Inmathematics, specificallyalgebraic geometry, ascheme is astructure that enlarges the notion of analgebraic variety in several ways, such as taking account ofmultiplicities (the equationsx = 0 andx² = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over anycommutative ring (for example,Fermat curves are defined over theintegers).

Scheme theory was introduced byAlexander Grothendieck in 1960 in his treatiseÉléments de géométrie algébrique (EGA); one of its aims was developing the formalism needed to solve deep problems ofalgebraic geometry, such as theWeil conjectures (the last of which was proved byPierre Deligne).^[1] Strongly based oncommutative algebra, scheme theory allows a systematic use of methods oftopology andhomological algebra. Scheme theory also unifies algebraic geometry with much ofnumber theory, which eventually led toWiles's proof of Fermat's Last Theorem.

Schemes elaborate the fundamental idea that an algebraic variety is best analyzed through thecoordinate ring of regular algebraic functions defined on it (or on its subsets), and each subvariety corresponds to theideal of functions which vanish on the subvariety. Intuitively, a scheme is atopological space consisting of closed points which correspond to geometric points, together with non-closed points which aregeneric points of irreducible subvarieties. The space is covered by anatlas 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 aringed space or asheaf of rings. The cases of main interest are theNoetherian schemes, in which the coordinate rings areNoetherian rings.

Formally, a scheme is a ringed space covered by affine schemes. An affine scheme is thespectrum of a commutative ring; its points are theprime ideals of the ring, and its closed points aremaximal ideals. The coordinate ring of an affine scheme is the ring itself, and the coordinate rings of open subsets arerings of fractions.

Therelative point of view is that much of algebraic geometry should be developed for a morphismX →Y of schemes (called a schemeXover the baseY ), rather than for an individual scheme. For example, in studyingalgebraic surfaces, it can be useful to consider families of algebraic surfaces over any schemeY. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as amoduli space.

For some of the detailed definitions in the theory of schemes, see theglossary of scheme theory.

The origins of algebraic geometry mostly lie in the study ofpolynomial equations over thereal numbers. By the 19th century, it became clear (notably in the work ofJean-Victor Poncelet andBernhard Riemann) that algebraic geometry over the real numbers is simplified by working over thefield ofcomplex numbers, which has the advantage of beingalgebraically closed.^[2] 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 positivecharacteristic, and more generally overnumber rings like the integers, where the tools of topology andcomplex analysis used to study complex varieties do not seem to apply?

Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed fieldk : themaximal ideals in thepolynomial ringk[x₁, ... ,x_n] are in one-to-one correspondence with the setkⁿ ofn-tuples of elements ofk, and theprime ideals correspond to the irreducible algebraic sets inkⁿ, known as affine varieties. Motivated by these ideas,Emmy Noether andWolfgang Krull developed commutative algebra in the 1920s and 1930s.^[3] 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 thedimension of a commutative ring in terms of prime ideals and, at least when the ring isNoetherian, 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 toaffine algebraic varieties. However, many arguments in algebraic geometry work better forprojective varieties, essentially because they arecompact. From the 1920s to the 1940s,B. L. van der Waerden,André Weil andOscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (orquasi-projective) varieties.^[4] In particular, theZariski 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 themetric 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 anabstract variety (not embedded inprojective space), by gluing affine varieties along open subsets, on the model of abstractmanifolds in topology. He needed this generality for his construction of theJacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil,Chow andMatsusaka.)

