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).
F(A) | G(A) | |
F(B) | G(B) |
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 (a_{0}, a_{1},..., a_{n}) which generate the ring A upto multiplication by units. Equivalently, one can think of all surjective A-module homomorphisms A^{n + 1}A 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 (a_{0} : a_{1} : ^{ ... } : a_{n}) to denote the equivalence class under unit multiples of the n + 1-tuple (a_{0}, d_{1},..., a_{n}) which gives rise to an element in ^{n}(A).
Now, if a = (a_{0} : a_{1} : ^{ ... } : a_{p}) and b = (b_{0} : b_{1} : ^{ ... } : b_{q}) are elements in ^{p}(A) and ^{q}(A) respectively, then we can form the (p + 1)^{ . }(q + 1)-tuple consisting of c_{ij} = a_{j}^{ . }b_{j}; 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 = (c_{ij})_{i = 0, j = 0}^{p, q} is replaced by its unit multiple (uv)c. Thus, we have a natural transformation ^{p}×^{q}^{pq + p + q}. Moreover, one easily checks that the resulting map on sets
For each positive integer d we can associate to a = (a_{0} : a_{1} : ^{ ... } : a_{p}) 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 b_{ij} = a_{i}a_{j}. 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(X_{0},..., X_{p}) be any homogeneous polynomial in the variables X_{0},...,X_{p} (in other words all the monomials in F have the same degree). While the value of F at a p + 1-tuple (a_{0},..., a_{p}) 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 (a_{1},..., a_{q}) is any q-tuple, then the collection (1, a_{1},..., a_{q}) generates the ring A so that this defines an element (1 : a_{1} : ^{ ... } : a_{q}) of ^{q}(A). Conversely, if (a_{0} : a_{1} : ^{ ... } : a_{q}) is an element of ^{q}(A), such that a_{0} is a unit then this is the same as (1 : a_{1}/a_{0} : ^{ ... } : a_{q}/a_{0}), which in turn corresponds to the point (a_{1}/a_{0},..., a_{q}/a_{0}) in ^{q}.
A generalisation of the above is the notion of a quasi-projective scheme. In addition to the homogeneous polynomials F_{i} considered above let G_{1}(X_{0},..., X_{p}), ..., G_{m}(X_{0},..., X_{p}) 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 F_{1},...,F_{n} 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 F_{j}^{ . }M where j varies between 1 and n and M varies over all monomials of degree d - deg(F_{j}). 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(F_{1},..., F_{n}) is not the functor that assigns to each A the set-theoretic complement ^{p}(A) V(F_{1},..., F_{n})(A), but in fact, when F_{i}'s have the same degree it is the quasi-projective scheme V(0;F_{1},..., F_{n})(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]/(X^{2}), 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 F_{1},..., F_{n} be a bunch of homogeneous polynomials of the same degree. The intersection X V(F_{1},..., F_{n};1) is clearly a subscheme of X and such subschemes are called closed subschemes of X. The intersection X V(0;F_{1},..., F_{n}) is also a subscheme of X and such subschemes are called open subschemes of X. More generally, the intersection of V(D_{1},..., D_{m};E_{1},..., E_{n}) and V(F_{1},..., F_{k};G_{1},..., G_{l}) 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 X_{i}Y_{j} = X_{j}Y_{i} for 0 i, j p. For any p < q we can exhibit ^{p} as the closed subscheme of ^{q} given by X_{i} = 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(X_{1}) and M = V(X_{2}) is easily seen to be V(X_{1}X_{2}); 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(X_{1}X_{2}) in ^{2}) is the union of the two lines V(X_{1}) and V(X_{2}). Now even a proper closed subscheme Y X can be ``essentially'' all of X; for example consider the closed subscheme Y = V(X_{2}^{2}) of the scheme X = V(X_{2}^{3}). 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(X_{1}X_{2}) is reduced but not irreducible, while V(X_{1}^{2}) 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(X_{1}^{2}, X_{1}X_{2}) in ^{2}. One can show that that L is the union of the closed subschemes M = V(X_{1}) and N = V(X_{1}^{2}, X_{1}X_{2}, X_{2}^{2}). But L can also be written as the union of M and K = V(X_{1}^{2}, X_{0}X_{2}, X_{1}X_{2}, X_{2}^{2}); moreover N and K are distinct schemes.