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| In [[mathematics]], '''schemes''' connect the fields of [[algebraic geometry]], [[commutative algebra]] and [[number theory]]. Schemes were introduced by [[Alexander Grothendieck]]{{when|date=January 2014}} so as to broaden the notion of [[algebraic variety]]; some consider schemes to be the basic object of study of modern algebraic geometry. Technically, a scheme is a [[topological space]] together with [[commutative ring]]s for all of its open sets, which arises from gluing together [[spectrum of a ring|spectra]] (spaces of [[prime ideal]]s) of commutative rings along their open subsets.
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| == Types of schemes ==
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| 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 ''[[Grothendieck's relative point of view|relative point of view]]'', doesn't lose anything.
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| For detail on the development of scheme theory, which quickly becomes technically demanding, see first [[glossary of scheme theory]].
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| == History and motivation ==
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| The algebraic geometers of the [[Italian school of algebraic geometry|Italian school]] had often used the somewhat foggy concept of "[[generic point]]" when proving statements about [[algebraic varieties]]. 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 ideal]]s of this ring will correspond to ordinary points of the variety (under suitable conditions), and the non-maximal [[prime ideal]]s will correspond to the various generic points, one for each subvariety. By taking all prime ideals, one thus gets the whole collection of ordinary and generic points. Noether did not pursue this approach.
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| 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. Others did not see the point of this generality and Krull abandoned it.
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| [[André Weil]] was especially interested in algebraic geometry over [[finite field]]s and other rings. 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]]. 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''.
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| In 1944 [[Oscar Zariski]] defined an abstract [[Zariski–Riemann 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 ring]]s as points.
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| 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. According to [[Pierre Cartier (mathematician)|Pierre Cartier]], the word ''scheme'' was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas; and it was [[André Martineau]] who suggested to Serre the move to the current [[spectrum of a ring]] in general.
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| ==Modern definitions of the objects of algebraic geometry==
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| {{see also|Spectrum of a ring}}
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| [[Alexander Grothendieck]] then gave the decisive definition, bringing to a conclusion a generation of experimental suggestions and partial developments.{{fact|date=January 2014}} He defined the [[spectrum of a ring|spectrum]] of a commutative ring as the space of prime ideals with Zariski topology, but augments it with a [[sheaf (mathematics)|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. 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 general varieties can be obtained by gluing together affine varieties.
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| The generality of the scheme concept was initially criticized: some schemes are removed from having straightforward geometrical interpretation, which made the concept difficult to grasp. However, admitting arbitrary schemes makes the whole category of schemes better-behaved. Moreover, natural considerations regarding, for example, [[moduli space]]s, lead to schemes that are "non-classical". The occurrence of these schemes that are not varieties (nor built up simply from varieties) in problems that could be posed in classical terms made for the gradual acceptance of the new foundations of the subject.
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| Subsequent work on [[algebraic space]]s and [[algebraic stack]]s by [[Pierre Deligne|Deligne]], [[David Mumford|Mumford]], and [[Michael Artin]], originally in the context of [[moduli problem]]s, has further enhanced the geometric flexibility of modern algebraic geometry. Grothendieck advocated certain types of [[ringed topos]]es as generalisations of schemes, and following his proposals [[relative scheme]]s over ringed toposes were developed by M. Hakim. Recent ideas about [[higher algebraic stack]]s 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]].
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| == Definitions ==
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| An [[affine scheme]] is a [[locally ringed space]] isomorphic to the spectrum of a commutative ring. We denote the spectrum of a commutative ring ''A'' by Spec(''A''). A '''scheme''' is a locally ringed space ''X'' admitting a covering by open sets ''U''<sub>''i''</sub>, such that the restriction of the structure sheaf ''O''<sub>''X''</sub> to each ''U''<sub>''i''</sub> is an affine scheme. Therefore 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]].
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| In the early days, this was called a ''prescheme'', and a scheme was defined to be a [[separated scheme|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 [[David Mumford|Mumford]]'s {{doi-inline|10.1007/b62130|''Red Book''}}.