The algebraic geometers of theItalian school had often used the somewhat foggy concept of thegeneric point of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil'sFoundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called auniversal domain.^[4] 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 andJean-Pierre Serre, motivated in part by theWeil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The wordscheme was first used in the 1956 Chevalley Seminar, in which Chevalley pursued Zariski's ideas.^[5] According toPierre Cartier, it wasAndré Martineau who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.^[6]

The theory took its definitive form in Grothendieck'sÉléments de géométrie algébrique (EGA) and the laterSéminaire de géométrie algébrique (SGA), bringing to a conclusion a generation of experimental suggestions and partial developments.^[7] Grothendieck defined thespectrum${\displaystyle X}$ of acommutative ring${\displaystyle R}$ as the space ofprime ideals of${\displaystyle R}$ with a natural topology (known as the Zariski topology), but augmented it with asheaf of rings: to every open subset${\displaystyle U}$ he assigned a commutative ring${\displaystyle {\mathcal{O}}_X(U)}$, which may be thought of as the coordinate ring of regular functions on${\displaystyle U}$. These objects${\displaystyle \mathrm{Spec}(R)}$ 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${\displaystyle k}$, 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.

Anaffine scheme is alocally ringed space isomorphic to thespectrum${\displaystyle \mathrm{Spec}(R)}$ of a commutative ring${\displaystyle R}$. Ascheme is a locally ringed space${\displaystyle X}$ admitting a covering by open sets${\displaystyle U_i}$, such that each${\displaystyle U_i}$ (as a locally ringed space) is an affine scheme.^[8] In particular,${\displaystyle X}$ comes with a sheaf${\displaystyle {\mathcal{O}}_X}$, which assigns to every open subset${\displaystyle U}$ a commutative ring${\displaystyle {\mathcal{O}}_X(U)}$ called thering of regular functions on${\displaystyle U}$. 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 aprescheme, and a scheme was defined to be aseparated 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" andMumford's "Red Book".^[9] The sheaf properties of${\displaystyle {\mathcal{O}}_X(U)}$ mean that its elements, which are not necessarily functions, can nevertheless be patched together from their restrictions in the same way as functions.

A basic example of an affine scheme isaffine${\displaystyle n}$-space over a field${\displaystyle k}$, for anatural number${\displaystyle n}$. By definition,${\displaystyle A_k^n}$ is the spectrum of the polynomial ring${\displaystyle k[x_1,\dots ,x_n]}$. In the spirit of scheme theory, affine${\displaystyle n}$-space can in fact be defined over any commutative ring${\displaystyle R}$, meaning${\displaystyle \mathrm{Spec}(R[x_1,\dots ,x_n])}$.

Schemes form acategory, with morphisms defined as morphisms of locally ringed spaces. (See also:morphism of schemes.) For a schemeY, a schemeXoverY (or aY-scheme) means a morphismX →Y of schemes. A schemeXover a commutative ringR means a morphismX → Spec(R).

An algebraic variety over a fieldk can be defined as a scheme overk with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that avariety overk means anintegral separated scheme offinite type overk.^[10]

A morphismf:X →Y of schemes determines apullback 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 homomorphismsB →A.^[11] In this sense, scheme theory completely subsumes the theory of commutative rings.

SinceZ is aninitial object in thecategory of commutative rings, the category of schemes has Spec(Z) as aterminal object.

For a schemeX over a commutative ringR, anR-point ofX means asection of the morphismX → Spec(R). One writesX(R) for the set ofR-points ofX. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations ofX with values inR. WhenR is a fieldk,X(k) is also called the set ofk-rational points ofX.

More generally, for a schemeX over a commutative ringR and any commutativeR-algebraS, anS-point ofX means a morphism Spec(S) →X overR. One writesX(S) for the set ofS-points ofX. (This generalizes the old observation that given some equations over a fieldk, one can consider the set of solutions of the equations in anyfield extensionE ofk.) For a schemeX overR, the assignmentS ↦X(S) is afunctor from commutativeR-algebras to sets. It is an important observation that a schemeX overR is determined by thisfunctor of points.^[12]

Thefiber product of schemes always exists. That is, for any schemesX andZ with morphisms to a schemeY, thecategorical fiber product${\displaystyle X{\times }_{Y}Z}$ exists in the category of schemes. IfX andZ are schemes over a fieldk, their fiber product over Spec(k) may be called theproductX ×Z in the category ofk-schemes. For example, the product of affine spaces${\displaystyle {\mathbb{A}}^m}$ and${\displaystyle {\mathbb{A}}^n}$ overk is affine space${\displaystyle {\mathbb{A}}^{m+n}}$ overk.

Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finitelimits.

Here and below, all the rings considered are commutative.

Letk be an algebraically closed field. The affine space${\displaystyle \overline{X}={\mathbb{A}}_k^n}$ is the algebraic variety of all points${\displaystyle a=(a_1,\dots ,a_n)}$ with coordinates ink; its coordinate ring is the polynomial ring${\displaystyle R=k[x_1,\dots ,x_n]}$. The corresponding scheme${\displaystyle X=\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)}$ is a topological space with the Zariski topology, whose closed points are the maximal ideals${\displaystyle {\mathfrak{m}}_a=(x_1-a_1,\dots ,x_n-a_n)}$, the set of polynomials vanishing at${\displaystyle a}$. The scheme also contains a non-closed point for each non-maximal prime ideal${\displaystyle \mathfrak{p}\subset R}$, whose vanishing defines an irreducible subvariety${\displaystyle \overline{V}=\overline{V}(\mathfrak{p})\subset \overline{X}}$; the topological closure of the scheme point${\displaystyle \mathfrak{p}}$ is the subscheme${\displaystyle V(\mathfrak{p})=\{\mathfrak{q}\in X\text{with}\mathfrak{p}\subset \mathfrak{q}\}}$, specially including all the closed points of the subvariety, i.e.${\displaystyle {\mathfrak{m}}_a}$ with${\displaystyle a\in \overline{V}}$, or equivalently${\displaystyle \mathfrak{p}\subset {\mathfrak{m}}_a}$.

The scheme${\displaystyle X}$ has a basis of open subsets given by the complements of hypersurfaces,$${\displaystyle U_f=X\setminus V(f)=\{\mathfrak{p}\in X\text{with}f\notin \mathfrak{p}\}}$$for irreducible polynomials${\displaystyle f\in R}$. This set is endowed with its coordinate ring of regular functions$${\displaystyle {\mathcal{O}}_X(U_f)=R[f^{-1}]=\left\{\frac{r}{f^m}\text{for}r\in R,m\in {\mathbb{Z}}_{\ge 0}\right\}.}$$This induces a unique sheaf${\displaystyle {\mathcal{O}}_X}$ which gives the usual ring of rational functions regular on a given open set${\displaystyle U}$.

Each ring element${\displaystyle r=r(x_1,\dots ,x_n)\in R}$, a polynomial function on${\displaystyle \overline{X}}$, also defines a function on the points of the scheme${\displaystyle X}$ whose value at${\displaystyle \mathfrak{p}}$ lies in the quotient ring${\displaystyle R/\mathfrak{p}}$, theresidue ring. We define${\displaystyle r(\mathfrak{p})}$ as the image of${\displaystyle r}$ under the natural map${\displaystyle R\to R/\mathfrak{p}}$. A maximal ideal${\displaystyle {\mathfrak{m}}_a}$ gives theresidue field${\displaystyle k({\mathfrak{m}}_a)=R/{\mathfrak{m}}_a\cong k}$, with the natural isomorphism${\displaystyle x_i\mapsto a_i}$, so that${\displaystyle r({\mathfrak{m}}_a)}$ corresponds to the original value${\displaystyle r(a)}$.

The vanishing locus of a polynomial${\displaystyle f=f(x_1,\dots ,x_n)}$ is ahypersurface subvariety${\displaystyle \overline{V}(f)\subset {\mathbb{A}}_k^n}$, corresponding to theprincipal ideal${\displaystyle (f)\subset R}$. The corresponding scheme is$V(f)=\mathrm{Spec}(R/(f))$, a closed subscheme of affine space. For example, takingk to be the complex or real numbers, the equation${\displaystyle x^2=y^2(y+1)}$ defines anodal cubic curve in the affine plane${\displaystyle {\mathbb{A}}_k^2}$, corresponding to the scheme${\displaystyle V=\mathrm{Spec}k[x,y]/(x^2-y^2(y+1))}$.

