Giving a system T that is ``derived'' from the system S by substituting the variables by polynomial functions of another set of r variables is a natural operation on systems of equations. The analogous notion is that of a morphism of functors (also called a natural transformation) FG. This is a way of giving a map F(A)G(A) so that for any ring homomorphism AB we get a commutative diagram (any element in the top left corner has the same image in the bottom right corner independent of the route followed).
For those who have studied affine schemes earlier in a slightly different way we offer the following result which is proved in the second appendix.
A slightly different example (but one which is fundamental) is the functor that associates with a ring A the collection of all n + 1-tuples (a0, a1,..., an) which generate the ring A upto multiplication by units. Equivalently, one can think of all surjective A-module homomorphisms An + 1A modulo the equivalence induced by multiplication by units. This functor is denoted n and is conceptualised as the projective n-dimensional space. We use the symbol (a0 : a1 : ... : an) to denote the equivalence class under unit multiples of the n + 1-tuple (a0, d1,..., an) which gives rise to an element in n(A).
Now, if a = (a0 : a1 : ... : ap) and b = (b0 : b1 : ... : bq) are elements in p(A) and q(A) respectively, then we can form the (p + 1) . (q + 1)-tuple consisting of cij = aj . bj; this tuple generates the ring A as well. Clearly, when a and b are replaced by unit multiples ua and vb for some units u and v in A, the tuple c = (cij)i = 0, j = 0p, q is replaced by its unit multiple (uv)c. Thus, we have a natural transformation p×qpq + p + q. Moreover, one easily checks that the resulting map on sets
For each positive integer d we can associate to a = (a0 : a1 : ... : ap) the tuple of all monomials of degree exactly d with the entries from a. For example, if d = 2 then we take the -tuple consisting of bij = aiaj. As above this gives a natural transformation of functors p - 1. For each finite ring A the resulting map on sets
The two examples above are special cases of projective subschemes defined as follows. Let F(X0,..., Xp) be any homogeneous polynomial in the variables X0,...,Xp (in other words all the monomials in F have the same degree). While the value of F at a p + 1-tuple (a0,..., ap) can change if we multiply the latter by a unit, this multiplication does nothing if the value is 0. Thus, the set
There is also a natural way of thinking of affine schemes in terms of subfunctors of n for a suitable n. As we saw above any affine scheme is a subscheme of q, so it is enough to exhibit q as a subfunctor of n for a suitable n. Now it is clear that if (a1,..., aq) is any q-tuple, then the collection (1, a1,..., aq) generates the ring A so that this defines an element (1 : a1 : ... : aq) of q(A). Conversely, if (a0 : a1 : ... : aq) is an element of q(A), such that a0 is a unit then this is the same as (1 : a1/a0 : ... : aq/a0), which in turn corresponds to the point (a1/a0,..., aq/a0) in q.
A generalisation of the above is the notion of a quasi-projective scheme. In addition to the homogeneous polynomials Fi considered above let G1(X0,..., Xp), ..., Gm(X0,..., Xp) be homogeneous polynomials of the same degree. We define a quasi-projective scheme
One can go further and define the notion of an abstract algebraic scheme but for our purposes the notion defined above of a quasi-projective scheme (of finite type over integers or of ``arithmetic'' type) will suffice.
Let F1,...,Fn be a collection of equations which define a projective scheme and d be no smaller than the maximum of their degrees. It is clear that the same projective scheme is defined by the larger collection of the form Fj . M where j varies between 1 and n and M varies over all monomials of degree d - deg(Fj). Thus we can always assume that a projective scheme is defined by homogeneous equations of the same degree.
The complement of the subscheme of V(F1,..., Fn) is not the functor that assigns to each A the set-theoretic complement p(A) V(F1,..., Fn)(A), but in fact, when Fi's have the same degree it is the quasi-projective scheme V(0;F1,..., Fn)(A). The reason for this choice becomes clear as we study schemes more. For the moment it is enough to note that if A is the ring p = p[X]/(X2), then the element (1 : : ... : ) is in the set-theoretic complement of (1 : 0 : ... : 0) in p(A) but is not in the scheme-theoretic complement that we have defined above.
Finally, let X p be a quasi-projective scheme, and let F1,..., Fn be a bunch of homogeneous polynomials of the same degree. The intersection X V(F1,..., Fn;1) is clearly a subscheme of X and such subschemes are called closed subschemes of X. The intersection X V(0;F1,..., Fn) is also a subscheme of X and such subschemes are called open subschemes of X. More generally, the intersection of V(D1,..., Dm;E1,..., En) and V(F1,..., Fk;G1,..., Gl) is the scheme
One very useful example of a closed subscheme is the subscheme p p×p, which is the diagonal; this is a closed subscheme of the scheme p×p defined by the conditions XiYj = XjYi for 0 i, j p. For any p < q we can exhibit p as the closed subscheme of q given by Xi = 0 for p < i q.
Like the case of set-theoretic complement, the set-theoretic union of closed subschemes is in general not a closed subschemes. For example the smallest closed subscheme of 2 that contains L = V(X1) and M = V(X2) is easily seen to be V(X1X2); but it is possible for the product of two elements of a finite ring to be 0 without either of them being zero. Thus we can define the scheme-theoretic union of a collection of closed subschemes to be the smallest closed subscheme that contains the set-theoretic union (the set-theoretic union defines a subfunctor); such a scheme exists by Hilbert's basis theorem. From now on when we use the term union of schemes we shall always mean the scheme theoretic union.
A closed subscheme Y X is said to be a proper closed subscheme if for some finite ring A, the subset Y(A) X(A) is a proper subset. A scheme is said to be reducible if it can be written as the union of two distinct (but not necessarily disjoint!) proper closed subschemes. For example V(X1X2) in 2) is the union of the two lines V(X1) and V(X2). Now even a proper closed subscheme Y X can be ``essentially'' all of X; for example consider the closed subscheme Y = V(X22) of the scheme X = V(X23). For any finite field F, we have Y(F) = X(F). A scheme X is said to be reduced if it has no proper closed subscheme Y such that Y(F) = X(F) for all finite fields F. Note that the scheme V(X1X2) is reduced but not irreducible, while V(X12) is irreducible but not reduced. Hilbert's Basis theorem can also be used to show that any scheme X has a closed subscheme Y so that Y is reduced and Y(F) = X(F) for finite fields F. As a consequence of the Lasker-Noether Primary Decomposition theorem any scheme can be written as the union of a finite collection of irreducible closed subschemes; moreover, the underlying reduced schemes of these closed subschemes are uniquely determined. For example, consider the scheme L = V(X12, X1X2) in 2. One can show that that L is the union of the closed subschemes M = V(X1) and N = V(X12, X1X2, X22). But L can also be written as the union of M and K = V(X12, X0X2, X1X2, X22); moreover N and K are distinct schemes.