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2 Definitions 3 The category of schemes 4 Types of schemes 5 OX modules |
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.
Andre 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, 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 and Chevalley-Nagata, motivated 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 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 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 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 quite the end of the road; but the subsequent definitions of algebraic space and algebraic stack by Michael Artin for use in moduli problems are of restricted technical application.
A scheme X is a locally ringed space with a covering by open sets Ui, such that the restriction of the structure sheaf OX to each Ui gives a locally ringed space of type Spec(Ai) (where Ai is some commutative ring), up to isomorphism of locally ringed spaces.
(NB There was a shift in axioms: in the early days the above was called a prescheme and a separation axiom was required for a scheme.)
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.
Schemes form a 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 pair: For every scheme X and every commutative ring A we have a natural equivalence
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 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).
Just like 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. (See locally ringed space for a definition of OX-modules.) The category of OX-modules is abelian. Of particular importance are the 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.History and motivation
Definitions
The category of schemes
Since Z is an initial object in the category of rings, the category of schemes has Spec(Z) as final object.Types of schemes
missingOX modules