The ring of integers${\displaystyle \mathbb{Z}}$ can be considered as the coordinate ring of the scheme${\displaystyle Z=\mathrm{Spec}(\mathbb{Z})}$. The Zariski topology has closed points${\displaystyle {\mathfrak{m}}_p=(p)}$, the principal ideals of the prime numbers${\displaystyle p\in \mathbb{Z}}$; as well as the generic point${\displaystyle {\mathfrak{p}}_0=(0)}$, the zero ideal, whoseclosure 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${\displaystyle p\in \mathbb{Z}}$ is${\displaystyle U_p=Z\setminus \{{\mathfrak{m}}_p\}}$, with coordinate ring${\displaystyle {\mathcal{O}}_Z(U_p)=\mathbb{Z}[p^{-1}]=\{\frac{n}{p^m}\text{for}n\in \mathbb{Z},m\ge 0\}}$. For the open set${\displaystyle U=Z\setminus \{{\mathfrak{m}}_{p_1},\dots ,{\mathfrak{m}}_{p_{\ell }}\}}$, this induces${\displaystyle {\mathcal{O}}_Z(U)=\mathbb{Z}[p_1^{-1},\dots ,p_{\ell }^{-1}]}$.

A number${\displaystyle n\in \mathbb{Z}}$ corresponds to a function on the scheme${\displaystyle Z}$, a function whose value at${\displaystyle {\mathfrak{m}}_p}$ lies in the residue field${\displaystyle k({\mathfrak{m}}_p)=\mathbb{Z}/(p)={\mathbb{F}}_p}$, thefinite field of integers modulo${\displaystyle p}$: the function is defined by${\displaystyle n({\mathfrak{m}}_p)=n\text{mod}p}$, and also${\displaystyle n({\mathfrak{p}}_0)=n}$ in the generic residue ring${\displaystyle \mathbb{Z}/(0)=\mathbb{Z}}$. The function${\displaystyle n}$ is determined by its values at the points${\displaystyle {\mathfrak{m}}_p}$ only, so we can think of${\displaystyle n}$ as a kind of "regular function" on the closed points, a very special type among the arbitrary functions${\displaystyle f}$ with${\displaystyle f({\mathfrak{m}}_p)\in {\mathbb{F}}_p}$.

Note that the point${\displaystyle {\mathfrak{m}}_p}$ is the vanishing locus of the function${\displaystyle n=p}$, the point where the value of${\displaystyle p}$ is equal to zero in the residue field. The field of "rational functions" on${\displaystyle Z}$ is the fraction field of the generic residue ring,${\displaystyle k({\mathfrak{p}}_0)=\mathrm{Frac}(\mathbb{Z})=\mathbb{Q}}$. A fraction${\displaystyle a/b}$ has "poles" at the points${\displaystyle {\mathfrak{m}}_p}$ corresponding to prime divisors of the denominator.

This also gives a geometric interpretation ofBezout's lemma stating that if the integers${\displaystyle n_1,\dots ,n_r}$ have no common prime factor, then there are integers${\displaystyle a_1,\dots ,a_r}$ with${\displaystyle a_1{n}_1+\cdots +a_r{n}_r=1}$. Geometrically, this is a version of the weakHilbert Nullstellensatz for the scheme${\displaystyle Z}$: if the functions${\displaystyle n_1,\dots ,n_r}$ have no common vanishing points${\displaystyle {\mathfrak{m}}_p}$ in${\displaystyle Z}$, then they generate the unit ideal${\displaystyle (n_1,\dots ,n_r)=(1)}$ in the coordinate ring${\displaystyle \mathbb{Z}}$. Indeed, we may consider the terms${\displaystyle {\rho }_i=a_i{n}_i}$ as forming a kind ofpartition of unity subordinate to the covering of${\displaystyle Z}$ by the open sets${\displaystyle U_i=Z\setminus V(n_i)}$.

