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
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.
|related to scheme structure|
|irreducible||A scheme X is said to be 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)||Spec k[x]/(x2)|
|integral||A scheme that is both reduced and irreducible is called integral. Equivalently, a connected scheme that is 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]/(x2+x3-y3)=|
|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 such 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 all chains between two irreducible subschemes have the same length.||(Virtually everything, e.g. varieties over a field)|
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 is called dominant, if the image f(Y) is dense. A morphism of affine schemes Spec A → Spec B is dense if and only if the kernel of the corresponding map B → A is contained in the nilradical of A.
A morphism is called quasi-compact, if for some (equivalently: every) open affine cover of X by some Ui = Spec Bi, the preimages f-1(Ui) are 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, and finite morphisms.
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.
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
Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is 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 points 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
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.