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EspañolGLOSSARY OF SCHEME THEORY
(Redirected from Closed subscheme)
This is a 'glossary of scheme theory'. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory. See also list of algebraic geometry topics.
A scheme is a locally ringed space, so ''a fortiori'' a topological space, but the meanings of ''point of '' are threefold:
#a point of the underlying topological space;
#a -valued point of is a morphism from to , for any scheme ;
#a ''geometric point'', where is defined over (is equipped with a morphism to) , where is a field, is a morphism from to where is an algebraic closure of .
Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The -valued points are thought of, via Yoneda's lemma, as a way of identifying with the representable functor it sets up. Historically there was a process by which projective geometry added more points (''e.g.'' complex points, line at infinity) to simplify the geometry by refining the basic objects. The -valued points were a massive further step.
As part of the predominating Grothendieck approach, there are three corresponding notions of ''fiber'' of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a 'geometric fiber' of a morphism is thought of as
:.
This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.
Most important properties of schemes are ''local in nature'', i.e. a scheme ''X'' has a certain property ''P'' if and only for any cover of ''X'' by open subschemes ''Xi'', i.e. ''X''= ''Xi'', every ''Xi'' has the property ''P''. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is ''Zariski-local'', if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology.
Consider a scheme ''X'' and a cover by affine open subschemes ''Spec Ai''. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings ''Ai''. A property ''P'' is local in the above sense, iff the corresponding property of rings is stable under localization.
For example, we can speak of ''locally noetherian'' schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a noetherian ring are still noetherian then means that the property of a scheme of being locally noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations.
An example for a non-local property is ''separatedness'' (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme.
The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let ''X'' = ''Spec Ai'' be a covering of a scheme by open affine subschemes. For definiteness, let ''k'' denote a field in the following. Most of the examples also work with the integers 'Z' as a base, though, or even more general bases.
One of Grothendieck's fundamental ideas is to emphasize ''relative'' notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring of integers; so that any scheme is ''over'' , and in a unique way.
For the following definitions, we take as standard notation
:
to be a morphism of schemes. Parallel to the properties of schemes above, the following properties of morphisms are also of local nature, i.e. if there is an open covering of by some open subschemes , such that the restriction of to has the property, then has it, as well.
A morphism of schemes is called ''open'' (''closed'') , if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of ''X'' are mapped to open subschemes of ''Y'' (and similarly for closed). For example, flat morphisms are open and proper maps are closed, see below.
A morphism is called ''dominant'', if the image ''f''(''X'') is dense. A morphism of affine schemes ''Spec A'' → ''Spec B'' is dense if and only if the corresponding map ''B'' → ''A'' is injective.
A morphism is called ''quasi-compact'', if for some (equivalently: every) open affine cover of ''Y'' by some ''Yi'' = ''Spec Bi'', the preimages ''f''-1(''Yi'') is quasi-compact.
A morphism is an ''open immersion'' if locally on the target it is of the form of an inclusion of an open subset.
A closed immersion morphism is one defined by the vanishing of a global ideal of -algebras, i.e. closed immersions correspond locally to morphisms of rings , where is the ideal of the closed subscheme . Equivalently, a morphism of schemes is a closed immersion if and only if induces a homeomorphism from sp(''Y''), the underlying topological space of ''Y'', onto a closed subset of sp(''X''), and if furthermore the induced morphism is surjective.
An ''immersion'' is an isomorphism of ''Y'' to an open subscheme of a closed subscheme of ''X''.
Note, that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: may be homeomorphic to , without . When specifying a closed subset of a scheme without mentioning the scheme structure, mostly the so-called ''reduced'' scheme-structure is meant, i.e. (locally) should have no nilpotent elements, which uniquely determines the closed subscheme.
A morphism is called affine, if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the global 'Spec' construction for sheaves of ''OX''-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles.
Projective morphisms are defined similarly, but in practice they turn out to be more important than affine morphisms: is called ''projective'', if it factors as a closed immersion followed by the projection of a projective space to . Again, one may say, that is projective if it is given by the global 'Proj' construction on graded commutative ''OX''-Algebras.
A separated morphism is a morphism such that the fiber product of with itself along has its diagonal as a closed subscheme — in other words, the diagonal map is a ''closed immersion''.
As a consequence, a scheme is 'separated' when the diagonal of within the ''scheme product'' of with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism is separated.
Notice that for a topological space ''Y'' is Hausdorff iff the diagonal embedding
:
is closed. In algebraic geometry, the above formulation is used because a scheme is a Hausdorff space if and only if it is zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) , which is different from the product of topological spaces.
Any ''affine'' scheme ''Spec A'' is separated, because the diagonal corresponds to the surjective map of rings (hence a closed immersion of schemes):
:''''.
While the separatedness is of rather technical nature, ''properness'' has deep geometrical meaning.