The affine space${\displaystyle {\mathbb{A}}_{\mathbb{Z}}^1=\{a\text{for}a\in \mathbb{Z}\}}$ is a variety with coordinate ring${\displaystyle \mathbb{Z}[x]}$, the polynomials with integer coefficients. The corresponding scheme is${\displaystyle Y=\mathrm{Spec}(\mathbb{Z}[x])}$, whose points are all of the prime ideals${\displaystyle \mathfrak{p}\subset \mathbb{Z}[x]}$. The closed points are maximal ideals of the form${\displaystyle \mathfrak{m}=(p,f(x))}$, where${\displaystyle p}$ is a prime number, and${\displaystyle f(x)}$ is a non-constant polynomial with no integer factor and which is irreducible modulo${\displaystyle p}$. Thus, we may picture${\displaystyle Y}$ as two-dimensional, with a "characteristic direction" measured by the coordinate${\displaystyle p}$, and a "spatial direction" with coordinate${\displaystyle x}$.

A given prime number${\displaystyle p}$ defines a "vertical line", the subscheme${\displaystyle V(p)}$ of the prime ideal${\displaystyle \mathfrak{p}=(p)}$: this contains${\displaystyle \mathfrak{m}=(p,f(x))}$ for all${\displaystyle f(x)}$, the "characteristic${\displaystyle p}$ points" of the scheme. Fixing the${\displaystyle x}$-coordinate, we have the "horizontal line"${\displaystyle x=a}$, the subscheme${\displaystyle V(x-a)}$ of the prime ideal${\displaystyle \mathfrak{p}=(x-a)}$. We also have the line${\displaystyle V(bx-a)}$ corresponding to the rational coordinate${\displaystyle x=a/b}$, which does not intersect${\displaystyle V(p)}$ for those${\displaystyle p}$ which divide${\displaystyle b}$.

A higher degree "horizontal" subscheme like${\displaystyle V(x^2+1)}$ corresponds to${\displaystyle x}$-values which are roots of${\displaystyle x^2+1}$, namely${\displaystyle x=\pm \sqrt{-1}}$. This behaves differently under different${\displaystyle p}$-coordinates. At${\displaystyle p=5}$, we get two points${\displaystyle x=\pm 2\text{mod}5}$, since${\displaystyle (5,x^2+1)=(5,x-2)\cap (5,x+2)}$. At${\displaystyle p=2}$, we get oneramified double-point${\displaystyle x=1\text{mod}2}$, since${\displaystyle (2,x^2+1)=(2,(x-1{)}^2)}$. And at${\displaystyle p=3}$, we get that${\displaystyle \mathfrak{m}=(3,x^2+1)}$ is a prime ideal corresponding to${\displaystyle x=\pm \sqrt{-1}}$ in an extension field of${\displaystyle {\mathbb{F}}_3}$; since we cannot distinguish between these values (they are symmetric under theGalois group), we should picture${\displaystyle V(3,x^2+1)}$ as two fused points. Overall,${\displaystyle V(x^2+1)}$ is a kind of fusion of two Galois-symmetric horizonal lines, a curve of degree 2.

