Scheme

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A scheme is a ringed space that is locally isomorphic to an affine scheme. More precisely, a scheme consists of a topological space $X$ (the underlying space of the scheme) and a sheaf $\def\cO{ {\mathcal O}}\cO_X$ of commutative rings with a unit on $X$ (the structure sheaf of the scheme); moreover, an open covering $(X_i)_{i\in I}$ of $X$ must exist such that $(X_i,\cO_X|_{X_i})$ is isomorphic to the affine scheme $\def\Spec{ {\rm Spec}\;}\def\G{\Gamma} \Spec\G(X_i,\cO_X)$ of the ring of sections of $\cO$ over $X_i$. A scheme is a generalization of the concept of an algebraic variety. For the history of the concept of a scheme, see [Di], [Sh], [Do].

Basic concepts and properties.

Let $(X,\cO_X)$ be a scheme. For every point $x\in X$, the stalk $\cO_{X,x}$ at $x$ of the sheaf is a local ring; the residue field of this ring is denoted by $k(x)$ and is called the residue field of the point $X$. As the topological properties of the scheme the properties of the underlying space $x$ are considered (for example, quasi-compactness, connectedness, irreducibility). If $P$ is a property of affine schemes (i.e. a property of rings), then one says that a scheme has property $P$ locally if any of its points has an open affine neighbourhood that has this property. The property of being locally Noetherian is an example of this (see Noetherian scheme). A scheme is regular if all its local rings are regular (cf. Regular ring (in commutative algebra)). Other schemes defined in the same way include normal and reduced schemes, as well as Cohen–Macaulay schemes.

A morphism of schemes is a morphism between them as locally ringed spaces. In other words, a morphism $f$ of a scheme $X$ into a scheme $Y$ consists of a continuous mapping $f:X\to Y$ and a homomorphism of the sheaves of rings $f^* : \cO_Y\to f_*\cO_X$, where for any point $x\in X$, the homomorphism of local rings $\cO_{Y,f(x)}\to f_*\cO_{X,x}$ must map maximal ideals to maximal ideals. For any ring $A$, the morphisms of $X$ into $\Spec A$ are in bijective correspondence with the ring homomorphisms $A\to\G(X,\cO)$. For any point $x\in X$, its imbedding in $X$ can also be considered as a morphism of schemes $\Spec k(x)\to X$. An important property is the existence in the category of schemes of direct and fibre products (cf. Fibre product of objects in a category), which generalize the concept of the tensor product of rings. The underlying topological space of the product of two schemes $X$ and $Y$ differs, generally speaking, from the product of the underlying spaces $X\times Y$.

A scheme $X$ endowed with a morphism into a scheme $S$ is called an $S$-scheme, or a scheme over $S$. A morphism $h:X\to Y$ is called a morphism of $S$-schemes $f:X\to S$ and $g:Y\to S$ if $f=g\circ h$. Any scheme can be seen as a scheme over $\Spec \Z$. A morphism of base change $S'\to S$ permits a transition from the $S$-scheme $X$ to the $S'$-scheme $X_{S'} = X\times_S S'$ — the fibre product of $X$ and $S'$. If the underlying scheme $S$ is the spectrum of a ring $k$, then one also speaks of a $k$-scheme. A $k$-scheme $X$ is called a $k$-scheme of finite type if a finite affine covering $(X_i)_{i\in I}$ of $X$ exists such that the $k$-algebras $\G(X_i,\cO_X)$ are generated by a finite number of elements. A scheme of finite type over a field, sometimes requiring separability and completeness, is usually called an algebraic variety. A morphism of $k$-schemes $\Spec k\to X$ is called a rational point of the $k$-scheme $X$; the set of such points is denoted by $X(k)$.

For an $S$-scheme $f:X\to S$ and a point $s\in S$, the $k(s)$-scheme $f^{-1}(s) = X_s$, obtained from $X$ by a base change $\Spec k(s) \to X$, is called a stalk (or fibre) of the morphism $f$ over $s$. If, instead of the field $k(s)$ in this definition one takes its algebraic closure, then the concept of a geometric fibre is obtained. Thereby, the $S$-scheme $X$ can be considered as a family of schemes $X_s$ parametrized by the scheme $S$. Often, when speaking of families, it is also required that the morphism $f$ be flat (cf. Flat morphism).

Concepts relating to schemes over $S$ are often said to be relative, as opposed to the absolute concepts relating to schemes. In fact, for every concept that is used for schemes there is a relative variant. For example, an $S$-scheme $X$ is said to be separated if the diagonal imbedding $X\to X\times_S X$ is closed; a morphism $f:Z\to S$ is said to be smooth if it is flat and all its geometric fibres are regular. Other morphisms defined in the same way include affine, projective, proper, finite, étale, non-ramified, finite-type, etc. A property of a morphism is said to be universal if it is preserved under any base change.

Cohomology of schemes.

Studies of schemes and related algebraic-geometric objects can often be divided into two problems — local and global. Local problems are usually linearized and their data are described by some coherent sheaf or by sheaf complexes. For example, in the study of the local structure of a morphism $X\to S$, the sheaves $\def\O{\Omega}\O_{X/S}^P$ of relative differential forms (cf. Differential form) are of some importance. The global part is usually related to the cohomology of these sheaves (see, for example, deformation of an algebraic variety). Finiteness theorems are useful here, as are theorems on the vanishing of the cohomology spaces (see Kodaira theorem), duality, the Künneth formula, the Riemann–Roch theorem, etc.

