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 proper morphism between
schemes is an analog of a
proper map between
complex analytic spaces.
Some authors call a proper
variety
Variety may refer to:
Arts and entertainment Entertainment formats
* Variety (radio)
* Variety show, in theater and television
Films
* ''Variety'' (1925 film), a German silent film directed by Ewald Andre Dupont
* ''Variety'' (1935 film), ...
over a
field ''k'' a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). This ca ...
. For example, every
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
over a field ''k'' is proper over ''k''. A scheme ''X'' of
finite type over the
complex numbers (for example, a variety) is proper over C if and only if the space ''X''(C) of complex points with the classical (Euclidean) topology is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in Britis ...
and
Hausdorff.
A
closed immersion is proper. A morphism is
finite if and only if it is proper and
quasi-finite.
Definition
A
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
''f'': ''X'' → ''Y'' of schemes is called universally closed if for every scheme ''Z'' with a morphism ''Z'' → ''Y'', the projection from the
fiber product
:
is a
closed map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, ...
of the underlying
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
s. A morphism of schemes is called proper if it is
separated, of
finite type, and universally closed (
GAII, 5.4.
. One also says that ''X'' is proper over ''Y''. In particular, a variety ''X'' over a field ''k'' is said to be proper over ''k'' if the morphism ''X'' → Spec(''k'') is proper.
Examples
For any natural number ''n'',
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P
''n'' over a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
''R'' is proper over ''R''.
Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a
smooth proper complex variety of dimension 3 which is not projective over C.
Affine varieties of positive dimension over a field ''k'' are never proper over ''k''. More generally, a proper
affine morphism of schemes must be finite. For example, it is not hard to see that the
affine line
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties rela ...
''A''
1 over a field ''k'' is not proper over ''k'', because the morphism ''A''
1 → Spec(''k'') is not universally closed. Indeed, the pulled-back morphism
:
(given by (''x'',''y'') ↦ ''y'') is not closed, because the image of the closed subset ''xy'' = 1 in ''A''
1 × ''A''
1 = ''A''
2 is ''A''
1 − 0, which is not closed in ''A''
1.
Properties and characterizations of proper morphisms
In the following, let ''f'': ''X'' → ''Y'' be a morphism of schemes.
* The composition of two proper morphisms is proper.
* Any
base change of a proper morphism ''f'': ''X'' → ''Y'' is proper. That is, if ''g'': Z → ''Y'' is any morphism of schemes, then the resulting morphism ''X'' ×
''Y'' ''Z'' → ''Z'' is proper.
* Properness is a
local property on the base (in the Zariski topology). That is, if ''Y'' is covered by some open subschemes ''Y
i'' and the restriction of ''f'' to all ''f
−1(Y
i)'' is proper, then so is ''f''.
* More strongly, properness is local on the base in the
fpqc topology. For example, if ''X'' is a scheme over a field ''k'' and ''E'' is a field extension of ''k'', then ''X'' is proper over ''k'' if and only if the base change ''X''
''E'' is proper over ''E''.
*
Closed immersions are proper.
* More generally, finite morphisms are proper. This is a consequence of the
going up theorem.
* By
Deligne, 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.
* For ''X'' proper over a scheme ''S'', and ''Y'' separated over ''S'', the image of any morphism ''X'' → ''Y'' over ''S'' is a closed subset of ''Y''. This is analogous to the theorem in topology that the image of a continuous map from a compact space to a Hausdorff space is a closed subset.
* The
Stein factorization 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 ...
theorem states that any proper morphism to a locally noetherian scheme can be factored as ''X'' → ''Z'' → ''Y'', where ''X'' → ''Z'' is proper, surjective, and has geometrically connected fibers, and ''Z'' → ''Y'' is finite.
*
Chow's lemma
Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following:
:If X ...
says that proper morphisms are closely related to
projective morphisms. One version is: if ''X'' is proper over a
quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
scheme ''Y'' and ''X'' has only finitely many irreducible components (which is automatic for ''Y'' noetherian), then there is a projective surjective morphism ''g'': ''W'' → ''X'' such that ''W'' is projective over ''Y''. Moreover, one can arrange that ''g'' is an isomorphism over a dense open subset ''U'' of ''X'', and that ''g''
−1(''U'') is dense in ''W''. One can also arrange that ''W'' is integral if ''X'' is integral.
*
Nagata's compactification theorem In algebraic geometry, Nagata's compactification theorem, introduced by , implies that every abstract variety can be embedded in a complete variety, and more generally shows that a separated and finite type morphism to a Noetherian scheme ''S'' ...
, as generalized by Deligne, says that a separated morphism of finite type between quasi-compact and
quasi-separated schemes factors as an open immersion followed by a proper morphism.
* Proper morphisms between locally noetherian schemes preserve coherent sheaves, in the sense that the
higher direct images ''R
if''
∗(''F'') (in particular the
direct image ''f''
∗(''F'')) of a
coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
''F'' are coherent (EGA III, 3.2.1). (Analogously, for a proper map between complex analytic spaces,
Grauert and
Remmert showed that the higher direct images preserve coherent analytic sheaves.) As a very special case: the ring of regular functions on a proper scheme ''X'' over a field ''k'' has finite dimension as a ''k''-vector space. By contrast, the ring of regular functions on the affine line over ''k'' is the polynomial ring ''k''
'x'' which does not have finite dimension as a ''k''-vector space.
*There is also a slightly stronger statement of this: let
be a morphism of finite type, ''S'' locally noetherian and
a
-module. If the support of ''F'' is proper over ''S'', then for each
the
higher direct image is coherent.