The residue field at${\displaystyle \mathfrak{m}=(p,f(x))}$ is${\displaystyle k(\mathfrak{m})=\mathbb{Z}[x]/\mathfrak{m}={\mathbb{F}}_p[x]/(f(x))\cong {\mathbb{F}}_p(\alpha )}$, a field extension of${\displaystyle {\mathbb{F}}_p}$ adjoining a root${\displaystyle x=\alpha }$ of${\displaystyle f(x)}$; this is a finite field with${\displaystyle p^d}$elements,${\displaystyle d=\mathrm{deg}(f)}$. A polynomial${\displaystyle r(x)\in \mathbb{Z}[x]}$ corresponds to a function on the scheme${\displaystyle Y}$ with values${\displaystyle r(\mathfrak{m})=r\mathrm{m}\mathrm{o}\mathrm{d}\mathfrak{m}}$, that is${\displaystyle r(\mathfrak{m})=r(\alpha )\in {\mathbb{F}}_p(\alpha )}$. Again each${\displaystyle r(x)\in \mathbb{Z}[x]}$ is determined by its values${\displaystyle r(\mathfrak{m})}$ at closed points;${\displaystyle V(p)}$ is the vanishing locus of the constant polynomial${\displaystyle r(x)=p}$; and${\displaystyle V(f(x))}$ contains the points in each characteristic${\displaystyle p}$ corresponding to Galois orbits of roots of${\displaystyle f(x)}$ in the algebraic closure${\displaystyle {\overline{\mathbb{F}}}_p}$.

The scheme${\displaystyle Y}$ is notproper, so that pairs of curves may fail tointersect with the expected multiplicity. This is a major obstacle to analyzingDiophantine equations withgeometric tools.Arakelov theory overcomes this obstacle by compactifying affine arithmetic schemes, adding points at infinity corresponding tovaluations.

If we consider a polynomial${\displaystyle f\in \mathbb{Z}[x,y]}$ then the affine scheme${\displaystyle X=\mathrm{Spec}(\mathbb{Z}[x,y]/(f))}$ has a canonical morphism to${\displaystyle \mathrm{Spec}\mathbb{Z}}$ and is called anarithmetic surface. The fibers${\displaystyle X_p=X{\times }_{\mathrm{Spec}(\mathbb{Z})}\mathrm{Spec}({\mathbb{F}}_p)}$ are then algebraic curves over the finite fields${\displaystyle {\mathbb{F}}_p}$. If${\displaystyle f(x,y)=y^2-x^3+a{x}^2+bx+c}$ is anelliptic curve, then the fibers over its discriminant locus, where$${\displaystyle {\mathrm{\Delta }}_f=-4a^{3}c+a^2{b}^2+18abc-4b^3-27c^2=0\text{mod}p,}$$are all singular schemes.^[13] For example, if${\displaystyle p}$ is a prime number and$${\displaystyle X=\mathrm{Spec}\frac{\mathbb{Z}[x,y]}{(y^2-x^3-p)}}$$ then its discriminant is${\displaystyle -27p^2}$. This curve is singular over the prime numbers${\displaystyle 3,p}$.

