_{\mathbb{C}}\left(\frac{\mathbb{C}[x,y]}{(f=y^2-x^3-x-1)}\right)\to \mathrm{Spec}(\mathbb{C})}$

It is smooth because of the Jacobian condition: theJacobian matrix

${\displaystyle [3x^2-1,y]}$

vanishes at the points${\displaystyle (1/\sqrt{3},0),(-1/\sqrt{3},0)}$ which has an empty intersection with the polynomial, since

which are both non-zero.

Given asmooth scheme${\displaystyle Y}$ the projection morphism

${\displaystyle Y\times X\to X}$

is smooth.

Everyvector bundle${\displaystyle E\to X}$ over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of${\displaystyle \mathcal{O}(k)}$ over${\displaystyle {\mathbb{P}}^n}$ is the weighted projective space minus a point

${\displaystyle O(k)=\mathbb{P}(1,\dots ,1,k)-\{[0:\cdots :0:1]\}\to {\mathbb{P}}^n}$

sending

${\displaystyle [x_0:\cdots :x_n:x_{n+1}]\to [x_0:\cdots :x_n]}$

Notice that the direct sum bundles${\displaystyle O(k)\oplus O(l)}$ can be constructed using the fiber product

${\displaystyle O(k)\oplus O(l)=O(k){\times }_{X}O(l)}$

Recall that afield extension${\displaystyle K\to L}$ is calledseparable iff given a presentation

${\displaystyle L=\frac{K[x]}{(f(x))}}$

we have that${\displaystyle gcd(f(x),f^{\prime }(x))=1}$. We can reinterpret this definition in terms ofKähler differentials as follows: the field extension is separable iff

${\displaystyle {\mathrm{\Omega }}_{L/K}=0.}$

Notice that this includes everyperfect field:finite fields and fields ofcharacteristic 0.

If we consider${\displaystyle \mathrm{Spec}}$ of the underlying algebra${\displaystyle R}$ for a projective variety${\displaystyle X}$, called the affine cone of${\displaystyle X}$, then the point at the origin is always singular. For example, consider theaffine cone of a quintic${\displaystyle 3}$-fold given by

${\displaystyle x_0^5+x_1^5+x_2^5+x_3^5+x_4^5}$

Then the Jacobian matrix is given by

which vanishes at the origin, hence the cone is singular. Affine hypersurfaces like these are popular insingularity theory because of their relatively simple algebra but rich underlying structures.

Another example of a singular variety is theprojective cone of a smooth variety: given a smooth projective variety${\displaystyle X\subset {\mathbb{P}}^n}$ its projective cone is the union of all lines in${\displaystyle {\mathbb{P}}^{n+1}}$ intersecting${\displaystyle X}$. For example, the projective cone of the points

${\displaystyle \text{Proj}\left(\frac{\mathbb{C}[x,y]}{(x^4+y^4)}\right)}$

is the scheme

${\displaystyle \text{Proj}\left(\frac{\mathbb{C}[x,y,z]}{(x^4+y^4)}\right)}$

If we look in the${\displaystyle z\ne 0}$ chart this is the scheme

${\displaystyle \mathrm{Spec}\left(\frac{\mathbb{C}[X,Y]}{(X^4+Y^4)}\right)}$

and project it down to the affine line${\displaystyle {\mathbb{A}}_Y^1}$, this is a family of four points degenerating at the origin. The non-singularity of this scheme can also be checked using the Jacobian condition.

Consider the flat family

${\displaystyle \mathrm{Spec}\left(\frac{\mathbb{C}[t,x,y]}{(xy-t)}\right)\to {\mathbb{A}}_t^1}$

Then the fibers are all smooth except for the point at the origin. Since smoothness is stable under base-change, this family is not smooth.

For example, the field${\displaystyle {\mathbb{F}}_p(t^p)\to {\mathbb{F}}_p(t)}$ is non-separable, hence the associated morphism of schemes is not smooth. If we look at the minimal polynomial of the field extension,

${\displaystyle f(x)=x^p-t^p}$

then${\displaystyle df=0}$, hence the Kähler differentials will be non-zero.

One can define smoothness without reference to geometry. We say that anS-schemeX isformally smooth if for any affineS-schemeT and a subscheme${\displaystyle T_0}$ ofT given by anilpotent ideal,${\displaystyle X(T)\to X(T_0)}$ is surjective where we wrote${\displaystyle X(T)={\mathrm{Hom}}_S(T,X)}$. Then a morphism locally of finite presentation is smooth if and only if it is formally smooth.

In the definition of "formally smooth", if we replace surjective by "bijective" (resp. "injective"), then we get the definition offormally étale (resp.formally unramified).

LetS be a scheme and${\displaystyle \mathrm{char}(S)}$ denote the image of the structure map${\displaystyle S\to \mathrm{Spec}\mathbb{Z}}$. Thesmooth base change theorem states the following: let${\displaystyle f:X\to S}$ be aquasi-compact morphism,${\displaystyle g:S^{\prime }\to S}$ a smooth morphism and${\displaystyle \mathcal{F}}$ atorsion sheaf on${\displaystyle X_{\text{et}}}$. If for every${\displaystyle 0\ne p}$ in${\displaystyle \mathrm{char}(S)}$,${\displaystyle p:\mathcal{F}\to \mathcal{F}}$ is injective, then thebase change morphism${\displaystyle g^{\ast }(R^i{f}_{\ast }\mathcal{F})\to R^i{f}_{\ast }^{\prime }(g^{\prime \ast }\mathcal{F})}$ is an isomorphism.

• smooth algebra
• regular embedding

• J. S. Milne (2012). "Lectures on Étale Cohomology"
• J. S. Milne.Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.

• Morphisms of schemes

• Articles lacking in-text citations from January 2025
• All articles lacking in-text citations