In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a morphism
between
schemes is said to be smooth if
*(i) it is
locally of finite presentation
*(ii) it is
flat
Flat or flats may refer to:
Architecture
* Flat (housing), an apartment in the United Kingdom, Ireland, Australia and other Commonwealth countries
Arts and entertainment
* Flat (music), a symbol () which denotes a lower pitch
* Flat (soldier), ...
, and
*(iii) for every
geometric point
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry.
...
the fiber
is regular.
(iii) means that each geometric fiber of ''f'' is a
nonsingular variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In cas ...
(if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
If ''S'' is the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
of an algebraically closed
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
and ''f'' is of finite type, then one recovers the definition of a nonsingular variety.
Equivalent definitions
There are many equivalent definitions of a smooth morphism. Let
be locally of finite presentation. Then the following are equivalent.
# ''f'' is smooth.
# ''f'' is formally smooth (see below).
# ''f'' is flat and the
sheaf of relative differentials
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper ser ...
is locally free of rank equal to the
relative dimension
In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.
In linear algebra, given a quotient space (linear algebra), quotient map V \to Q, the difference dim ''V'' − dim ''Q'' is the relat ...
of
.
# For any
, there exists a neighborhood
of x and a neighborhood
of
such that
and the ideal generated by the ''m''-by-''m'' minors of
is ''B''.
# Locally, ''f'' factors into
where ''g'' is étale.
# Locally, ''f'' factors into
where ''g'' is étale.
A morphism of finite type is
étale if and only if it is smooth and
quasi-finite.
A smooth morphism is stable under base change and composition.
A smooth morphism is universally
locally acyclic.
Examples
Smooth morphisms are supposed to geometrically correspond to smooth
submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by
Ehresmann's theorem).
Smooth Morphism to a Point
Let
be the morphism of schemes
:
It is smooth because of the Jacobian condition: the Jacobian matrix
:
vanishes at the points
which has an empty intersection with the polynomial, since
:
which are both non-zero.
Trivial Fibrations
Given a smooth scheme
the projection morphism
:
is smooth.
Vector Bundles
Every vector bundle
over a scheme is a smooth morphism. For example, it can be shown that the associated vector bundle of
over
is the weighted projective space minus a point
:
sending
: