Last update: 3 June 2013
The subject-matter of algebraic geometry, from the time of Descartes onwards, has been the study of the solutions of systems of polynomial equations in several variables:
Originally the were taken to have real coefficients, and one looked for real solutions. However, fairly soon it was realised that it made better sense to include complex solutions, since there was then a better chance of their existence (e.g., has no real solutions, but plenty of complex ones).
Equally, one of the main preoccupations of number theory has been Diophantine problems, i.e., the solutions (if any) of a system of equations (1) in rational integers, the now being supposed to have integer coefficients: for example, 'Fermat's last theorem' , the equation As this example indicates, the problem thus set was often too hard, so it was natural to modify it by asking either for rational solutions or for solutions mod. a prime number), i.e., to regard the equations (1) as having their coefficients in the rational field or the finite field and to ask for solutions in that field. More generally, we may reduce the equations (1) mod. thereby replacing the coefficient domain by the Artin local ring and we may then pass to the ring of integers or its field of fractions
Thus it is natural to consider systems of equations (1) with coefficient domains other than the fields of real or complex numbers, and these coefficient domains may not always, be fields. However, if we stick to a coefficient field, we had better let it be quite arbitrary if we want a theory which is of sufficient generality for its applications. In particular, our field should be allowed to have positive characteristic (e.g., the finite fields). So we are led to study the solutions of (1), where the are now polynomials over an a arbitrary field As already observed, it is not enough to consider only the solutions in because there may not be any, or at any rate not enough: we should therefore take an algebraically closed field and consider the solutions of (1) in This is roughly the point of view of Weil (Foundations of Algebraic Geometry). If we agree to ignore questions of rationality, we can jettison and use only But this is inadequate for many purposes, e.g., Weil's conjectures on the number of points of an algebraic variety over a finite field.
Let be a field, an algebraically closed field containing and let be a subset of the polynomial ring (which we shall abbreviate to The variety defined by is the set of all such that for all If is the ideal generated by in then clearly Now let be the ideal consisting of all which vanish at every point of Clearly and the inclusion may be strict (for example, The relationship between and is given by a theorem of Hilbert (the Nullstellensatz) which asserts that is the radical of that is to say it is the set of all polynomials some power of which lies in is an affine
Each polynomial in determines a function on with values in and the restriction of this function to is called a regular function The regular functions form a ring clearly isomorphic to this ring is called the coordinate ring (or affine algebra) of Obviously is finitely generated as a and from Hilbert's theorem it follows immediately that has no non-zero nilpotent elements. Conversely, every finitely generated with no nilpotent elements arises as the coordinate ring of some in (for some we have only to take a set of generators of which defines a homomorphism of onto the kernel of this homomorphism is an ideal which is equal to its own radical, and is the variety sought. But there is a more intrinsic way of getting from namely, the points of are in one-to-one correspondence with the of into For if then is a and conversely, if is a let then is a point of Thus an affine algebraic variety is determined by its coordinate ring.
If are affine say a mapping is if it is induced by a mapping of into We how have a category of affine varieties and regular maps (we shall drop the prefix from now on). If are the coordinate rings of respectively, then the regular maps correspond one-to-one to the homomorphisms if (i.e., is regular) then is regular and thus we have a mapping of into which of course is a homomorphism. Moreover, this correspondence is functorial: if corresponds to (where is the coordinate ring of then corresponds to In this way it appears that the category of affine is equivalent to the dual of the category of finitely-generated with no nilpotent elements. In other words, the theory of affine algebraic varieties over is equivalent to the theory of a rather special class of commutative rings, and one can compile a dictionary for translating statements about affine vatieties into statements of commutative algebra. Thus, in the hands of the German school of the 1920's and 1930's, algebraic geometry became the study of ideals in polynomial rings.
Let be an affine its coordinate ring. The elements of are functions from to If is any subset of let denote the set of common zeros of the functions in then it is easily verified that by taking the as closed sets we have a topology on called the Zariski topology (strictly, the From the topologist's point of view, this is a very bad topology: in general it is not even (unless when it is (but not If the closure of the set in the Zariski topology is the intersection of all the closed sets which contain it is what Weil calls the locus of and its points are the specializations of Thus is a specialization of if and only if
If are two affine varieties then is an affine variety, the product of and (it is the product of and in the category of all affine that is to say it satisfies the usual universal mapping property in this category). If are the coordinate rings of respectively then one might hope that the coordinate ring of would be the tensor product Unfortunately it isn't, in general, because may well have nilpotent elements (unless is perfect), and to get the coordinate ring of one has to factor out the ideal of nilpotent elements in the tensor product. This is one example where the exclusion of nilpotent elements leads to an unsatisfactory situation.
