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In
algebraic geometry
, a finite morphism between two
affine varieties
X, Y
is a dense
regular map
which induces isomorphic inclusion
k\left
\right
hookrightarrow k\left
\right
/math> between their
coordinate rings
, such that
k\left
\right
/math> is
integral over
In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' i ...
k\left
\right
/math>. This definition can be extended to the
quasi-projective varieties
In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
, such that a
regular map
f\colon X\to Y
between quasiprojective varieties is finite if any point like
y\in Y
has an affine neighbourhood V such that
U=f^(V)
is affine and
f\colon U\to V
is a finite map (in view of the previous definition, because it is between affine varieties).
Definition by Schemes
A morphism ''f'': ''X'' → ''Y'' of
scheme
s is a finite morphism if ''Y'' has an
open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
by
affine schemes
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with th ...
:
V_i = \mbox \; B_i
such that for each ''i'', :
f^(V_i) = U_i
is an open affine subscheme Spec ''A''
''i''
, and the restriction of ''f'' to ''U''
''i''
, which induces a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition prese ...
:
B_i \rightarrow A_i,
makes ''A''
''i''
a
finitely generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts i ...
over ''B''
''i''
. One also says that ''X'' is finite over ''Y''. In fact, ''f'' is finite if and only if for ''every'' open affine open subscheme ''V'' = Spec ''B'' in ''Y'', the inverse image of ''V'' in ''X'' is affine, of the form Spec ''A'', with ''A'' a finitely generated ''B''-module. For example, for any
field
''k'',
\text(k
,x
(x^n-t)) \to \text(k
is a finite morphism since
k
,x
(x^n-t) \cong k
oplus k
cdot x \oplus\cdots \oplus k
cdot x^
as
k
/math>-modules. Geometrically, this is obviously finite since this is a ramified n-sheeted cover of the affine line which degenerates at the origin. By contrast, the inclusion of ''A''
1
− 0 into ''A''
1
is not finite. (Indeed, the
Laurent polynomial
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field \mathbb is a linear combination of positive and negative powers of the variable with coefficients in \mathbb. Laurent polynomials in ''X'' f ...
ring ''k''
−1">'y'', ''y''
−1
is not finitely generated as a module over ''k''
'y''
) This restricts our geometric intuition to surjective families with finite fibers.
Properties of finite morphisms
* The composition of two finite morphisms is finite. * Any
base change
of a finite morphism ''f'': ''X'' → ''Y'' is finite. That is, if ''g'': Z → ''Y'' is any morphism of schemes, then the resulting morphism ''X'' ×
''Y''
''Z'' → ''Z'' is finite. This corresponds to the following algebraic statement: if ''A'' and ''C'' are (commutative) ''B''-algebras, and ''A'' is finitely generated as a ''B''-module, then the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an e ...
''A'' ⊗
''B''
''C'' is finitely generated as a ''C''-module. Indeed, the generators can be taken to be the elements ''a''
''i''
⊗ 1, where ''a''
''i''
are the given generators of ''A'' as a ''B''-module. *
Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies ''Z'' as a closed subset of ''X'' such that locally, regular functions on ''Z'' can be extended to ''X''. The latter condition can be formal ...
s are finite, as they are locally given by ''A'' → ''A''/''I'', where ''I'' is the
ideal
corresponding to the closed subscheme. * Finite morphisms are closed, hence (because of their stability under base change)
proper
. This follows from the
going up
theorem of Cohen-Seidenberg in commutative algebra. * Finite morphisms have finite fibers (that is, they are
quasi-finite
). This follows from the fact that for a field ''k'', every finite ''k''-algebra is an
Artinian ring
In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
. A related statement is that for a finite surjective morphism ''f'': ''X'' → ''Y'', ''X'' and ''Y'' have the same
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
. * By
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
, a morphism of schemes is finite if and only if it is proper and quasi-finite. This had been shown by
Grothendieck
if the morphism ''f'': ''X'' → ''Y'' is
locally of finite presentation
, which follows from the other assumptions if ''Y'' is
Noetherian
. * Finite morphisms are both projective and
affine
Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
.
.
See also
*
Glossary of algebraic geometry
*
Finite algebra
Notes
References
* * * *
External links
*{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/
Algebraic geometry
Morphisms