• For any commutative ringR and natural numbern,projective space${\displaystyle {\mathbb{P}}_R^n}$ can be constructed as a scheme by gluingn + 1 copies of affinen-space overR 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${\displaystyle {\mathbb{P}}_R^n}$ isproper overR; this is an algebro-geometric version of compactness. Indeed,complex projective space${\displaystyle \mathbb{C}{\mathbb{P}}^n}$ is a compact space in the classical topology, whereas${\displaystyle {\mathbb{C}}^n}$ is not.
• Ahomogeneous polynomialf of positive degree in the polynomial ringR[x₀, ...,x_n] determines a closed subschemef = 0 in projective space${\displaystyle {\mathbb{P}}_R^n}$, called aprojective hypersurface. In terms of theProj construction, this subscheme can be written as$${\displaystyle \mathrm{Proj}R[x_0,\dots ,x_n]/(f).}$$ For example, the closed subschemex³ +y³ =z³ of${\displaystyle {\mathbb{P}}_{\mathbb{Q}}^2}$ is anelliptic curve over therational numbers.
• Theline with two origins (over a fieldk) is the scheme defined by starting with two copies of the affine line overk, 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.^[14]
• A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let${\displaystyle X={\mathbb{A}}^n\setminus \{0\}}$, say over the complex numbers${\displaystyle \mathbb{C}}$; thenX is not affine forn ≥ 2. (However, the affine line minus the origin is isomorphic to the affine scheme${\displaystyle \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{C}[x,x^{-1}]}$. To showX is not affine, one computes that every regular function onX extends to a regular function on${\displaystyle {\mathbb{A}}^n}$ whenn ≥ 2: this is analogous toHartogs's lemma in complex analysis, though easier to prove. That is, the inclusion${\displaystyle f:X\to {\mathbb{A}}^n}$ induces an isomorphism from${\displaystyle O({\mathbb{A}}^n)=\mathbb{C}[x_1,\dots ,x_n]}$ to${\displaystyle O(X)}$. IfX were affine, it would follow thatf is an isomorphism, butf is not surjective and hence not an isomorphism. Therefore, the schemeX is not affine.^[15]
• Letk be a field. Then the scheme$\mathrm{Spec}\left(\prod _{n=1}^{\mathrm{\infty }}k\right)$ is an affine scheme whose underlying topological space is theStone–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 theultrafilters on the positive integers, with the ideal$\prod _{m\ne n}k$ corresponding to the principal ultrafilter associated to the positive integern.^[16] This topological space iszero-dimensional, and in particular, each point is anirreducible component. Since affine schemes arequasi-compact, this is an example of a non-Noetherian quasi-compact scheme with infinitely many irreducible components. (By contrast, aNoetherian scheme has only finitely many irreducible components.)

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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.

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 inn variables over a fieldk, one can study the setX(k) of solutions of the equations in the product setkⁿ. If the fieldk is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such asX(k): define the Zariski topology onX(k), consider polynomial mappings between different sets of this type, and so on. But ifk is not algebraically closed, then the setX(k) is not rich enough. Indeed, one can study the solutionsX(E) of the given equations in any field extensionE ofk, but these sets are not determined byX(k) in any reasonable sense. For example, the plane curveX over the real numbers defined byx² +y² = −1 hasX(R) empty, butX(C) not empty. (In fact,X(C) can be identified withC − 0.) By contrast, a schemeX over a fieldk has enough information to determine the setX(E) ofE-rational points for every extension fieldE ofk. (In particular, the closed subscheme of A²
  _R defined byx² +y² = −1 is a nonempty topological space.)
• Generic point. The points of the affine line A¹
  _C, 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)) → A¹
  _C, whereC(x) is the field ofrational functions in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
• LetX be the plane curvey² =x(x−1)(x−5) over the complex numbers. This is a closed subscheme of A²
  _C. It can be viewed as aramified double cover of the affine line A¹
  _C by projecting to thex-coordinate. The fiber of the morphismX → A¹ over the generic point of A¹ is exactly the generic point ofX, yielding the morphism$${\displaystyle \mathrm{Spec}\mathbf{C}(x)\left(\sqrt{x(x-1)(x-5)}\right)\to \mathrm{Spec}\mathbf{C}(x).}$$ This in turn is equivalent to thedegree-2 extension of fields$${\displaystyle \mathbf{C}(x)\subset \mathbf{C}(x)\left(\sqrt{x(x-1)(x-5)}\right).}$$ 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 offunction fields. This generalizes to a relation between thefundamental group (which classifiescovering spaces in topology) and theGalois group (which classifies certainfield extensions). Indeed, Grothendieck's theory of theétale fundamental group treats the fundamental group and the Galois group on the same footing.

