scheme theory
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, specifically
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, a scheme is a
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
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 solution set, set of solutions of a system of polynomial equations over the real number, ...
in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different schemes) and allowing "varieties" defined over any
commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
(for example, Fermat curves are defined over the
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise '' Éléments de géométrie algébrique'' (EGA); one of its aims was developing the formalism needed to solve deep problems of
algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, such as the Weil conjectures (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 Crafoor ...
). Strongly based on
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, scheme theory allows a systematic use of methods of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, which eventually led to Wiles's proof of Fermat's Last Theorem. Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through the coordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to the ideal of functions which vanish on the subvariety. Intuitively, a scheme is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
consisting of closed points which correspond to geometric points, together with non-closed points which are generic points of irreducible subvarieties. The space is covered by an
atlas An atlas is a collection of maps; it is typically a bundle of world map, maps of Earth or of a continent or region of Earth. Advances in astronomy have also resulted in atlases of the celestial sphere or of other planets. Atlases have traditio ...
of open sets, each endowed with a coordinate ring of regular functions, with specified coordinate changes between the functions over intersecting open sets. Such a structure is called a ringed space or a sheaf of rings. The cases of main interest are the Noetherian schemes, in which the coordinate rings are Noetherian rings. Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of a commutative ring; its points are the prime ideals of the ring, and its closed points are maximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets are rings of fractions. The relative point of view is that much of algebraic geometry should be developed for a morphism of schemes (called a scheme over the base ), 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 . 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.


Development

The origins of algebraic geometry mostly lie in the study of polynomial 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
) that algebraic geometry over the real numbers is simplified by working over the field 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 for ...
s, which has the advantage of being algebraically closed. The early 20th century saw analogies between algebraic geometry and number theory, suggesting the question: can algebraic geometry be developed over other fields, such as those with positive characteristic, and more generally over number rings like the integers, where 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 functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
used to study complex varieties do not seem to apply? Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field : the maximal ideals in the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
are in one-to-one correspondence with the set of -tuples of elements of , and the prime ideals correspond to the irreducible algebraic sets in , known as affine varieties. Motivated by these ideas,
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
and Wolfgang Krull developed commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction, generalizing the study of points (maximal ideals in a polynomial ring) to the study of prime ideals in any commutative ring. For example, Krull defined the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of a commutative ring in terms of prime ideals and, at least when the ring is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...
, he proved that this definition satisfies many of the intuitive properties of geometric 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 they are compact. From the 1920s to the 1940s, B. L. van der Waerden, 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 metric 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 abstract
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 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''. This worked awkwardly: 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, Masayoshi Nagata 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 pursued 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

The theory took its definitive form in Grothendieck's ''Éléments de géométrie algébrique'' (EGA) and the later ''Séminaire de géométrie algébrique'' (SGA), bringing to a conclusion a generation of experimental suggestions and partial developments. Grothendieck defined the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
X of a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), 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 prope ...
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 \mathcal_X(U), which may be thought of as the coordinate ring of regular functions on U. These objects \operatorname(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, most often over the complex numbers. Grothendieck developed a large body of theory for arbitrary schemes extending much of the geometric intuition for varieties. 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. Applying Grothendieck's theory to schemes over the integers and other number fields led to powerful new perspectives in number theory.


Definition

An affine scheme is a locally ringed space isomorphic to the
spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
\operatorname(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 \mathcal_X, which assigns to every open subset U a commutative ring \mathcal_X(U) called the ring of regular functions on U. One can think of a scheme as being covered by "coordinate charts" that 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". The sheaf properties of \mathcal_X(U) mean that its elements'','' which are not necessarily functions, can neverthess be patched together from their restrictions in the same way as functions. 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 the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
n. By definition, A_k^n is the spectrum of the polynomial ring k _1,\dots,x_n/math>. In the spirit of scheme theory, affine n-space can in fact be defined over any commutative ring R, meaning \operatorname(R _1,\dots,x_n.