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| == The category of schemes ==
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| Schemes form a [[category theory|category]] if we take as morphisms the morphisms of [[Ringed space|locally ringed space]]s.
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| Morphisms from schemes to affine schemes are completely understood in terms of ring homomorphisms by the following contravariant [[adjoint functor|adjoint pair]]: For every scheme ''X'' and every commutative ring ''A'' we have a natural equivalence
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| :<math>\operatorname{Hom}_{\rm Schemes}(X, \operatorname{Spec}(A)) \cong \operatorname{Hom}_{\rm CRing}(A, {\mathcal O}_X(X)).</math> | |
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| Since '''[[integer|Z]]''' is an [[initial object]] in the [[category of rings]], the category of schemes has Spec('''Z''') as a [[final object]].
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| The category of schemes has finite [[product (category theory)|products]], but one has to be careful: the underlying topological space of the product scheme of (''X'', ''O<sub>X</sub>'') and (''Y'', ''O<sub>Y</sub>'') is normally ''not'' equal to the [[product topology|product]] of the topological spaces ''X'' and ''Y''. In fact, the underlying topological space of the product scheme often has more points than the product of the underlying topological spaces. For example, if ''K'' is the field with nine elements, then Spec ''K'' × Spec ''K'' ≈ Spec (''K'' ⊗<sub>'''Z'''</sub> ''K'') ≈ Spec (''K'' ⊗<sub>'''Z'''/3'''Z'''</sub> ''K'') ≈ Spec (''K'' × ''K''), a set with two elements, though Spec ''K'' has only a single element.
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| For a scheme <math>S</math>, the category of schemes over <math>S</math> has also [[fibre product]]s, and since it has a final object <math>S</math>, it follows that it has finite [[Limit (category theory)|limit]]s.
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| == ''O<sub>X</sub>'' modules ==
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| Just as the ''R''-[[module (mathematics)|modules]] are central in [[commutative algebra]] when studying the [[commutative ring]] ''R'', so are the ''O<sub>X</sub>''-modules central in the study of the scheme ''X'' with structure sheaf ''O<sub>X</sub>''. (See [[locally ringed space]] for a definition of ''O<sub>X</sub>''-modules.) The category of ''O<sub>X</sub>''-modules is [[abelian category|abelian]]. Of particular importance are the [[coherent sheaf|coherent sheaves]] on ''X'', which arise from finitely generated (ordinary) modules on the affine parts of ''X''. The category of coherent sheaves on ''X'' is also abelian.
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| ==Generalizations==
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| A commonly used generalization of schemes are the [[algebraic stack]]s. All schemes are algebraic stacks, but the category of algebraic stacks is richer in that it contains many quotient objects and [[moduli space]]s that cannot be constructed as schemes; stacks can also have negative dimension. Standard constructions of scheme theory, such as [[sheaf (mathematics)|sheaves]] and [[étale cohomology]], can be extended to algebraic stacks.
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| See also: [[derived scheme]].
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| ==References==
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| *{{cite book
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| | author = [[David Eisenbud]]
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| | coauthors = [[Joe Harris (mathematician)|Joe Harris]]
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| | year = 1998
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| | title = The Geometry of Schemes
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| | publisher = [[Springer Science+Business Media|Springer-Verlag]]
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| | isbn = 0-387-98637-5
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| }}
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| *{{cite book
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| | author = [[Robin Hartshorne]]
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| | year = 1997
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| | title = Algebraic Geometry
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| | publisher = Springer-Verlag
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| | isbn = 0-387-90244-9
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| }}
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| *{{cite book
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| | author = [[David Mumford]]
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| | year = 1999
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| | title = The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
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| | edition = 2nd ed.
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| | publisher = Springer-Verlag
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| | doi = 10.1007/b62130
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| | isbn = 3-540-63293-X
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| }}
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| *{{cite book
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| | author = Qing Liu
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| | year = 2002
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| | title = Algebraic Geometry and Arithmetic Curves
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| | publisher = [[Oxford University Press]]
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| | isbn = 0-19-850284-2
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| }}
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| [[Category:Scheme theory]]
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