Last update: 3 June 2013
If is a ringed space, an open subset of is said to be an affine open set if the ringed space is isomorphic to some affine scheme.
Definition. A prescheme is a ringed space such that every has an affine open neighbourhood, i.e., it is a locally affine ringed space.
Let be a prescheme.
|(i)||The affine open sets form a basis of the topology of|
|(ii)||If is any open set in the ringed space is a prescheme, called the restriction of to|
|(iv)||Every irreducible closed subset of has a unique generic point and is a one-one correspondence between the points of and the irreducible closed subsets of|
(i) Let be an open set in and for each let be an affine open neighbourhood of then is the union of the sets each is open in and is therefore a union of basic open sets contained in by (3.3); and these basic open sets are affine by (5.1). Hence is a union of affine open sets.
(ii) follows from (i).
(iii) Let If and are not in the same affine open set, it is clear that the condition is satisfied. If they are in the same affine open set, use the fact (Chapter 3) that an affine scheme is a
(iv) Let and let be an affine open neighbourhood of in Then is dense in (since is irreducible) and is itself irreducible, hence is the closure in of some Hence if is the closure of in we have (since but hence hence since is irreducible. Hence The uniqueness of the generic point follows from (iii), for the is equivalent to the statement:
Let and be preschemes. A morphism of ringed spaces is a morphism of preschemes if, for each is a local homomorphism Hence defines a field monomorphism so that is an extension of the field
Let be a fixed prescheme. (Strictly speaking, we should write but from now on we shall drop the structure sheaf from the notation.) An is a pair where is a prescheme and is a morphism of preschemes. If is the affine scheme of a ring we speak of an
If and are an is a morphism of preschemes such that the diagram
The 'base prescheme' may be considered as a generalization of the ground field of algebraic geometry: if is the coordinate ring of an affine then is a with identity element, so we have a homomorphism hence Thus is a (Of course, consists of only one potnt, so the map is trivial as a map of topological spaces; but a morphism of preschemes comprises also a map of the structure sheaves.)
Every prescheme may be considered canonically as a Namely the 'characteristic morphism' (Ex. 5, Chapter 3) is a morphism and hence one defines a morphism of preschemes for any prescheme (do it on the affine open sets).
Let be any category and for any two objects in let denote the set of all morphisms in For fixed and variable is a contravariant functor category of all sets).
Let be any contravariant functor on with values in is said to be representable if there exists an object in and a functorial isomorphism (for all If exists, it must be unique up to isomorphism.
A product of two objects in is an object which (if it exists) represents the functor in other words, there is a functorial isomorphism for all objects in
Products exist in many categories: in the category of groups. ('direct products'), topological spaces, algebraic varieties, modules over a fixed ring (here the product is 'direct sum' etc. The dual concept is that of sum: in the category of groups, for example, sum is 'free product'; in the category of commutative where is a fixed commutative ring, sum is tensor product over Since the category of affine schemes over is dual to the category of is a product of and in the category of affine schemes over (here are any two
Theorem (6.2). Let be a fixed prescheme, and two Then the product exists in the category of
The proof is tedious but not essentially difficult (EGA, I, 3.2.6), and we shall not reproduce it here. Locally, as we have just observed, it corresponds to the tensor product of rings, and it is a question of sticking things together so that it all fits.
This product of course has the usual associativity and commutativity properties, as in any category in which products exist.
The existence of products is fundamental, and arises in many contexts:
(1) Change of base. If is an and if is a morphism of preschemes, then the product is denoted by and is said to be obtained by extension of the base-prescheme from to We have a commutative diagram
and is to be regarded as an
Base extension is a transitive operation, i.e. if are morphisms of preschemes and is an then is canonically isomorphic to
This operation generalizes the notion of extension of the ground-field in algebraic geometry: if is a say affine with coordinate ring and if is an extension field of then the embedding gives and gives rise to an affine variety defined over
(2) Geometrical points. If is an affine (Chapter 1) with coordinate ring then the points of are in one-one correspondence with the i.e. with the This motivates the following definition: if are the are called points of the with values in the Let denote the set of points of with values in then the product of two and is characterized by the formula for any
In particular, a geometrical point of is a point of with values in an algebraically closed field that is to say it is a morphism consists of a single point, whose image under is the locality of the geometrical point. Given the locality the geometrical point is determined by an embedding of the residue field in
Remark. The product is not the set-theoretic product of and nor even the fibre product of and over that is to say, if temporarily denotes the set underlying then in general we have However, there is a surjective mapping For if and lie over the same point then and are extensions of and can therefore both be embedded in an extension of hence we have and localized at and respectively, and therefore an localized at say Clearly the projections of are and i.e.
To show that is not in general injective, it is enough to take to be the spectra of fields respectively being extensions of then in general has more than one prime ideal.
In fact it is not difficult to show that if and lie over the same point then the points of such that are in one-one correspondence with the isomorphism types of composite extensions of and over (E.G.A. I, 3.4.9).
(3) Fibres. Let be a morphism of preschemes and let be a point of Then the projection is a homeomorphism of the space underlying onto the fibre (E.G.A, I, 3.6.1). Hence the fibre can be regarded as a prescheme over the field as such we denote it by If and where it turns out that the residue fields and are the same, i.e. the residue field is the same whether is regarded as a point of the prescheme or as a point of the prescheme
(4) Separated morphisms. Schemes. Whenever the product is defined in a category being an object of there is a well defined diagonal morphism is the element of corresponding to in identity morphism of Hence if is a morphism of preschemes, we have a diagonal morphism If then corresponds to the homomorphism which maps to this homomorphism is surjective and therefore is in this case a homeomorphism of onto a closed subset of the product.
If are arbitrary preschemes, let be the projection on the first factor; then is the identity map of and therefore is a homeomorphism of onto If is a covering of by affine open sets, then is the diagonal of hence closed in and is contained in hence is locally closed (Chapter 2) (but not necessarily closed) in
The morphism is said to be separated, or is separated over if is closed in
A prescheme is a scheme if it is separated over i.e. if the 'characteristic morphism' is separated. This is the formal analogue of Hausdorff's axiom, or of Serre's second axiom for algebraic varieties (see Chapter 1).
Remark. If then is surjective, as we have remarked above, and therefore the characteristic morphism is separated. This justifies the terminology 'affine scheme' rather than 'affine prescheme'.
If are two affine open sets in a prescheme then need not be affine. But if is a scheme, will be affine, for is isomorphic to hence is closed in and therefore affine.
(5) Proper morphisms. A morphism of preschemes is of finite type if is a union of affine open sets such that each is a finite union of affine open sets with the property that each ring is finitely generated as an algebra over (here, if is an affine scheme, denotes the associated ring). If and are both affine, say then is of finite type if and only if is finitely generated as a (E.G.A., I, 6.3.3).
A morphism is proper if
|(i)||is separated and of finite type;|
|(ii)||is universally closed, i.e. for every morphism the projection is a closed mapping.|
This is the generalization of the notion of completeness for an algebraic variety over a field (cf. Chapter I).
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".