A morphism is 'proper' if it is separated, ''universally closed'' (i.e. such that fiber products with it preserve closed immersions), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a ''Stein factorization'', namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
A morphism is 'finite' if may be covered by affine open sets such that each is affine -- say of the form -- and furthermore is finitely generated as a -module. See finite morphism.
The morphism is 'locally of finite type' if may be covered by affine open sets such that each inverse image is covered by affine open sets where each is finitely generated as a -algebra.
The morphism is 'finite type' if may be covered by affine open sets such that each inverse image is covered by finitely many affine open sets where each is finitely generated as a -algebra.
The morphism has 'finite fibers' if the fiber over each point is a finite set. A morphism is 'quasi-finite' if it is of finite type and has finite fibers.
Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite.
A morphism is flat if it gives rise to a flat map on stalks. When viewing a morphism as a family of schemes parametrized by the poins of , the geometric meaning of flatness could roughly be described by saying, that the fibers do not vary too wildly.
For a point in , consider the corresponding morphism of local rings
:
Let be the maximal ideal of , and let
:
be the ideal generated by the image of in . The morphism is 'unramified' if for all in , is the maximal ideal of and the induced map
:
is a finite, separable field extension.
A morphism is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties and over a field, étale morphisms are precisely those inducing an isomorphism of tangent spaces , which coincides with the usual notion of étale map in differential geometry.
Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry.
This is a 'glossary of scheme theory'. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory. See also list of algebraic geometry topics.
Points
A scheme is a locally ringed space, so ''a fortiori'' a topological space, but the meanings of ''point of '' are threefold:
#a point of the underlying topological space;
#a -valued point of is a morphism from to , for any scheme ;
#a ''geometric point'', where is defined over (is equipped with a morphism to) , where is a field, is a morphism from to where is an algebraic closure of .
Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The -valued points are thought of, via Yoneda's lemma, as a way of identifying with the representable functor it sets up. Historically there was a process by which projective geometry added more points (''e.g.'' complex points, line at infinity) to simplify the geometry by refining the basic objects. The -valued points were a massive further step.
As part of the predominating Grothendieck approach, there are three corresponding notions of ''fiber'' of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a 'geometric fiber' of a morphism is thought of as
:.
This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.
Properties of schemes
Most important properties of schemes are ''local in nature'', i.e. a scheme ''X'' has a certain property ''P'' if and only for any cover of ''X'' by open subschemes ''Xi'', i.e. ''X''= ''Xi'', every ''Xi'' has the property ''P''. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is ''Zariski-local'', if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology.
Consider a scheme ''X'' and a cover by affine open subschemes ''Spec Ai''. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings ''Ai''. A property ''P'' is local in the above sense, iff the corresponding property of rings is stable under localization.
For example, we can speak of ''locally noetherian'' schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a noetherian ring are still noetherian then means that the property of a scheme of being locally noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations.
An example for a non-local property is ''separatedness'' (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme.
The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let ''X'' = ''Spec Ai'' be a covering of a scheme by open affine subschemes. For definiteness, let ''k'' denote a field in the following. Most of the examples also work with the integers 'Z' as a base, though, or even more general bases.
| notion | definition | example | non-example |
|---|---|---|---|
| related to scheme structure | |||
| irreducible | A scheme ''X'' is said ''irreducible'' when (as a topological space) it is not the union of two closed subsets except if one is equal to ''X''. Using the correspondence of prime ideals and points in an affine scheme, this means ''X'' is irreducible iff the affine schemes ''Spec Ai'' all have exactly one minimal prime ideal. Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. | affine space, projective space | ''Spec'' ''k''[''x,y'']/(''xy'') =  |
| reduced | The ''Ai'' are reduced rings. Equivalently, none of its rings of sections (''U'' any open subset of ''X'') has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations varieties to schemes. | varieties (by definition) | ''k''[''x'']/(''x''2) |
| integral | A scheme that is both reduced and irreducible is called ''integral''. Equivalently, a connected scheme that it covered by the spectra of integral domains. (Strictly speaking, this is not a local property, because the disjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property). | ''Spec k''[''t'']/''f'', ''f'' irreducible polynomial | ''Spec A'' ⊕ ''B''. (''A'', ''B'' ≠ 0) |
| normal | An integral scheme is called ''normal'', if the ''Ai'' are integrally closed domains. | regular schemes | singular curves |
| related to regularity | |||
| regular | The ''Ai'' are regular. | smooth varieties over a field | ''Spec k''[''x,y'']/(''x''2+''x''3-''y''3)= |
| Cohen-Macaulay | All local rings are Cohen-Macaulay. | regular schemes, ''Spec k''[''x,y'']/(''xy'') |  |
| related to "size" | |||
| locally noetherian | The ''Ai'' are Noetherian rings. If in addition a finite number of affine spectra covers ''X'', the scheme is called ''noetherian''. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. | (Virtually everything in algebraic geometry). | |
| dimension | The dimension, by definition the maximal length of a chain of irreducible subschemes, is a local property. See also Global dimension. | dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces. | |
| catenary | A scheme is catenary, if chains between two irreducible subschemes have all the same length. | (Virtually everything, e.g. varieties over a field) | |
Properties of scheme morphisms
One of Grothendieck's fundamental ideas is to emphasize ''relative'' notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring of integers; so that any scheme is ''over'' , and in a unique way.
