# Smooth scheme

In algebraic geometry, a **smooth scheme** over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology.

## Definition

First, let *X* be an affine scheme of finite type over a field *k*. Equivalently, *X* has a closed immersion into affine space *A ^{n}* over

*k*for some natural number

*n*. Then

*X*is the closed subscheme defined by some equations

*g*

_{1}= 0, ...,

*g*

_{r}= 0, where each

*g*is in the polynomial ring

_{i}*k*[

*x*

_{1},...,

*x*

_{n}]. The affine scheme

*X*is

**smooth**of dimension

*m*over

*k*if

*X*has dimension at least

*m*in a neighborhood of each point, and the matrix of derivatives (∂

*g*

_{i}/∂

*x*

_{j}) has rank at least

*n*−

*m*everywhere on

*X*.

^{[1]}(It follows that

*X*has dimension equal to

*m*in a neighborhood of each point.) Smoothness is independent of the choice of embedding of

*X*into affine space.

The condition on the matrix of derivatives is understood to mean that the closed subset of *X* where all (*n*−*m*) × (*n* − *m*) minors of the matrix of derivatives are zero is the empty set. Equivalently, the ideal in the polynomial ring generated by all *g*_{i} and all those minors is the whole polynomial ring.

In geometric terms, the matrix of derivatives (∂*g*_{i}/∂*x*_{j}) at a point *p* in *X* gives a linear map *F*^{n} → *F*^{r}, where *F* is the residue field of *p*. The kernel of this map is called the Zariski tangent space of *X* at *p*. Smoothness of *X* means that the dimension of the Zariski tangent space is equal to the dimension of *X* near each point; at a singular point, the Zariski tangent space would be bigger.

More generally, a scheme *X* over a field *k* is **smooth** over *k* if each point of *X* has an open neighborhood which is a smooth affine scheme of some dimension over *k*. In particular, a smooth scheme over *k* is locally of finite type.

There is a more general notion of a smooth morphism of schemes, which is roughly a morphism with smooth fibers. In particular, a scheme *X* is smooth over a field *k* if and only if the morphism *X* → Spec *k* is smooth.

## Properties

A smooth scheme over a field is regular and hence normal. In particular, a smooth scheme over a field is reduced.

Define a **variety** over a field *k* to be an integral separated scheme of finite type over *k*. Then any smooth separated scheme of finite type over *k* is a finite disjoint union of smooth varieties over *k*.

For a smooth variety *X* over the complex numbers, the space *X*(**C**) of complex points of *X* is a complex manifold, using the classical (Euclidean) topology. Likewise, for a smooth variety *X* over the real numbers, the space *X*(**R**) of real points is a real manifold, possibly empty.

For any scheme *X* that is locally of finite type over a field *k*, there is a coherent sheaf Ω^{1} of differentials on *X*. The scheme *X* is smooth over *k* if and only if Ω^{1} is a vector bundle of rank equal to the dimension of *X* near each point.^{[2]} In that case, Ω^{1} is called the cotangent bundle of *X*. The tangent bundle of a smooth scheme over *k* can be defined as the dual bundle, *TX* = (Ω^{1})^{*}.

Smoothness is a geometric property, meaning that for any field extension *E* of *k*, a scheme *X* is smooth over *k* if and only if the scheme *X _{E}* :=

*X*×

_{Spec k}Spec

*E*is smooth over

*E*. For a perfect field

*k*, a scheme

*X*is smooth over

*k*if and only if

*X*is locally of finite type over

*k*and

*X*is regular.

## Generic smoothness

A scheme *X* is said to be **generically smooth** of dimension *n* over *k* if *X* contains an open dense subset that is smooth of dimension *n* over *k*. Every variety over a perfect field (in particular an algebraically closed field) is generically smooth.^{[3]}

## Examples

- Affine space and projective space are smooth schemes over a field
*k*. - An example of a smooth hypersurface in projective space
**P**^{n}over*k*is the Fermat hypersurface*x*_{0}^{d}+ ... +*x*_{n}^{d}= 0, for any positive integer*d*that is invertible in*k*. - An example of a singular (non-smooth) scheme over a field
*k*is the closed subscheme*x*^{2}= 0 in the affine line*A*^{1}over*k*. - An example of a singular (non-smooth) variety over
*k*is the cuspidal cubic curve*x*^{2}=*y*^{3}in the affine plane*A*^{2}, which is smooth outside the origin (*x*,*y*) = (0,0). - A 0-dimensional variety
*X*over a field*k*is of the form*X*= Spec*E*, where*E*is a finite extension field of*k*. The variety*X*is smooth over*k*if and only if*E*is a separable extension of*k*. Thus, if*E*is not separable over*k*, then*X*is a regular scheme but is not smooth over*k*. For example, let*k*be the field of rational functions**F**_{p}(*t*) for a prime number*p*, and let*E*=**F**_{p}(*t*^{1/p}); then Spec*E*is a variety of dimension 0 over*k*which is a regular scheme, but not smooth over*k*. - Schubert varieties are in general not smooth.

## Notes

- ↑ The definition of smoothness used in this article is equivalent to Grothendieck's definition of smoothness by Theorems 30.2 and Theorem 30.3 in: Matsumura, Commutative Ring Theory (1989).
- ↑ Theorem 30.3, Matsumura, Commutative Ring Theory (1989).
- ↑ Lemma 1 in section 28 and Corollary to Theorem 30.5, Matsumura, Commutative Ring Theory (1989).

## References

- D. Gaitsgory's notes on flatness and smoothness at http://www.math.harvard.edu/~gaitsgde/Schemes_2009/BR/SmoothMaps.pdf
- Hartshorne, Robin (1977),
*Algebraic Geometry*, Graduate Texts in Mathematics,**52**, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 - Matsumura, Hideyuki (1989),
*Commutative Ring Theory*, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461