• Nilpotent elements. LetX be the closed subscheme of the affine line A¹
  _C defined byx² = 0, sometimes called afat point. The ring of regular functions onX isC[x]/(x²); in particular, the regular functionx onX isnilpotent but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction toX if and only if they have the same valueand firstderivative at the origin. Allowing such non-reduced schemes brings the ideas ofcalculus andinfinitesimals into algebraic geometry.
• Nilpotent elements arise naturally inintersection theory. For example in the plane${\displaystyle {\mathbb{A}}_k^2}$ over a field${\displaystyle k}$, with coordinate ring${\displaystyle k[x,y]}$, consider thex-axis, which is the variety${\displaystyle V(y)}$, and the parabola${\displaystyle y=x^2}$, which is${\displaystyle V(x^2-y)}$. Their scheme-theoretic intersection is defined by the ideal${\displaystyle (y)+(x^2-y)=(x^2,y)}$. Since the intersection is nottransverse, this is not merely the point${\displaystyle (x,y)=(0,0)}$ defined by the ideal${\displaystyle (x,y)\subset k[x,y]}$, but rather a fat point containing thex-axis tangent direction (the common tangent of the two curves) and having coordinate ring:$${\displaystyle \frac{k[x,y]}{(x^2,y)}\cong \frac{k[x]}{(x^2)}.}$$Theintersection multiplicity of 2 is defined as thelength of this${\displaystyle k[x,y]}$-module, i.e. its dimension as a${\displaystyle k}$-vector space.
• For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in asmooth complex varietyY. Such a subscheme consists of either two distinct complex points ofY, or else a subscheme isomorphic toX = SpecC[x]/(x²) as in the previous paragraph. Subschemes of the latter type are determined by a complex pointy ofY together with a line in thetangent space T_yY.^[17] This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.

A central part of scheme theory is the notion ofcoherent sheaves, generalizing the notion of (algebraic)vector bundles. For a schemeX, one starts by considering theabelian category ofO_X-modules, which are sheaves of abelian groups onX that form amodule over the sheaf of regular functionsO_X. In particular, a moduleM over a commutative ringR determines anassociatedO_X-module~M onX = Spec(R). Aquasi-coherent sheaf on a schemeX means anO_X-module that is the sheaf associated to a module on each affine open subset ofX. Finally, acoherent sheaf (on a Noetherian schemeX, say) is anO_X-module that is the sheaf associated to afinitely generated module on each affine open subset ofX.

Coherent sheaves include the important class ofvector bundles, which are the sheaves that locally come from finitely generatedfree modules. An example is thetangent bundle of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subschemeY ofX can be viewed as a coherent sheaf onX that is zero outsideY (by thedirect image construction). In this way, coherent sheaves on a schemeX include information about all closed subschemes ofX. Moreover,sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves. The resulting theory ofcoherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.^[18][19]

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 theétale topology.Michael Artin defined analgebraic 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, theArtin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.^[20]

A further generalization is the idea of astack. Crudely speaking,algebraic stacks generalize algebraic spaces by having analgebraic group attached to each point, which is viewed as the automorphism group of that point. For example, anyaction of an algebraic groupG on an algebraic varietyX determines aquotient stack [X/G], which remembers thestabilizer subgroups for the action ofG. 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 ofdescent. In that formulation, stacks are (informally speaking) sheaves of categories.^[21] From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These includeDeligne–Mumford stacks (similar toorbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. TheKeel–Mori theorem says that an algebraic stack with finite stabilizer groups has acoarse moduli space that is an algebraic space.

Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer tohomotopy theory. In this setting, known asderived 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 ofE-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 thatderived functors in homological algebra yield higher information about operations such astensor product and theHom functor on modules.

• Flat morphism,Smooth morphism,Proper morphism,Finite morphism,Étale morphism
• Stable curve
• Birational geometry
• Étale cohomology,Chow group,Hodge theory
• Group scheme,Abelian variety,Linear algebraic group,Reductive group
• Moduli of algebraic curves
• Gluing schemes

• David Mumford,Can one explain schemes to biologists?
• The Stacks Project Authors,The Stacks Project
• https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ – the comment section contains some interesting discussion on scheme theory (including posts fromTerence Tao).

• Scheme theory

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