mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
ringed space
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
which is locally a spectrum of a commutative ring.
The relative point of view is that much of algebraic geometry should be developed for a morphism ''X'' → ''Y'' of schemes (called a scheme ''X'' over ''Y''), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme ''Y''. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.
For some of the detailed definitions in the theory of schemes, see the
glossary of scheme theory
This is a glossary of algebraic geometry.
See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometr ...
.
Development
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry was simplified by working over the field of complex numbers, which has the advantage of being algebraically closed. Two issues gradually drew attention in the early 20th century, motivated by problems in number theory: how can algebraic geometry be developed over any algebraically closed field, especially in positive characteristic? (The tools of topology and complex analysis used to study complex varieties do not seem to apply here.) And what about algebraic geometry over an arbitrary field?
Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field ''k'': the maximal ideals in the polynomial ring ''k'' 1,...,''x''''n''">'x''1,...,''x''''n''are in one-to-one correspondence with the set ''k''''n'' of ''n''-tuples of elements of ''k'', and the prime ideals correspond to the irreducible algebraic sets in ''k''''n'', known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed the subject of commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension.
Noether and Krull's commutative algebra can be viewed as an algebraic approach to ''affine'' algebraic varieties. However, many arguments in algebraic geometry work better for projective varieties, essentially because projective varieties are
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 ...
. From the 1920s to the 1940s,
B. L. van der Waerden
Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics.
Biography
Education and early career
Van der Waerden learned advanced mathematics at the University of Amst ...
, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties. In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the topology of the complex numbers).
For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an ''abstract variety'' (not embedded in projective space), by gluing affine varieties along open subsets, on the model of manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.)
The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's ''Foundations of Algebraic Geometry'' (1946), generic points are constructed by taking points in a very large algebraically closed field, called a ''universal domain''. Although this worked as a foundation, it was awkward: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)
In the 1950s,
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a fou ...
,
Masayoshi Nagata
Masayoshi Nagata (Japanese: 永田 雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra.
Work
Nagata's compactification theorem shows that va ...
and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word ''scheme'' was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas. According to Pierre Cartier, it was André Martineau who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.
Origin of schemes
Grothendieck then gave the decisive definition of a scheme, bringing to a conclusion a generation of experimental suggestions and partial developments. He defined the spectrum ''X'' of a commutative ring ''R'' as the space of prime ideals of ''R'' with a natural topology (known as the Zariski topology), but augmented it with a sheaf of rings: to every open subset ''U'' he assigned a commutative ring ''O''''X''(''U''). These objects Spec(''R'') are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
Much of algebraic geometry focuses on projective or quasi-projective varieties over a field ''k''; in fact, ''k'' is often taken to be the complex numbers. Schemes of that sort are very special compared to arbitrary schemes; compare the examples below. Nonetheless, it is convenient that Grothendieck developed a large body of theory for arbitrary schemes. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Also, applications to number theory rapidly lead to schemes over the integers which are not defined over any field.
Definition
An affine scheme is a locally ringed space isomorphic to the spectrum Spec(''R'') of a commutative ring ''R''. A scheme is a locally ringed space ''X'' admitting a covering by open sets ''U''''i'', such that each ''U''''i'' (as a locally ringed space) is an affine scheme. In particular, ''X'' comes with a sheaf ''O''''X'', which assigns to every open subset ''U'' a commutative ring ''O''''X''(''U'') called the ring of regular functions on ''U''. One can think of a scheme as being covered by "coordinate charts" which are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.
In the early days, this was called a ''prescheme'', and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and Mumford's "Red Book".
A basic example of an affine scheme is affine ''n''-space over a field ''k'', for a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
Schemes form a category, with morphisms defined as morphisms of locally ringed spaces. (See also: morphism of schemes.) For a scheme ''Y'', a scheme ''X'' over ''Y'' (or a ''Y''-scheme) means a morphism ''X'' → ''Y'' of schemes. A scheme ''X'' over a commutative ring ''R'' means a morphism ''X'' → Spec(''R'').
An algebraic variety over a field ''k'' can be defined as a scheme over ''k'' with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a variety over ''k'' means an integral separated scheme of finite type over ''k''..