The category of schemes

Schemes form a category, 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 generali ...
.) 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 Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
''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 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. The fiber product of schemes always exists. That is, for any schemes ''X'' and ''Z'' with morphisms to a scheme ''Y'', the categorical fiber product X\times_Y Z 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 \mathbb^m and \mathbb^n over ''k'' is affine space \mathbb^ over ''k''. Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.


Examples

Here and below, all the rings considered are commutative.


Affine space

Let be an algebraically closed field. The affine space \bar X = \mathbb^n_k is the algebraic variety of all points a=(a_1,\ldots,a_n) with coordinates in ; its coordinate ring is the polynomial ring R = k _1,\ldots,x_n/math>. The corresponding scheme X = \mathrm(R) is a topological space with the Zariski topology, whose closed points are the maximal ideals \mathfrak_a = (x_1-a_1,\ldots,x_n-a_n), the set of polynomials vanishing at a. The scheme also contains a non-closed point for each non-maximal prime ideal \mathfrak\subset R , whose vanishing defines an irreducible subvariety \bar V=\bar V(\mathfrak)\subset \bar X; the topological closure of the scheme point \mathfrak is the subscheme V(\mathfrak)=\, specially including all the closed points of the subvariety, i.e. \mathfrak_a with a\in \bar V, or equivalently \mathfrak\subset\mathfrak_a. The scheme X has a basis of open subsets given by the complements of hypersurfaces, U_f = X\setminus V(f) = \ for irreducible polynomials f\in R. This set is endowed with its coordinate ring of regular functions \mathcal_X(U_f) = R ^= \left\. This induces a unique sheaf \mathcal_X which gives the usual ring of rational functions regular on a given open set U. Each ring element r=r(x_1,\ldots,x_n)\in R, a polynomial function on \bar X, also defines a function on the points of the scheme X whose value at \mathfrak lies in the quotient ring R/\mathfrak, the ''residue ring''. We define r(\mathfrak) as the image of r under the natural map R\to R/\mathfrak. A maximal ideal \mathfrak_a gives the ''residue field'' k(\mathfrak_a)=R/\mathfrak_a\cong k, with the natural isomorphism x_i\mapsto a_i, so that r(\mathfrak_a) corresponds to the original value r(a). The vanishing locus of a polynomial f = f(x_1,\ldots,x_n) is a hypersurface subvariety \bar V(f) \subset \mathbb^n_k, corresponding to the principal ideal (f)\subset R. The corresponding scheme is V(f)=\operatorname(R/(f)), a closed subscheme of affine space. For example, taking to be the complex or real numbers, the equation x^2=y^2(y+1) defines a nodal cubic curve in the affine plane \mathbb^2_k, corresponding to the scheme V = \operatorname k ,y(x^2-y^2(y+1)).


Spec of the integers

The ring of integers \mathbb can be considered as the coordinate ring of the scheme Z = \operatorname( \mathbb ) . The Zariski topology has closed points \mathfrak_p = (p) , the principal ideals of the prime numbers p\in\mathbb; as well as the generic point \mathfrak_0 = (0) , the zero ideal, whose closure is the whole scheme. Closed sets are finite sets, and open sets are their complements, the cofinite sets; any infinite set of points is dense. The basis open set corresponding to the irreducible element p \in \mathbb is U_p = Z\smallsetminus\, with coordinate ring \mathcal_Z (U_p) = \mathbb ^= \. For the open set U = Z\smallsetminus\, this induces \mathcal_Z (U) = \mathbb _1^,\ldots,p_\ell^/math>. A number n\in \mathbb corresponds to a function on the scheme Z, a function whose value at \mathfrak_p lies in the residue field k(\mathfrak_p)=\mathbb/(p) = \mathbb_p, the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
of integers modulo p '':'' the function is defined by n(\mathfrak_p) = n \ \text\ p, and also n(\mathfrak_0)=n in the generic residue ring \mathbb/(0) = \mathbb. The function n is determined by its values at the points \mathfrak_p only, so we can think of n as a kind of "regular function" on the closed points, a very special type among the arbitrary functions f with f(\mathfrak_p)\in \mathbb_p. Note that the point \mathfrak_p is the vanishing locus of the function n=p , the point where the value of p is equal to zero in the residue field. The field of "rational functions" on Z is the fraction field of the generic residue ring, k(\mathfrak_0)=\operatorname(\mathbb) = \mathbb. A fraction a/b has "poles" at the points \mathfrak_p corresponding to prime divisors of the denominator. This also gives a geometric interpretaton of Bezout's lemma stating that if the integers n_1,\ldots, n_r have no common prime factor, then there are integers a_1,\ldots,a_r with a_1 n_1+\cdots + a_r n_r = 1. Geometrically, this is a version of the weak Hilbert Nullstellensatz for the scheme Z: if the functions n_1,\ldots, n_r have no common vanishing points \mathfrak_p in Z, then they generate the unit ideal (n_1,\ldots,n_r) = (1) in the coordinate ring \Z. Indeed, we may consider the terms \rho_i = a_i n_i as forming a kind of partition of unity subordinate to the covering of Z by the open sets U_i = Z\smallsetminus V(n_i).


