between schemes is said to be smooth if (iii) means that each geometric fiber of f is a nonsingular variety (if it is separated).
Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties.
If S is the spectrum of an algebraically closed field and f is of 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.
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.
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).
which has an empty intersection with the polynomial, since which are both non-zero.
is the weighted projective space minus a point sending Notice that the direct sum bundles
can be constructed using the fiber product Recall that a field extension
is called separable iff given a presentation we have that
-fold given by Then the Jacobian matrix is given by which vanishes at the origin, hence the cone is singular.
Affine hypersurfaces like these are popular in singularity theory because of their relatively simple algebra but rich underlying structures.
its projective cone is the union of all lines in
For example, the projective cone of the points is the scheme If we look in the
chart this is the scheme and project it down to the affine line
, 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 Then the fibers are all smooth except for the point at the origin.
is non-separable, hence the associated morphism of schemes is not smooth.
If we look at the minimal polynomial of the field extension, then
One can define smoothness without reference to geometry.
In the definition of "formally smooth", if we replace surjective by "bijective" (resp.
"injective"), then we get the definition of formally étale (resp.
denote the image of the structure map
The smooth base change theorem states the following: let
is injective, then the base change morphism