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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 (mathematics). The concern here is to list the fundamental technical definitions and properties of scheme theory. See also list of algebraic geometry topics. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In mathematics, a projective space is a fundamental construction from any vector space. ...
In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
This is a list of algebraic geometry topics, by Wikipedia page. ...
Points
A scheme S is a locally ringed space, so a fortiori a topological space, but the meaning of point of S are threefold: In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
- a point P of the underlying topological space;
- a T-valued point of S is a morphism from T to S, for any scheme T;
- a geometric point, where S is defined over (is equipped with a morphism to) Spec(K), where K is a field, is a morphism from Spec(K*) to S where 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 hS it sets up. Historically there was a process by which projective geometry added more and points (complex points, line at infinity) to simplify the geometry by refining the basic objects. The T-valued points were a massive further step. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
In mathematics, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...
This article is about algebraic varieties. ...
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
Oscar Zariski was one of the most influential mathematicians working in the field of algebraic geometry in the twentieth century. ...
André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century, a founding member of the influential Bourbaki group. ...
In category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ...
Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...
In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ...
Local properties A property P of (commutative) rings is local in nature, for the Zariski topology if it remains stable under finite localization. Such a property automatically gives a property of schemes which is local in nature. We can require that the scheme is covered by affine open sets whose rings of coordinates have the property P in question. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. ...
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ...
For example, we can speak of locally noetherian schemes, namely those which are covered by the spectra of Noetherian rings; and we say that a scheme is noetherian when it is covered by 'finitely' many spectra of noetherian rings. Unfortunately, while it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. Much of algebraic geometry is concerned only about noetherian (or, at any rate, locally noetherian) schemes, but non-noetherian and even non-locally noetherian schemes do turn up. In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...
Another example of a local property is for a scheme to be reduced: this means that none of its rings of sections has any nilpotent element other than zero, or that it is covered by the spectra of reduced rings (viz. rings having no nonzero nilpotent elements). In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. ...
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0. ...
Here are some local properties of rings:
In ring theory, a ring R is said to be reduced if it has no non-zero nilpotent elements. ...
In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ...
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ...
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension. ...
In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. ...
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ...
Properties of the underlying space Naturally, since a scheme has an underlying topological space, one can also apply to schemes whatever properties apply to topological spaces. For example, one might say of a scheme that it is connected which simply means that the underlying topological space is connected. Things aren't always that simple, however. A scheme is rarely a Hausdorff space. The word separated is used in a way (separated morphism) not familiar from topological separation axioms as conventionally stated. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology and related branches of mathematics, a topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. ...
In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...
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. (Any noetherian scheme can be written uniquely as the union of finitely many irreducible non-empty closed subsets, called its irreducible components.) A scheme that is both reduced and irreducible is called integral; this is equivalent to saying that the scheme is connected and that it covered by the spectra of integral domains (and a ring is an integral domain if and only if its spectrum, an affine scheme, is integral). In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ...
Of course, these are only a small subset of what adjectives can be applied to the word "scheme".
For the following definitions, we take as standard notation
- f:Y → X
to be a morphism of schemes.
Properties of scheme morphisms
Affine morphism An affine morphism is one defined by the global Spec construction for sheaves of OX-Algebras, defined by analogy with the spectrum of a ring. In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
Closed immersion A closed immersion morphism is one defined by the vanishing of a global Ideal of OX-Algebras.
Étale morphism A morphism f is étale if it is finite, flat and unramified.
Finite morphism A morphism f is finite if, locally on X, it is represented by an integral extension of commutative rings. See finite morphism. A morphism of scheme (mathematics) is called finite, if there is a covering of by affine schemes , such that the preimages can be covered by a finite number of affine schemes , such (via the induced morphism ) is a finitely generated -module. ...
Flat morphism A morphism f is flat if it gives rise to a flat map on stalks. In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i. ...
In ring theory, a homomorphism f from a ring R to a ring S is flat if S becomes a flat R-module when the action of R on S is given by f. ...
Open immersion A morphism f is an open immersion if locally it is of the form of an inclusion of an affine open subset.
Projective morphism A morphism f is projective if it is given by the global Proj construction on graded commutative OX-Algebras.
Proper morphism See main article on proper morphisms This is a glossary of scheme theory. ...
A morphism f is proper if it is separated, universally closed (i.e. such that fiber products with it preserve closed immersions), and of finite type. A projective morphism is proper; but the converse is not in general true. See also complete variety. This is a glossary of some terms used in the branch of mathematics known as topology. ...
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...
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.
Separated morphism A separated morphism is a morphism f such that the fiber product of Y with itself along f has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion. In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ...
In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ...
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. Any affine scheme is separated.
This compares with the criterion (closed diagonal) for a topological space to be Hausdorff; with the usual topological space product used.
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