In
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 ...
, a scheme is a
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additional ...
that enlarges the notion of
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
in several ways, such as taking account of
multiplicities
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
The notion of multip ...
(the equations ''x'' = 0 and ''x''
2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
(for example,
Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation
:X^n + Y^n = Z^n.\
Therefore, in terms of the affine plane its equation is
:x^n ...
s are defined over the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s).
Scheme theory was introduced by
Alexander Grothendieck in 1960 in his treatise "
Éléments de géométrie algébrique
The ''Éléments de géométrie algébrique'' ("Elements of algebraic geometry, Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French language, French, on algebraic ge ...
"; one of its aims was developing the formalism needed to solve deep problems of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, such as the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
(the last of which was proved by
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
). Strongly based on
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...
, scheme theory allows a systematic use of methods of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
and
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
. Scheme theory also unifies algebraic geometry with much of
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, which eventually led to
Wiles's proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's ...
.
Formally, a scheme is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s) of commutative rings along their open subsets. In other words, it is a
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 of ...
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 surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of di ...
s, 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
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
.
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 geometry.
...
.
Development
The origins of algebraic geometry mostly lie in the study of
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
equations over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s. By the 19th century, it became clear (notably in the work of
Jean-Victor Poncelet
Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work ''Tr ...
and
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
) that algebraic geometry was simplified by working over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, which has the advantage of being
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
. 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
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
used to study complex varieties do not seem to apply here.) And what about algebraic geometry over an arbitrary field?
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
suggests an approach to algebraic geometry over any algebraically closed field ''k'': the
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
s in the
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
''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 ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s correspond to the irreducible algebraic sets in ''k''
''n'', known as affine varieties. Motivated by these ideas,
Emmy Noether
Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noethe ...
and
Wolfgang Krull
Wolfgang Krull (26 August 1899 – 12 April 1971) was a German mathematician who made fundamental contributions to commutative algebra, introducing concepts that are now central to the subject.
Krull was born and went to school in Baden-Baden. H ...
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
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
of any commutative ring in terms of prime ideals. At least when the ring is
Noetherian In mathematics, the adjective Noetherian is used to describe Category_theory#Categories.2C_objects.2C_and_morphisms, objects that satisfy an ascending chain condition, ascending or descending chain condition on certain kinds of subobjects, meaning t ...
, 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
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective spaces, projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n' ...
, 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 British ...
. 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 Amsterd ...
,
André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
and
Oscar Zariski
, birth_date =
, birth_place = Kobrin, Russian Empire
, death_date =
, death_place = Brookline, Massachusetts, United States
, nationality = American
, field = Mathematics
, work_institutions = ...
applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or
quasi-projective In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in ...
) varieties. In particular, the
Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
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
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
), by gluing affine varieties along open subsets, on the model of
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s in topology. He needed this generality for his construction of the
Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian vari ...
of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil,
Chow and
Matsusaka
is a city located in Mie Prefecture, Japan. , the city had an estimated population of 157,235 in 66,018 households and a population density of 250 persons per km². The total area of the city is . The city is famous for Matsusaka beef.
Geography ...
.)
The algebraic geometers of the
Italian school The Italian School refers to several different Italian schools of thought, including:
* Italian School (art)
* Italian School (philosophy)
*Italian school of algebraic geometry
*Italian school of swordsmanship
*Italian school of criminology
The ...
had often used the somewhat foggy concept of the
generic point
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point.
In classical algebraic g ...
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 foundin ...
,
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 var ...
and
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
, motivated in part by the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
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
André Martineau (born 14 May 1930 – 4 May 1972) was a French mathematician, specializing in mathematical analysis.
Martineau studied at the École Normale Supérieure and received there, with Laurent Schwartz as supervisor, his Ph.D. with a the ...
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
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
''X'' of a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
''R'' as the space of
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s of ''R'' with a natural topology (known as the Zariski topology), but augmented it with a
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper ser ...
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
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 of ...
isomorphic to the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
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 n ...
''n''. By definition, A is the spectrum of the polynomial ring ''k''
1,...,''x''''n''">'x''1,...,''x''''n'' In the spirit of scheme theory, affine ''n''-space can in fact be defined over any commutative ring ''R'', meaning Spec(''R''
1,...,''x''''n''">'x''1,...,''x''''n''.
The category of schemes
Schemes form a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
, with morphisms defined as morphisms of locally ringed spaces. (See also:
morphism of schemes In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.
A morphism of algebraic stacks generalizes a ...
.) 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 category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
in the
category of commutative rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is ...
, the category of schemes has Spec(Z) as a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
.
For a scheme ''X'' over a commutative ring ''R'', an ''R''-point of ''X'' means a
section
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sign ...
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 point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
s of ''X''.
More generally, for a scheme ''X'' over a commutative ring ''R'' and any commutative ''R''-
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
''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
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
''E'' of ''k''.) For a scheme ''X'' over ''R'', the assignment ''S'' ↦ ''X''(''S'') is a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
from commutative ''R''-algebras to sets. It is an important observation that a scheme ''X'' over ''R'' is determined by this
functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...
.
The
fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field determin ...
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
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
) 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:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''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
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
. 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
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
over ''R''; this is an algebro-geometric version of compactness. A related observation is that
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
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
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; t ...
''f'' of positive degree in the polynomial ring determines a closed subscheme in projective space P
''n'' over ''R'', called a
projective hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
. In terms of the
Proj construction
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functo ...
, 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. If ...
over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s.
* 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 A
1 − 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 In the mathematical discipline of general topology, Stone–Čech compactification (or ÄŒech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...
of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
s on the positive integers, with the ideal
corresponding to the principal ultrafilter associated to the positive integer ''n''. This topological space is
zero-dimensional
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical ...
, and in particular, each point is an
irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (for ...
. Since affine schemes are
quasi-compact
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
, this is an example of a quasi-compact scheme with infinitely many irreducible components. (By contrast, a
Noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus ...
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.
Arithmetic surfaces
If we consider a polynomial