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In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern algebraic geometry. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Technically, a scheme is a topological space together with commutative rings for all its open sets, which arises from "gluing together" spectra (spaces of prime ideals) of commutative rings. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers

Contents

History and motivation

The algebraic geometers of the Italian school had often used the somewhat foggy concept of "generic point" when proving statements about algebraic varieties. In Mathematics, in the fields of General topology and particularly of Algebraic geometry, a generic point P of a Topological space This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety What is true for the generic point is true for all points of the variety except a small number of special points. In the 1920s, Emmy Noether had first suggested a way to clarify the concept: start with the coordinate ring of the variety (the ring of all polynomial functions defined on the variety); the maximal ideals of this ring will correspond to ordinary points of the variety (under suitable conditions), and the non-maximal prime ideals will correspond to the various generic points. The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals By taking all prime ideals, one thus gets the whole collection of ordinary and generic points. Noether did not pursue this approach.

In the 1930s, Wolfgang Krull turned things around and took a radical step: start with any commutative ring, consider the set of its prime ideals, turn it into a topological space by introducing the Zariski topology, and study the algebraic geometry of these quite general objects. Year 1930 ( MCMXXX) was a Common year starting on Wednesday (link will display 1930 calendar of the Gregorian calendar. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic Others did not see the point of this generality and Krull abandoned it.

André Weil was especially interested in algebraic geometry over finite fields and other rings. André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In the 1940s he returned to the prime ideal approach; he needed an abstract variety (outside projective space) for foundational reasons, particularly for the existence in an algebraic setting of the Jacobian variety. Year 1940 ( MCMXL) was a Leap year starting on Monday (link will display the full 1940 calendar of the Gregorian calendar. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which In Mathematics, the Jacobian variety of a non-singular Algebraic curve C of genus g &ge 1 is a particular Abelian variety In Weil's main foundational book (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain. In Mathematics, a field F is said to be algebraically closed if every Polynomial in one Variable of degree at least 1 with Coefficients

Around 1942 Oscar Zariski had defined an abstract Zariski space from the function field of an algebraic variety, for the needs of birational geometry: this is like a direct limit of ordinary varieties (under 'blowing up'), and the construction, reminiscent of locale theory, used valuation rings as points. Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety In Mathematics, birational geometry is a part of the subject of Algebraic geometry, that deals with the geometry of an Algebraic variety that is dependent In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Mathematics, pointless topology (also called point-free or pointfree topology is an approach to Topology which avoids the mentioning of points In Abstract algebra, a valuation ring is an Integral domain D such that for every element x of its Field of fractions F

In the 1950s, Jean-Pierre Serre, Claude Chevalley and Masayoshi Nagata, motivated largely by the Weil conjectures relating number theory and algebraic geometry, pursued similar approaches with prime ideals as points. Year 1950 ( MCML) was a Common year starting on Sunday (link will display the full calendar of the Gregorian calendar. Claude Chevalley ( 11 February 1909, Johannesburg, South Africa - 28 June 1984, Paris) was a French Masayoshi Nagata ( Japanese: 永田 雅宜 Nagata Masayoshi; born 1927 in Aichi Japan (February 9 1927–August 27 2008 is a Japanese mathematician In Mathematics, the Weil conjectures, which had become theorems by 1974 were some highly-influential proposals from the late 1940s by André Weil on the Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with According to Pierre Cartier, the word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas; and it was Martineau who suggested to Serre the move to the current spectrum of a ring in general. Pierre Cartier (born in Sedan France in 1932 is a Mathematician. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of

Alexander Grothendieck then gave the decisive definition. Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany He defines the spectrum of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a sheaf of rings: to every Zariski-open set he assigns a commutative ring, thought of as the ring of "polynomial functions" defined on that set. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. These objects are the "affine schemes"; a general scheme is then obtained by "gluing together" several such affine schemes, in analogy to the fact that projective varieties can be obtained by gluing together affine varieties.

See also the article on spectrum of a ring for a motivation of the paradigm "points are prime ideals". In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of

The generality of the scheme concept was initially criticized: some schemes are extremely far removed from having any geometrical interpretation. Grothendieck and Dieudonné studied the category of all schemes, and Grothendieck's student Pierre Deligne later wrote that admitting bizarre schemes made the whole category of schemes much nicer. Jean Alexandre Eugène Dieudonné ( July 1 1906, Lille - November 29 1992, Nice) was a French mathematician Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician.

The evolution of the scheme concept was not the end of the road. Subsequent work on algebraic spaces and algebraic stacks by Deligne, Mumford, and Michael Artin, originally in the context of moduli problems, have significantly enhanced the geometric flexibility of modern algebraic geometry. In Mathematics, an algebraic space is a generalization of the schemes of Algebraic geometry introduced by Michael Artin for use in Deformation In Algebraic geometry, a branch of Mathematics, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and Pierre René Viscount Deligne (born 3 October 1944 in Brussels) is a Belgian Mathematician. David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of Recent ideas about higher algebraic stacks and homotopical or derived algebraic geometry have regard to further expanding the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to homotopy theory. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical

Definitions

An affine scheme is a locally ringed space isomorphic to Spec(A) for some commutative ring A. In Abstract algebra and Algebraic geometry, the spectrum of a Commutative ring R, denoted by Spec( R) is defined to be the set of In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on A scheme is a locally ringed space X admitting a covering by open sets Ui, such that the restriction of the structure sheaf OX to each Ui is an affine scheme. In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf 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 . Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany The Éléments de géométrie algébrique ("Elements of Algebraic Geometry " by Alexander Grothendieck (assisted by Jean Dieudonné David Bryant Mumford (born 11 June 1937) is a Mathematician known for distinguished work in Algebraic geometry, and then for research into

One may think of a scheme as being covered by "coordinate charts" of affine schemes. The above formal definition means exactly that schemes are obtained by glueing together affine schemes for the Zariski topology. In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic

The category of schemes

Schemes form a category if we take as morphisms the morphisms of locally ringed spaces. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on

Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant adjoint pair: For every scheme X and every commutative ring A we have a natural equivalence

\operatorname{Hom}_{\rm Schemes}(X, \operatorname{Spec}(A)) \cong \operatorname{Hom}_{\rm CRing}(A, {\mathcal O}_X(X)).

Since Z is an initial object in the category of rings, the category of schemes has Spec(Z) as final object. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms

The category of schemes has finite products, but one has to be careful: the underlying topological space of the product scheme of (X, OX) and (Y, OY) is normally not equal to the product of the topological spaces X and Y. In Category theory, the product of two (or more objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In fact, the two are not even the same set in general. For example, if K is the field with nine elements, then Spec K × Spec K ≈ Spec (K ⊗Z K) ≈ Spec (K ⊗Z/3Z K) ≈ Spec (K × K), a set with two elements, though Spec K has only a single element.

Types of schemes

There are many ways one can qualify a scheme. According to a basic idea of Grothendieck, conditions should be applied to a morphism of schemes. Any scheme S has a unique morphism to Spec(Z), so this attitude, part of the relative point of view, doesn't lose anything.

For detail on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory. This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf

OX modules

Just as the R-modules are central in commutative algebra when studying the commutative ring R, so are the OX-modules central in the study of the scheme X with structure sheaf OX. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property (See locally ringed space for a definition of OX-modules. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on ) The category of OX-modules is abelian. In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist Of particular importance are the coherent sheaves on X, which arise from finitely generated (ordinary) modules on the affine parts of X. In Mathematics, especially in Algebraic geometry and the theory of Complex manifolds coherent sheaves are specific class of sheaves having The category of coherent sheaves on X is also abelian.

References

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