In
algebraic geometry, a branch of
mathematics, a
morphism ''f'' : ''X'' → ''Y'' of
schemes is quasi-finite if it is of
finite type and satisfies any of the following equivalent conditions:
* Every point ''x'' of ''X'' is isolated in its fiber ''f''
−1(''f''(''x'')). In other words, every fiber is a discrete (hence finite) set.
* For every point ''x'' of ''X'', the scheme is a finite κ(''f''(''x'')) scheme. (Here κ(''p'') is the residue field at a point ''p''.)
* For every point ''x'' of ''X'',
is finitely generated over
.
Quasi-finite morphisms were originally defined by
Alexander Grothendieck in
SGA 1 and did not include the finite type hypothesis. This hypothesis was added to the definition in
EGA II 6.2 because it makes it possible to give an algebraic characterization of quasi-finiteness in terms of
stalks.
For a general morphism and a point ''x'' in ''X'', ''f'' is said to be quasi-finite at ''x'' if there exist open affine neighborhoods ''U'' of ''x'' and ''V'' of ''f''(''x'') such that ''f''(''U'') is contained in ''V'' and such that the restriction is quasi-finite. ''f'' is locally quasi-finite if it is quasi-finite at every point in ''X''. A quasi-compact locally quasi-finite morphism is quasi-finite.
Properties
For a morphism ''f'', the following properties are true.
[EGA II, Proposition 6.2.4.]
* If ''f'' is quasi-finite, then the induced map ''f''
red between
reduced scheme
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.
...
s is quasi-finite.
* If ''f'' is a closed immersion, then ''f'' is quasi-finite.
* If ''X'' is noetherian and ''f'' is an immersion, then ''f'' is quasi-finite.
* If , and if is quasi-finite, then ''f'' is quasi-finite if any of the following are true:
*#''g'' is separated,
*#''X'' is noetherian,
*# is locally noetherian.
Quasi-finiteness is preserved by base change. The composite and fiber product of quasi-finite morphisms is quasi-finite.
If ''f'' is
unramified at a point ''x'', then ''f'' is quasi-finite at ''x''. Conversely, if ''f'' is quasi-finite at ''x'', and if also
, the local ring of ''x'' in the fiber ''f''
−1(''f''(''x'')), is a field and a finite separable extension of κ(''f''(''x'')), then ''f'' is unramified at ''x''.
Finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
s are quasi-finite. A quasi-finite
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 ...
locally of finite presentation is finite. Indeed, a morphism is finite if and only if it is proper and locally quasi-finite.
A generalized form of
Zariski Main Theorem is the following:
[EGA IV3, Théorème 8.12.6.] Suppose ''Y'' is
quasi-compact and quasi-separated. Let ''f'' be quasi-finite, separated and of finite presentation. Then ''f'' factors as
where the first morphism is an open immersion and the second is finite. (''X'' is open in a finite scheme over ''Y''.)
See also
* The
quasi-finite fundamental group scheme
Notes
References
*
*
*{{cite journal
, last = Grothendieck
, first = Alexandre
, author-link = Alexandre Grothendieck
, author2=Jean Dieudonné
, author2-link=Jean Dieudonné
, year = 1966
, title = Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
, journal = Publications Mathématiques de l'IHÉS
, volume = 28
, pages = 5–255
, url = http://www.numdam.org:80/numdam-bin/feuilleter?id=PMIHES_1966__28_
Morphisms of schemes