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Scheme (mathematics)
*** Shopping-Tip: Scheme (mathematics)
In
mathematics, a '''scheme''' is an important concept connecting the fields of
algebraic geometry,
commutative algebra and
number theory. 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. Technically, a scheme is a
topological space together with
commutative rings for all its open sets, which arises from "gluing together"
spectrum of a ring spectra (spaces of
prime ideals) of commutative rings.
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. 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. 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. 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. 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''.
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.
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, 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.
Alexander Grothendieck then gave the decisive definition. He defines 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 defines 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 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".
The generality of the scheme concept was initially criticized: some schemes are extremely far removed from having any geometrical interpretation. Grothendieck and
Jean Dieudonné 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.
The evolution of the scheme concept was not the end of the road. Subsequent work on
algebraic spaces and
algebraic stacks by
Pierre Deligne Deligne,
David Mumford Mumford, and
Michael Artin, originally in the context of
moduli problems, have significantly enhanced the geometric flexibility of modern algebraic geometry. Recent ideas about
higher algebraic stacks and
derived algebraic geometry have regard to further expanding the algebraic reach of geometric intuition, bringing algebraic geometry closer in spirit to
homotopy theory.
Definitions
A '''scheme''' ''X'' is a
locally ringed space with a covering by open sets ''U''
''i'', such that the restriction of the structure sheaf ''O''
''X'' to each ''U''
''i'' gives a locally ringed space of type
spectrum of a ring Spec(''A''
''i'') (where ''A''
''i'' is some commutative ring),
up to isomorphism of
locally ringed spaces.
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
David Mumford Mumford's ''Red Book''.
Schemes isomorphic to Spec(''A'') for a commutative ring ''A'' are called
affine schemes. One may think of a scheme as covered by "coordinate charts" of affine schemes: the above formal definition means exactly that schemes are obtained by gluing together affine schemes for the
Zariski topology.
The category of schemes
Schemes form a
category theory category if we take as morphisms the morphisms of
locally ringed spaces.
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
:
Since
integer '''Z''' is an
initial object in the category of rings, the category of schemes has Spec('''Z''') as
final object.
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
X'') and (''Y'', ''O
Y'') is normally ''not'' equal to the
product topology product of the topological spaces ''X'' and ''Y''. In fact one can look at Spec ('''Z'''[X,Y]) for an example. In the category of commutative rings, '''Z'''[X,Y] is the
coproduct of '''Z'''[X] and '''Z'''[Y]; this means that Spec ('''Z'''[X,Y]) is the product in the category of
affine schemes of Spec ('''Z'''[X]) and Spec ('''Z'''[Y]) (and the inclusion into the category of schemes respects this). But all proper closed subsets of Spec ('''Z'''[X]) are finite. On the other hand Spec ('''Z'''[X,Y]) has many closed subsets V corresponding to polynomials P(X,Y) that are irreducible and of total degree higher than one: these are not in any sense derived from the two factors (and the underlying set of prime ideals isn't a cartesian product, either).
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 ''
Grothendieck's relative point of view 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.
''OX'' modules
Just like the ''R''-
module (mathematics) modules are central in
commutative algebra when studying the
commutative ring ''R'', so are the ''O
X''-modules central in the study of the scheme ''X'' with structure sheaf ''O
X''. (See
locally ringed space for a definition of ''O
X''-modules.) The category of ''O
X''-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.
References
* David Eisenbud, Joe Harris, ''The Geometry of Schemes'', (2000) Graduate Texts in Mathematics '''197''', Springer, New York ISBN 0-387-98638-3
* R. Hartshorne, ''Algebraic Geometry,'' Springer-Verlag: New York, 1977.
* David Mumford "The Red Book of Varieties and Schemes" Lecture Notes in Mathematics, Springer, New York ISBN 3-540-63293-X
* Qing Liu, "Algebraic Geometry and Arithmetic Curves" Oxford University Press, ISBN 0-19-850284-2
Category:Scheme theory
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