*For a scheme ''X'' of finite type over the complex numbers, the set ''X''(C) of complex points is a
complex analytic space, using the classical (Euclidean) topology. For ''X'' and ''Y'' separated and of finite type over C, a morphism ''f'': ''X'' → ''Y'' over C is proper if and only if the continuous map ''f'': ''X''(C) → ''Y''(C) is proper in the sense that the inverse image of every compact set is compact.
* If ''f'': ''X''→''Y'' and ''g'': ''Y''→''Z'' are such that ''gf'' is proper and ''g'' is separated, then ''f'' is proper. This can for example be easily proven using the following criterion.
Valuative criterion of properness
There is a very intuitive criterion for properness which goes back to
Chevalley. It is commonly called the valuative criterion of properness. Let ''f'': ''X'' → ''Y'' be a morphism of finite type of
noetherian schemes. Then ''f'' is proper if and only if for all
discrete valuation rings ''R'' with
fraction field ''K'' and for any ''K''-valued point ''x'' ∈ ''X''(''K'') that maps to a point ''f''(''x'') that is defined over ''R'', there is a unique lift of ''x'' to
. (EGA II, 7.3.8). More generally, a quasi-separated morphism ''f'': ''X'' → ''Y'' of finite type (note: finite type includes quasi-compact) of *any* schemes ''X'', ''Y'' is proper if and only if for all
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
s ''R'' with
fraction field ''K'' and for any ''K''-valued point ''x'' ∈ ''X''(''K'') that maps to a point ''f''(''x'') that is defined over ''R'', there is a unique lift of ''x'' to
. (Stacks project Tags 01KF and 01KY). Noting that ''Spec K'' is the
generic point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
In classical algebraic g ...
of ''Spec R'' and discrete valuation rings are precisely the
regular local one-dimensional rings, one may rephrase the criterion: given a regular curve on ''Y'' (corresponding to the morphism ''s'': Spec ''R'' → ''Y'') and given a lift of the generic point of this curve to ''X'', ''f'' is proper if and only if there is exactly one way to complete the curve.
Similarly, ''f'' is separated if and only if in every such diagram, there is at most one lift
.
For example, given the valuative criterion, it becomes easy to check that projective space P
''n'' is proper over a field (or even over Z). One simply observes that for a discrete valuation ring ''R'' with fraction field ''K'', every ''K''-point
0,...,''x''''n''">'x''0,...,''x''''n''of projective space comes from an ''R''-point, by scaling the coordinates so that all lie in ''R'' and at least one is a unit in ''R''.
Geometric interpretation with disks
One of the motivating examples for the valuative criterion of properness is the interpretation of
as an infinitesimal disk, or complex-analytically, as the disk
. This comes from the fact that every power series
converges in some disk of radius
around the origin. Then, using a change of coordinates, this can be expressed as a power series on the unit disk. Then, if we invert
, this is the ring
which are the power series which may have a pole at the origin. This is represented topologically as the open disk
with the origin removed. For a morphism of schemes over
, this is given by the commutative diagram
Then, the valuative criterion for properness would be a filling in of the point
in the image of
.
Example
It's instructive to look at a counter-example to see why the valuative criterion of properness should hold on spaces analogous to closed compact manifolds. If we take
and
, then a morphism
factors through an affine chart of
, reducing the diagram to
where
is the chart centered around
on
. This gives the commutative diagram of commutative algebras
Then, a lifting of the diagram of schemes,
, would imply there is a morphism
sending
from the commutative diagram of algebras. This, of course, cannot happen. Therefore
is not proper over
.
Geometric interpretation with curves
There is another similar example of the valuative criterion of properness which captures some of the intuition for why this theorem should hold. Consider a curve
and the complement of a point
. Then the valuative criterion for properness would read as a diagram
with a lifting of
. Geometrically this means every curve in the scheme
can be completed to a compact curve. This bit of intuition aligns with what the scheme-theoretic interpretation of a morphism of topological spaces with compact fibers, that a sequence in one of the fibers must converge. Because this geometric situation is a problem locally, the diagram is replaced by looking at the local ring
, which is a DVR, and its fraction field
. Then, the lifting problem then gives the commutative diagram
where the scheme
represents a local disk around
with the closed point
removed.
Proper morphism of formal schemes
Let
be a morphism between
locally noetherian formal schemes. We say ''f'' is proper or
is proper over
if (i) ''f'' is an
adic morphism
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off of ...
(i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map
is proper, where
and ''K'' is the ideal of definition of
. The definition is independent of the choice of ''K''.
For example, if ''g'': ''Y'' → ''Z'' is a proper morphism of locally noetherian schemes, ''Z''
0 is a closed subset of ''Z'', and ''Y''
0 is a closed subset of ''Y'' such that ''g''(''Y''
0) ⊂ ''Z''
0, then the morphism
on formal completions is a proper morphism of formal schemes.
Grothendieck proved the coherence theorem in this setting. Namely, let
be a proper morphism of locally noetherian formal schemes. If ''F'' is a coherent sheaf on
, then the higher direct images
are coherent.
[Grothendieck, EGA III, Part 1, Théorème 3.4.2.]
See also
*
Proper base change theorem
*
Stein factorization 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 ...
References
*
*, section 5.3. (definition of properness), section 7.3. (valuative criterion of properness)
*
*, section 15.7. (generalizations of valuative criteria to not necessarily noetherian schemes)
*
*
*
External links
*
*{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/
Morphisms of schemes