Affine line over the integers

The affine space \mathbb^1_ = \ is a variety with coordinate ring \mathbb /math>, the polynomials with integer coefficients. The corresponding scheme is Y=\operatorname(\mathbb , whose points are all of the prime ideals \mathfrak\subset \mathbb /math>. The closed points are maximal ideals of the form \mathfrak=(p, f(x)), where p is a prime number, and f(x) is a non-constant polynomial with no integer factor and which is irreducible modulo p . Thus, we may picture Y as two-dimensional, with a "characteristic direction" measured by the coordinate p , and a "spatial direction" with coordinate x . A given prime number p defines a "vertical line", the subscheme V(p) of the prime ideal \mathfrak=(p) : this contains \mathfrak=(p, f(x)) for all f(x), the "characteristic p points" of the scheme. Fixing the x-coordinate, we have the "horizontal line" x=a , the subscheme V(x-a) of the prime ideal \mathfrak=(x-a) . We also have the line V(bx-a) corresponding to the rational coordinate x=a/b , which does not intersect V(p) for those p which divide b . A higher degree "horizontal" subscheme like V(x^2+1) corresponds to x-values which are roots of x^2+1 , namely x=\pm \sqrt . This behaves differently under different p -coordinates. At p=5, we get two points x=\pm 2\ \text\ 5 , since (5,x^2+1)=(5,x-2)\cap(5,x+2) . At p=2, we get one ramified double-point x=1\ \text\ 2 , since (2,x^2+1)=(2,(x-1)^2) . And at p=3, we get that \mathfrak=(3, x^2+1) is a prime ideal corresponding to x=\pm \sqrt in an extension field of \mathbb_3 ; since we cannot distinguish between these values (they are symmetric under the Galois group), we should picture V(3, x^2+1) as two fused points. Overall, V(x^2+1) is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2. The residue field at \mathfrak=(p, f(x)) is k(\mathfrak)=\Z \mathfrak = \mathbb_p (f(x))\cong \mathbb_(\alpha), a field extension of \mathbb_p adjoining a root x=\alpha of f(x) ; this is a finite field with p^d elements, d=\operatorname(f) . A polynomial r(x)\in\Z corresponds to a function on the scheme Y with values r(\mathfrak) = r \ \mathrm\ \mathfrak, that is r(\mathfrak) = r(\alpha)\in \mathbb_p(\alpha) . Again each r(x)\in\Z is determined by its values r(\mathfrak) at closed points; V(p) is the vanishing locus of the constant polynomial r(x)=p; and V(f(x)) contains the points in each characteristic p corresponding to Galois orbits of roots of f(x) in the algebraic closure \overline_p. The scheme Y is not proper, so that pairs of curves may fail to intersect with the expected multiplicity. This is a major obstacle to analyzing Diophantine equations with geometric tools. Arakelov theory overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding to valuations.


