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TODO :

• Prune the history & motivation, or at least make it clearer.
• Probably should do the diagram for schemes over a base.
• mess about with the category of schemes subsection
• Redo O_X modules section.

In mathematics, schemes connect the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck in 1960 in his treatise Éléments de géométrie algébrique, with the aim of developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).^[1] Schemes enlarge the notion of algebraic variety to include nilpotent elements (the equations x = 0 and x² = 0 define the same points, but different schemes), and "varieties" defined over any commutative ring.

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, 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.

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.

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 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 André Martineau who suggested to Serre the move to the current spectrum of a ring in general.

Alexander Grothendieck then gave the modern definition. He defined the spectrum of a commutative ring as the space of prime ideals with Zariski topology, augmenting it with a sheaf of rings, thought of as the ring of "polynomial functions". These objects are the "affine schemes"; a general scheme is then obtained by "gluing together" several such affine schemes, by analogy with the gluing together of affine varieties to from general varieties.

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 spaces, 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.

For more details on the development of scheme theory, which quickly becomes technically demanding, see first glossary of scheme theory.

If ${\displaystyle R}$ is a commutative ring, then the spectrum of ${\displaystyle R}$ is ${\displaystyle \operatorname {Spec} (R)=\{p\colon p{\text{ is a prime ideal of }}R\}}$. This is given the Zariski topology which has closed sets ${\displaystyle V_{I}=\{p\in \operatorname {Spec} (R)\colon p\subset I\}}$ for each ideal ${\displaystyle I}$ of ${\displaystyle R}$.

The sets ${\displaystyle D(f)=\{p\in \operatorname {Spec} (R):f\notin p\}}$ for ${\displaystyle f\in R}$ form a base for this topology. So a structure sheaf can be defined by ${\displaystyle O_{\operatorname {Spec} (R)}(D(f))=R_{f}}$, where ${\displaystyle R_{f}}$ is the localization of ${\displaystyle R}$ by the multiplicative system ${\displaystyle \{1,f,f^{2}...\}}$.

Then ${\displaystyle (\operatorname {Spec} (R),O_{\operatorname {Spec} (R)})}$ is an affine scheme. It is typically denoted as ${\displaystyle \operatorname {Spec} (R)}$ with the structure sheaf supressed.

A scheme is defined to be a locally ringed space ${\displaystyle (X,O_{X})}$ such that ${\displaystyle X}$ admits an open covering ${\displaystyle \{U_{i}\colon i\in I\}}$ where each ${\displaystyle (U_{i},O_{X}(U_{i}))}$ is isomorphic to an affine scheme (every affine scheme is a locally ringed space). Therefore one may think of a scheme as being covered by "coordinate charts" of affine schemes. ^{[nb 1]}

The category of schemes is formed by taking as morphisms the morphisms of locally ringed spaces (so the category of schemes is a full subcategory of the category of locally ringed spaces).

Morphisms between affine schemes are completely understood in terms of ring homomorphisms: given affine schemes ${\displaystyle \operatorname {Spec} (A),\operatorname {Spec} (B)}$ there is a natural equivalence

${\displaystyle \operatorname {Hom} _{\rm {Schemes}}(\operatorname {Spec} (A),\operatorname {Spec} (B))\cong \operatorname {Hom} _{\rm {CRing}}(B,A).}$

so the category of affine schemes is the opposite category to the category of commutative rings.

More generally there is a contravariant adjoint pair: For every scheme ${\displaystyle X}$ and every commutative ring ${\displaystyle A}$ there is a natural equivalence

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

For any scheme ${\displaystyle S}$, a scheme over ${\displaystyle S}$ is a scheme ${\displaystyle X}$ with a structure morphism ${\displaystyle \varphi _{X}:X\to S}$. A morphism between schemes ${\displaystyle X}$ and ${\displaystyle Y}$ over ${\displaystyle S}$ is a morphism ${\displaystyle f:X\to Y}$ of schemes such that ${\displaystyle \varphi _{Y}\circ f=\varphi _{X}}$.

Since ${\displaystyle \mathbb {Z} }$ is an initial object in the category of rings, the category of schemes has ${\displaystyle \operatorname {Spec} (\mathbb {Z} )}$ as a final object. So the category of schemes can be identified with the category of schemes over ${\displaystyle \operatorname {Spec} (\mathbb {Z} )}$. This approach forms part of what is known as Grothendieck's relative point of view, that conditions should be applied to morphisms of schemes rather than the schemes themselves.

For any scheme ${\displaystyle S}$, the category of schemes over ${\displaystyle S}$ has fibre products, and since it has a final object ${\displaystyle S}$, it follows that it has finite limits.

Just as the R-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. 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.

The sections of the structure sheaf O_X of X are called regular functions, which are defined on each open subsets U in X. The invertible subsheaf of O_X, denoted O^*_X, consists only of the invertible germs of regular functions under the multiplication. In most situations, the sheaf K_X is defined on an open affine subset Spec A of X as the total quotient rings Q(A) (though there are cases where the definition is more complicated).^[2] The sections of K_X are called rational functions on X. The invertible subsheaf of K_X is denoted by K^*_X. The equivalent class of this invertible sheaf turns to be an abelian group with tensor products and isomorphic to H¹(X, O^*_X), which is called Picard group. On projective varieties the sections of the structure sheaf O_X defined on each open subsets U of X are also called regular functions though there are no global sections except for constants.

Subsequent work on algebraic spaces and algebraic stacks by Deligne, Mumford, and Michael Artin, originally in the context of moduli problems, has further enhanced the geometric flexibility of modern algebraic geometry. All schemes are algebraic stacks, but the category of algebraic stacks is richer in that it contains many quotient objects and moduli spaces that cannot be constructed as schemes; stacks can also have negative dimension. Standard constructions of scheme theory, such as sheaves and étale cohomology, can be extended to algebraic stacks.

Grothendieck advocated certain types of ringed toposes as generalisations of schemes, and following his proposals relative schemes over ringed toposes were developed by M. Hakim. 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.

• David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5.
• Robin Hartshorne (1997). Algebraic Geometry. Springer-Verlag. ISBN 0-387-90244-9.
• David Mumford (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians (2nd ed. ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 3-540-63293-X. {{cite book}}: |edition= has extra text (help)
• Qing Liu (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press. ISBN 0-19-850284-2.