A morphism ''f'': ''X'' → ''Y'' of schemes determines a pullback homomorphism on the rings of regular functions, ''f''*: ''O''(''Y'') → ''O''(''X''). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(''A'') → Spec(''B'') of schemes and ring homomorphisms ''B'' → ''A''. In this sense, scheme theory completely subsumes the theory of commutative rings.
Since Z is an initial object in the category of commutative rings, the category of schemes has Spec(Z) as a terminal object.
For a scheme ''X'' over a commutative ring ''R'', an ''R''-point of ''X'' means a section of the morphism ''X'' → Spec(''R''). One writes ''X''(''R'') for the set of ''R''-points of ''X''. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of ''X'' with values in ''R''. When ''R'' is a field ''k'', ''X''(''k'') is also called the set of ''k''- rational points of ''X''.
More generally, for a scheme ''X'' over a commutative ring ''R'' and any commutative ''R''- algebra ''S'', an ''S''-point of ''X'' means a morphism Spec(''S'') → ''X'' over ''R''. One writes ''X''(''S'') for the set of ''S''-points of ''X''. (This generalizes the old observation that given some equations over a field ''k'', one can consider the set of solutions of the equations in any field extension ''E'' of ''k''.) For a scheme ''X'' over ''R'', the assignment ''S'' ↦ ''X''(''S'') is a functor from commutative ''R''-algebras to sets. It is an important observation that a scheme ''X'' over ''R'' is determined by this functor of points.
The fiber product of schemes always exists. That is, for any schemes ''X'' and ''Z'' with morphisms to a scheme ''Y'', the fiber product ''X''×''Y''''Z'' (in the sense of category theory) exists in the category of schemes. If ''X'' and ''Z'' are schemes over a field ''k'', their fiber product over Spec(''k'') may be called the product ''X'' × ''Z'' in the category of ''k''-schemes. For example, the product of affine spaces A''m'' and A''n'' over ''k'' is affine space A''m''+''n'' over ''k''.
Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite
limit
Limit or Limits may refer to:
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* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
s.
Examples
Here and below, all the rings considered are commutative:
* Every affine scheme Spec(''R'') is a scheme.
* A polynomial ''f'' over a field ''k'', , determines a closed subscheme in affine space A''n'' over ''k'', called an affine hypersurface. Formally, it can be defined as For example, taking ''k'' to be the complex numbers, the equation defines a singular curve in the affine plane A, called a nodal cubic curve.
* For any commutative ring ''R'' and natural number ''n'', projective space P can be constructed as a scheme by gluing ''n'' + 1 copies of affine ''n''-space over ''R'' along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that P is proper over ''R''; this is an algebro-geometric version of compactness. A related observation is that complex projective space CP''n'' is a compact space in the classical topology (based on the topology of C), whereas C''n'' is not (for ''n'' > 0).
* A homogeneous polynomial ''f'' of positive degree in the polynomial ring determines a closed subscheme in projective space P''n'' over ''R'', called a projective hypersurface. In terms of the Proj construction, this subscheme can be written as For example, the closed subscheme of P is an
elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. I ...
over the rational numbers.
* The line with two origins (over a field ''k'') is the scheme defined by starting with two copies of the affine line over ''k'', and gluing together the two open subsets A1 − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.
* A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let , say over the complex numbers C; then ''X'' is not affine for ''n'' ≥ 2. (The restriction on ''n'' is necessary: the affine line minus the origin is isomorphic to the affine scheme . To show that ''X'' is not affine, one computes that every regular function on ''X'' extends to a regular function on A''n'', when ''n'' ≥ 2. (This is analogous to Hartogs's lemma in complex analysis, though easier to prove.) That is, the inclusion induces an isomorphism from to . If ''X'' were affine, it would follow that ''f'' was an isomorphism. But ''f'' is not surjective and hence not an isomorphism. Therefore, the scheme ''X'' is not affine.
* Let ''k'' be a field. Then the scheme is an affine scheme whose underlying topological space is the Stone–Čech compactification of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the ultrafilters on the positive integers, with the ideal corresponding to the principal ultrafilter associated to the positive integer ''n''. This topological space is zero-dimensional, and in particular, each point is an irreducible component. Since affine schemes are quasi-compact, this is an example of a quasi-compact scheme with infinitely many irreducible components. (By contrast, a Noetherian scheme has only finitely many irreducible components.)
Examples of morphisms
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.