Arithmetic surfaces

If we consider a polynomial f \in \mathbb ,y/math> then the affine scheme X = \operatorname(\mathbb ,y(f)) has a canonical morphism to \operatorname\mathbb and is called an arithmetic surface. The fibers X_p = X \times_\operatorname(\mathbb_p) are then algebraic curves over the finite fields \mathbb_p. If f(x,y) = y^2 - x^3 + ax^2 + bx + c is an elliptic curve, then the fibers over its discriminant locus, where \Delta_f = -4a^3c + a^2b^2 + 18abc - 4b^3 - 27c^2 = 0 \ \text\ p,are all singular schemes. For example, if p is a prime number and X = \operatorname \frac then its discriminant is -27p^2. This curve is singular over the prime numbers 3, p.


Non-affine schemes

* For any commutative ring ''R'' and natural number ''n'', projective space \mathbb^n_R 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 \mathbb^n_R is proper over ''R''; this is an algebro-geometric version of compactness. Indeed, complex projective space \C\mathbb^n is a compact space in the classical topology, whereas \C^n is not. * A homogeneous polynomial ''f'' of positive degree in the polynomial ring determines a closed subscheme in projective space \mathbb^n_R, called a projective hypersurface. In terms of the Proj construction, this subscheme can be written as \operatorname R _0,\ldots,x_n(f). For example, the closed subscheme of \mathbb^2_\Q is an elliptic curve 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 (for example, The set of all ...
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 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 X=\mathbb^n\smallsetminus\, say over the complex numbers \C; then ''X'' is not affine for ''n'' ≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme \mathrm\,\C ,x^/math>. To show ''X'' is not affine, one computes that every regular function on ''X'' extends to a regular function on \mathbb^n when ''n'' ≥ 2: this is analogous to Hartogs's lemma in complex analysis, though easier to prove. That is, the inclusion f:X\to\mathbb^n induces an isomorphism from O(\mathbb^n)=\C _1,\ldots,x_n to O(X). If ''X'' were affine, it would follow that ''f'' is 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 \operatorname\left(\prod_^\infty k\right) 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 \prod_ k 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 non-Noetherian 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.


Motivation for schemes

Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance. *Field extensions. Given some polynomial equations in ''n'' variables over a field ''k'', one can study the set ''X''(''k'') of solutions of the equations in the product set ''k''''n''. If the field ''k'' is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as ''X''(''k''): define the Zariski topology on ''X''(''k''), consider polynomial mappings between different sets of this type, and so on. But if ''k'' is not algebraically closed, then the set ''X''(''k'') is not rich enough. Indeed, one can study the solutions ''X''(''E'') of the given equations in any field extension ''E'' of ''k'', but these sets are not determined by ''X''(''k'') in any reasonable sense. For example, the plane curve ''X'' over the real numbers defined by ''x''2 + ''y''2 = −1 has ''X''(R) empty, but ''X''(C) not empty. (In fact, ''X''(C) can be identified with C − 0.) By contrast, a scheme ''X'' over a field ''k'' has enough information to determine the set ''X''(''E'') of ''E''-rational points for every extension field ''E'' of ''k''. (In particular, the closed subscheme of A defined by ''x''2 + ''y''2 = −1 is a nonempty topological space.) *Generic point. The points of the affine line A, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(C(''x'')) → A, where C(''x'') is the field of rational functions in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example. *Let ''X'' be the plane curve ''y''2 = ''x''(''x''−1)(''x''−5) over the complex numbers. This is a closed subscheme of A. It can be viewed as a ramified double cover of the affine line A by projecting to the ''x''-coordinate. The fiber of the morphism ''X'' → A1 over the generic point of A1 is exactly the generic point of ''X'', yielding the morphism \operatorname \mathbf(x) \left (\sqrt \right )\to \operatorname\mathbf(x). This in turn is equivalent to the degree-2 extension of fields \mathbf(x) \subset \mathbf(x) \left (\sqrt \right ). Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of function fields. This generalizes to a relation between the fundamental group (which classifies
covering space In topology, a covering or covering projection is a continuous function, map between topological spaces that, intuitively, Local property, locally acts like a Projection (mathematics), projection of multiple copies of a space onto itself. In par ...
s in topology) and the Galois group (which classifies certain field extensions). Indeed, Grothendieck's theory of the étale fundamental group treats the fundamental group and the Galois group on the same footing. *Nilpotent elements. Let ''X'' be the closed subscheme of the affine line A defined by ''x''2 = 0, sometimes called a fat point. The ring of regular functions on ''X'' is C 'x''(''x''2); in particular, the regular function ''x'' on ''X'' is nilpotent but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to ''X'' if and only if they have the same value ''and first
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
'' at the origin. Allowing such non- reduced schemes brings the ideas of
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
and infinitesimals into algebraic geometry. *Nilpotent elements arise naturally in intersection theory. For example in the plane \mathbb^2_k over a field k, with coordinate ring k ,y/math>, consider the ''x-''axis, which is the variety V(y), and the parabola y=x^2, which is V(x^2-y). Their scheme-theoretic intersection is defined by the ideal (y)+(x^2-y)=(x^2,\, y). Since the intersection is not transverse, this is not merely the point (x,y) = (0,0) defined by the ideal (x,y)\subset k ,y/math>, but rather a fat point containing the ''x-''axis tangent direction (the common tangent of the two curves) and having coordinate ring:\frac \cong \frac.The intersection multiplicity of 2 is defined as the
length Length is a measure of distance. In the International System of Quantities, length is a quantity with Dimension (physical quantity), dimension distance. In most systems of measurement a Base unit (measurement), base unit for length is chosen, ...
of this k ,y/math>-module, i.e. its dimension as a k-vector space. *For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a smooth complex variety ''Y''. Such a subscheme consists of either two distinct complex points of ''Y'', or else a subscheme isomorphic to ''X'' = Spec C 'x''(''x''2) as in the previous paragraph. Subschemes of the latter type are determined by a complex point ''y'' of ''Y'' together with a line in the tangent space T''y''''Y''. This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.