For the following definitions, we take as standard notation
:
to be a morphism of schemes. Parallel to the properties of schemes above, the following properties of morphisms are also of local nature, i.e. if there is an open covering of by some open subschemes , such that the restriction of to has the property, then has it, as well.
Notions related to the topological structure
A morphism of schemes is called ''open'' (''closed'') , if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of ''X'' are mapped to open subschemes of ''Y'' (and similarly for closed). For example, flat morphisms are open and proper maps are closed, see below.
A morphism is called ''dominant'', if the image ''f''(''X'') is dense. A morphism of affine schemes ''Spec A'' → ''Spec B'' is dense if and only if the corresponding map ''B'' → ''A'' is injective.
A morphism is called ''quasi-compact'', if for some (equivalently: every) open affine cover of ''Y'' by some ''Yi'' = ''Spec Bi'', the preimages ''f''-1(''Yi'') is quasi-compact.
Open and closed immersions
A morphism is an ''open immersion'' if locally on the target it is of the form of an inclusion of an open subset.
A closed immersion morphism is one defined by the vanishing of a global ideal of -algebras, i.e. closed immersions correspond locally to morphisms of rings , where is the ideal of the closed subscheme . Equivalently, a morphism of schemes is a closed immersion if and only if induces a homeomorphism from sp(''Y''), the underlying topological space of ''Y'', onto a closed subset of sp(''X''), and if furthermore the induced morphism is surjective.
An ''immersion'' is an isomorphism of ''Y'' to an open subscheme of a closed subscheme of ''X''.
Note, that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: may be homeomorphic to , without . When specifying a closed subset of a scheme without mentioning the scheme structure, mostly the so-called ''reduced'' scheme-structure is meant, i.e. (locally) should have no nilpotent elements, which uniquely determines the closed subscheme.
Affine and projective morphisms
A morphism is called affine, if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the global 'Spec' construction for sheaves of ''OX''-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles.
Projective morphisms are defined similarly, but in practice they turn out to be more important than affine morphisms: is called ''projective'', if it factors as a closed immersion followed by the projection of a projective space to . Again, one may say, that is projective if it is given by the global 'Proj' construction on graded commutative ''OX''-Algebras.
Separated and proper morphism
A separated morphism is a morphism such that the fiber product of with itself along has its diagonal as a closed subscheme — in other words, the diagonal map is a ''closed immersion''.
As a consequence, a scheme is 'separated' when the diagonal of within the ''scheme product'' of with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism is separated.
Notice that for a topological space ''Y'' is Hausdorff iff the diagonal embedding
:
is closed. In algebraic geometry, the above formulation is used because a scheme is a Hausdorff space if and only if it is zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) , which is different from the product of topological spaces.
Any ''affine'' scheme ''Spec A'' is separated, because the diagonal corresponds to the surjective map of rings (hence a closed immersion of schemes):
:''''.
While the separatedness is of rather technical nature, ''properness'' has deep geometrical meaning.
A morphism is 'proper' if it is separated, ''universally closed'' (i.e. such that fiber products with it preserve closed immersions), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a ''Stein factorization'', namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
Finite, quasi-finite, and finite type morphisms
A morphism is 'finite' if may be covered by affine open sets such that each is affine -- say of the form -- and furthermore is finitely generated as a -module. See finite morphism.
The morphism is 'locally of finite type' if may be covered by affine open sets such that each inverse image is covered by affine open sets where each is finitely generated as a -algebra.
The morphism is 'finite type' if may be covered by affine open sets such that each inverse image is covered by finitely many affine open sets where each is finitely generated as a -algebra.
The morphism has 'finite fibers' if the fiber over each point is a finite set. A morphism is 'quasi-finite' if it is of finite type and has finite fibers.
Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite.
Flat morphism
A morphism is flat if it gives rise to a flat map on stalks. When viewing a morphism as a family of schemes parametrized by the poins of , the geometric meaning of flatness could roughly be described by saying, that the fibers do not vary too wildly.
Unramified and étale morphisms
For a point in , consider the corresponding morphism of local rings
:
Let be the maximal ideal of , and let
:
be the ideal generated by the image of in . The morphism is 'unramified' if for all in , is the maximal ideal of and the induced map
:
is a finite, separable field extension.
A morphism is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties and over a field, étale morphisms are precisely those inducing an isomorphism of tangent spaces , which coincides with the usual notion of étale map in differential geometry.
Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry.
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