In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein factorization contracts the connected components of the fibers of a mapping to points.
Statement
One version for schemes states the following:
Let ''X'' be a scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
, ''S'' a locally noetherian scheme and a proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces.
Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a field '' ...
. Then one can write
:
where is a finite morphism In algebraic geometry, a finite morphism between two affine varieties
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polyn ...
and is a proper morphism so that
The existence of this decomposition itself is not difficult. See below. But, by
Zariski's connectedness theorem In algebraic geometry, Zariski's connectedness theorem (due to Oscar Zariski) says that under certain conditions the fibers of a morphism of varieties are connected. It is an extension of Zariski's main theorem to the case when the morphism of vari ...
, the last part in the above says that the fiber
is connected for any
. It follows:
Corollary: For any
, the set of connected components of the fiber
is in bijection with the set of points in the fiber
.
Proof
Set:
:
where Spec
''S'' is the
relative Spec. The construction gives the natural map
, which is finite since
is coherent and ''f'' is proper. The morphism ''f'' factors through ''g'' and one gets
, which is proper. By construction,
. One then uses the
theorem on formal functions In algebraic geometry, the theorem on formal functions states the following:
:Let f: X \to S be a proper morphism of noetherian schemes with a coherent sheaf \mathcal on ''X''. Let S_0 be a closed subscheme of ''S'' defined by \mathcal and \widehat, ...
to show that the last equality implies
has connected fibers. (This part is sometimes referred to as Zariski's connectedness theorem.)
See also
*
Contraction morphism In algebraic geometry, a contraction morphism is a surjective projective morphism f: X \to Y between normal projective varieties (or projective schemes) such that f_* \mathcal_X = \mathcal_Y or, equivalently, the geometric fibers are all connected ...
References
*
*
*{{Citation , last1=Stein , first1=Karl , authorlink=Karl Stein (mathematician) , title=Analytische Zerlegungen komplexer Räume , doi=10.1007/BF01343331 , mr=0083045 , year=1956 , journal=
Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
, issn=0025-5831 , volume=132 , pages=63–93
Algebraic geometry