Coherent sheaves

A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic)
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
s. For a scheme ''X'', one starts by considering the abelian category of ''O''''X''-modules, which are sheaves of abelian groups on ''X'' that form a module over the sheaf of regular functions ''O''''X''. In particular, a module ''M'' over a commutative ring ''R'' determines an associated ''O''''X''-module on ''X'' = Spec(''R''). A quasi-coherent sheaf on a scheme ''X'' means an ''O''''X''-module that is the sheaf associated to a module on each affine open subset of ''X''. Finally, a coherent sheaf (on a Noetherian scheme ''X'', say) is an ''O''''X''-module that is the sheaf associated to a finitely generated module on each affine open subset of ''X''. Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the
tangent bundle A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme ''Y'' of ''X'' can be viewed as a coherent sheaf on ''X'' that is zero outside ''Y'' (by the direct image construction). In this way, coherent sheaves on a scheme ''X'' include information about all closed subschemes of ''X''. Moreover, sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of coherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.


Generalizations

Considered as its functor of points, a scheme is a functor that is a sheaf of sets for the Zariski topology on the category of commutative rings, and that, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor that is a sheaf in the étale topology and that, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.. A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group ''G'' on an algebraic variety ''X'' determines a quotient stack 'X''/''G'' which remembers the stabilizer subgroups for the action of ''G''. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified. Grothendieck originally introduced stacks as a tool for the theory of descent. In that formulation, stacks are (informally speaking) sheaves of categories. From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include Deligne–Mumford stacks (similar to orbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The Keel–Mori theorem says that an algebraic stack with finite stabilizer groups has a coarse moduli space that is an algebraic space. Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of E-infinity ring spectra). These sheaves admit algebraic operations that are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory that can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
and the
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between object (category theory), objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applicati ...
on modules.


See also

* Flat morphism, Smooth morphism,
Proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k ...
, Finite morphism, Étale morphism * Stable curve * Birational geometry * Étale cohomology,
Chow group In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ...
, Hodge theory * Group scheme, Abelian variety, Linear algebraic group, Reductive group * Moduli of algebraic curves * Gluing schemes


Citations


References

* * * * * * * * * *


External links

*David Mumford
Can one explain schemes to biologists?
* *https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ – the comment section contains some interesting discussion on scheme theory (including posts from Terence Tao). {{Authority control