A scheme of finite type over a field $\C$ can also be considered as a complex analytic space. Using transcendental methods, it is possible to calculate the cohomology of coherent sheaves; it is more important, however, that it is possible to speak of the complex, or strong, topology on $X(\C)$, the fundamental group, the Betti numbers, etc. The desire to find something similar for arbitrary schemes and the far-reaching arithmetical hypotheses put forward (see Zeta-function in algebraic geometry) have led to the construction of different topologies in the category of schemes, the best known of which is the étale topology (cf. Etale topology). This has made it possible to define the fundamental group of a scheme, other homotopy invariants, cohomology spaces with values in discrete sheaves, Betti numbers, etc. (see $l$-adic cohomology; Weil cohomology; Motives, theory of).

Construction of schemes.

In the construction of a concrete scheme one most frequently uses the concepts of an affine or projective spectrum (see Affine morphism; Projective scheme), including the definition of a subscheme by a sheaf of ideals. The construction of a projective spectrum makes it possible, in particular, to construct a monoidal transformation of schemes. Fibre products and glueing are also used in the construction of schemes. Less elementary constructions rely on the concept of a representable functor. By having at one's disposal a good concept of a family of objects parametrized by schemes, and by juxtaposing every scheme $S$ with a set $F(S)$ of families parametrized by $S$, a contravariant functor $F$ is obtained from the category of schemes into the category of sets (possibly with an additional structure). If the functor $F$ is representable, i.e. if a scheme $X$ exists such that $F(S)={\rm Hom}(S,X)$ for any $S$, then a universal family of objects parametrized by $X$ is obtained. The Picard scheme and Hilbert scheme are constructed in this way (see also Algebraic space; Moduli theory).

One other method of generating new schemes is transition to a quotient space by means of an equivalence relation on a scheme. As a rule, this quotient space exists as an algebraic space. A particular instance of this construction is the scheme of orbits $X/G$ under the action of a group scheme $G$ on a scheme $X$ (see Invariants, theory of).

One of the generalizations of the concept of a scheme is a formal scheme, which may be understood to be the inductive limit of schemes with one and the same underlying topological space.

Comments

In earlier terminology, e.g. the fundamental original book [GrDi], the phrase pre-scheme was used for a scheme as defined above; and scheme referred to a separated scheme, i.e. a scheme such that the diagonal $X\to X\times X$ is closed.

There are a large number of conditions, especially finiteness conditions, on morphisms between schemes that are considered. Some of these are as follows.

A morphism of schemes $f:X\to Y$ is a compact morphism (also called quasi-compact morphism) if there is an open covering of $Y$ by affine sets $V_i$ such that $f^{-1}(V_i)$ is compact for all $i$.

A morphism of schemes $f:X\to Y$ is a quasi-finite morphism if for every $y\in Y$, $f^{-1}(y)$ is a finite set.

A morphism $f:X\to Y$ is a quasi-separated morphism if the diagonal morphism $X\to X\times_Y X$ is compact.

A morphism $f:X\to Y$ is a morphism locally of finite type if there exists a covering of $Y$ by open affine sets $V_i=\Spec(B_i)$ such that for each $i$, $f^{-1}(V_i)$ can be covered by open affine sets $U_{ij} = \Spec(A_{ij}$ such that each $A_{ij}$ is a finitely-generated $B_i$-algebra. If, in addition, finitely many $U_{ij}$ suffice (for each $i$), then $f$ is a morphism of finite type.

A morphism $f:X\to Y$ is a finite morphism if there exists a covering of $Y$ by open affine sets $V_i=\Spec(B_i)$ such that each $f^{-1}(V_i)$ is affine, say $f^{-1}(V_i) = \Spec(A_i)$, and $A_i$ is a $B_i$-algebra which is finitely generated as a $B_i$-module.

Let $B$ be an algebra over a ring $R$. The algebra $B$ is said to be finitely presentable over $R$ if it is isomorphic to a quotient $R[T_1,\dots,T_n]/\def\fa{ {\mathfrak a}}\fa$, where $\fa$ is a finitely-generated ideal in $R[T_1,\dots,T_n]$. If $R$ is Noetherian, $B$ is finitely presentable if and only if $B$ is of finite type (i.e. finitely generated as an algebra over $R$).

Let $f:X\to Y$ be a morphism of (pre-) schemes, and $x\in X$, $y=f(x)$. Then $f$ is said to be finitely presentable in $x$ if there exists an open affine set $V\ni y$ and an open affine set $U\ni x$ such that $f(U)\subset V$ and such that the ring $A(U)$ is a finitely-presentable $A(V)$-algebra. The morphism $f$ is said to be locally finitely presentable if it is finitely presentable in each point $x$. If $Y$ is locally Noetherian, a morphism $f:X\to Y$ is locally finitely presentable if and only if it is locally of finite type. A morphism $f$ is finitely presentable if it is locally finitely presentable, quasi-compact and quasi-separated.

For some more important special conditions on morphisms of schemes and pre-schemes cf. Affine morphism; Smooth morphism (of schemes); Quasi-affine scheme; Separable mapping; Etale morphism; Proper morphism.

If $X\to Y$ is a morphism of such-and-such-a-type, then one often says that $X$ is a scheme of such-and-such-a-type over $Y$.

References

[Di]	J. Dieudonné, "Cours de géométrie algébrique", I, Presses Univ. France (1974) Zbl 1092.14500 Zbl 1085.14500
[Do]	I.V. Dolgachev, "Abstract algebraic geometry" J. Soviet Math., 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra Topol. Geom., 10 (1972) pp. 47–112 Zbl 1068.14059
[GrDi]	A. Grothendieck, J. Dieudonné, "Eléments de géometrie algébrique", I. Le langage des schémes, Springer (1971) MR0217085 {ZBL|0203.23301}}
[Ha]	R. Hartshorne, "Algebraic geometry", Springer (1977) MR0463157 Zbl 0367.14001
[Sh]	I.R. Shafarevich, "Basic algebraic geometry", Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001