Noetherian scheme

Glossary of scheme theory

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 S is a locally ringed space, so a fortiori a topological space, but the meanings of point of S are threefold:

  1. a point P of the underlying topological space;
  2. a T -valued point of S is a morphism from T to S , for any scheme T ;
  3. a geometric point, where S is defined over (is equipped with a morphism to) textrm{Spec}(K) , where K is a field, is a morphism from textrm{Spec} (overline{K}) to S where overline{K} is an algebraic closure of K.

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 P 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 T -valued points are thought of, via Yoneda's lemma, as a way of identifying S with the representable functor h_{S} 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 T -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 S^{prime} to S is thought of as

S^{prime} times_{S} textrm{Spec}(overline{K}) .

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 if for any cover of X by open subschemes Xi, i.e. X=cup 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 = cup 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 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 mathcal O_X(U) (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 AB. (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). GL_infty = cup GL_n
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)

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 mathbb{Z} of integers; so that any scheme S is over textrm{Spec} (mathbb{Z}) , and in a unique way.

For the following definitions, we take as standard notation

f: Y to X

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 X by some open subschemes U_i, such that the restriction of f to f^{-1}(U_i) has the property, then f 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 Y are mapped to open subschemes of X (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(Y) is dense. A morphism of affine schemes Spec ASpec B is dense if and only if the kernel of the corresponding map BA 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.

Open and closed immersions

A morphism f 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 mathcal{O}_{X} -algebras, i.e. closed immersions correspond locally to morphisms of rings A rightarrow A/I, where I is the ideal of the closed subscheme Y . Equivalently, a morphism f: Y to X of schemes is a closed immersion if and only if f 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 f^{#}: mathcal{O}_{X} to f_{*} mathcal{O}_{Y} 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: textrm{Spec} (A/I) may be homeomorphic to textrm{Spec} (A/I^{prime}), without I = I^{prime} . 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) A/I 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, and finite morphisms.

Projective morphisms are defined similarly, but in practice they turn out to be more important than affine morphisms: f is called projective, if it factors as a closed immersion followed by the projection of a projective space mathbb{P}^{n}_X := mathbb{P}^n times X to X . Again, one may say, that f is projective if it is given by the global Proj construction on graded commutative OX-Algebras.

Separated and proper morphisms

A separated morphism is a morphism f such that the fiber product of f with itself along f has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion.

As a consequence, a scheme X is separated when the diagonal of X within the scheme product of X 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 X rightarrow textrm{Spec} (mathbb{Z}) is separated.

Notice that for a topological space Y is Hausdorff iff the diagonal embedding

Y stackrel{Delta}{longrightarrow} Y times Y
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) X times_{textrm{Spec} (mathbb{Z})} X, 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 is a closed immersion of schemes):

A otimes_{mathbb Z} A rightarrow A, a otimes a' mapsto a cdot a'.

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 f: Y to X is finite if X may be covered by affine open sets text{Spec }B such that each f^{-1}(text{Spec }B) is affine -- say of the form text{Spec }A -- and furthermore A is finitely generated as a B -module. See finite morphism.

The morphism f is locally of finite type if X may be covered by affine open sets text{Spec }B such that each inverse image f^{-1}(text{Spec }B) is covered by affine open sets text{Spec }A where each A is finitely generated as a B-algebra.

The morphism f is finite type if X may be covered by affine open sets text{Spec }B such that each inverse image f^{-1}(text{Spec }B) is covered by finitely many affine open sets text{Spec }A where each A is finitely generated as a B-algebra.

The morphism f has finite fibers if the fiber over each point x in X 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 morphisms

A morphism f 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 Y , the geometric meaning of flatness could roughly be described by saying, that the fibers f^{-1}(y) do not vary too wildly.

Unramified and étale morphisms

For a point y in Y , consider the corresponding morphism of local rings

f^# colon mathcal{O}_{X, f(y)} to mathcal{O}_{Y, y}.

Let mathfrak{m} be the maximal ideal of mathcal{O}_{X,f(y)} , and let

mathfrak{n} = f^#(mathfrak{m}) mathcal{O}_{Y,y}

be the ideal generated by the image of mathfrak{m} in mathcal{O}_{Y,y} . The morphism f is unramified if for all y in Y , mathfrak{n} is the maximal ideal of mathcal{O}_{Y,y} and the induced map

mathcal{O}_{X,f(y)}/mathfrak{m} to mathcal{O}_{Y,y}/mathfrak{n}

is a finite, separable field extension. This is the geometric version (and generalization) of an unramified field extension in algebraic number theory.

A morphism f is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties X and Y over a field, étale morphisms are precisely those inducing an isomorphism of tangent spaces df: T_{x} X rightarrow T_{f(x)} Y, 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.

Smooth morphisms

The higher-dimensional analog of étale morphisms are smooth morphisms. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness:

  • for any yY, there are open affine neighborhoods V and U of y, x=f(y), respectively, such that the restriction of f to V factors in an étale morphism followed by the projection of affine n-space over U.
  • f is flat, and for every point x with algebraically closed residue field (a so-called geometric point), the fiber f-1(x) is a smooth n-dimensional variety in the sense of classical algebraic geometry.

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