It should also be remarked that the Zariski topology on is not (in general) the product topology: generally it is strictly finer than the product topology, i.e. it has more open sets. The standard example is the affine plane
It was realised early on that affine geometry is in many respects unsatisfactory. For example, two subvarieties of an affine variety may have empty intersection even if their dimensions are right, and Bézout's theorem does not hold without qualification; or a point or subvariety may escape 'to infinity'. This was rectified by 'completing' affine space by sticking on suitable 'points at infinity', as everyone knows, and the result is projective space From a geometrical point of view, projective space and projective varieties are much more satisfactory to deal with. The process outlined above of constructing coordinate rings etc. can be imitated in the projective case, but it doesn't work nearly as well. A projective variety in is given by a set of homogeneous polynomial equations (with coefficients in these generate a homogeneous ideal in the graded polynomial ring The radical of is again a homogeneous ideal, so we can form which is a graded But: (i) the elements of do not correspond to regular functions on because the only everywhere-defined regular functions on are in fact constants; and (ii) there is no longer a one-to-one correspondence (as in the affine case) between graded coordinate rings and projective varieties: non-isomorphic rings can give rise to isomorphic varieties. For example, the coordinate ring of and of a conic in are not isomorphic.
A different approach is the following. can be regarded as the union of a finite number of overlapping affine spaces - for example, the complements of hyperplanes with no common point - which are open sets in the Zariski topology, and hence any projective variety is the union of a finite number of overlapping affine varieties which are open sets in thus is 'locally affine'. The situation is analogous to that for a manifold, which is 'locally Euclidean', i,e., is obtained by sticking together overlapping Euclidean spaces in a suitable way. Thus it is natural to go further, as Weil did, and define an 'abstract variety' as one which is obtained by pasting together overlapping affine varieties. The resulting object mayor may not be projective (i.e., embeddable in a projective space). The characteristic 'good' property of projective varieties, that they are in some sense 'compact' or that they don't have bits missing at infinity, is then replaced by the property of completeness, which can be formulated in various ways. Probably the simplest of these is the following: an (abstract) variety is complete if, for every variety the projection is a closed map (with respect to the Zariski topology).
To give meaning to the definition of an abstract variety, it is necessary to specify how the affine varieties which make it up are to be stuck together. There are various ways of doing this: one is the following. If is an affine variety, say we associate with a structure sheaf which may be defined as follows. A rational function said to be regular at or defined at if u can be put in the form where are polynomials and (so that is well-defined). The domain of definition of a rational function is an open set in A rational function on is by definition the restriction to of a rational function on (so the domain of is an open set in If is any open set in the rational functions on which are defined at every point of form a ring and the assignment is a presheaf of rings on which is immediately verified to be a sheaf. This is the structure sheaf and it is, intrinsically related to i.e., it does not depend on the embedding of in an affine space. One then defines a prealgebraic variety to be a topological space together with a sheaf of rings this sheaf being a sheaf of germs of functions on with values in with the following property: there exists a finite open covering of such that each together with the restriction of to is isomorphic, sheaf and all, to an affine algebraic variety. is an (abstract) algebraic variety if in addition it satisfies a 'separation axiom' which is the formal analogue of Hausdorff's axiom for topological spaces, namely that the diagonal should be a closed subset of the product (only here, as we have already seen, the topology on is not the product topology).
This definition is due to Serre (Faisceaux algébriques cohérents). Thus the philosophy is this: an affine variety is equivalent to a commutative ring (of a rather restricted type) and an abstract variety is obtained by sticking a number of these together by means of their structure sheaves.
We have now more or less set the stage. Going back for a moment to the affine case, we have remarked that any situation or theorem relating to affine varieties can be transcribed into one relating to their coordinate rings, and it has been recognised for a long time that in this way one gets more general statements, for generally the theorems of commutative algebra that arise are valid under much less restrictive hypotheses on the rings in question: often it is enough that they should be Noetherian. So, to obtain a satisfactorily general theory, one should start with a quite arbitrary commutative ring and construct something like an 'affine variety' from it, and then stick these objects together by means of structure sheaves to obtain generalised abstract varieties or preschemes.
This is a typed excerpt of the book "Algebraic Geometry: Introduction to Schemes - I.G. Macdonald".