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## 8.3 Morphisms of schemes

We have already discussed natural transformations. However, not all natural transformations of functors are morphisms''; which we now define. It is in fact easier to first define the notion of a multi-valued'' morphism or correspondence.

Let L = V(F1,..., Fn;G1,..., Gm) be a quasi-projective scheme in p and K = V(D1,..., Dk;E1,..., El) be a quasi-projective scheme in q. As before we can and do assume that the collections {Di}, {Ei}, {Fi} and {Gi} have constant degrees. Let Xi's be the p + 1 variables for p and Yj be the q + 1 variables for q. If d1 is the degree of the Dt's then the bi-homogeneous polynomials of the form Dt . M where M is a monomial of degree d1 in the variables Xi can be written as polynomials in the variables Zij = XiYj (by choosing some arbitrary pairing of X's with Y's for each term). Let {} denote the resulting collection of polynomials in Zij as M varies over all possible monomials in the X's and Ft's vary. We have similar collections {}, {} and {}. One then checks quite easily that L(AK(A) is the subset of pq + p + q(A) defined by the conditions:

1. The equations ZijZkl - ZilZkj = 0 hold.
2. All the 's and the 's vanish.
3. The evaluation of the collection {} results in a tuple that generates the ring A.
In particular, this is also a quasi-projective scheme.

Thus, when X and Y are quasi-projective schemes, then so is X×Y. Hence, for a sub-functor Z of X×Y it makes sense say that it is a subscheme; or more specifically a closed or open subscheme. In particular, if W is a subscheme (resp. closed or open subscheme) of Y, we see that X×W is a subscheme (resp. closed or open subscheme) of X×Y. Similarly, for subschemes of X. Another useful closed subscheme is X×X, the diagonal subscheme, which is defined by intersecting X×X with the diagonal subscheme of q×q when X is given a a subscheme of q.

A correspondence from X to Y is a closed subscheme of X×Y. For any natural transformation f : XY the graph is the subfunctor of X×Y which gives for each finite ring A the graph of f (A) : X(A)Y(A). We say that f is a morphism if is a closed subscheme of X×Y. In other words, a morphism is a natural transformation which is also a correspondence. Alternatively, if Z X×Y is a correspondence so that the projection Z(A)X(A) is a bijection for all finite rings A, then Z is the graph of a morphism.

Now it follows easily that the identity natural transformation XX is a morphism with the diagonal as the associated correspondence. Moreover, each of the projections X×YX and X×YY is a morphism. It is also clear that if W X is a subscheme then the intersection of W×Y with gives the graph of the restriction of f : XY to W; as a result this restriction is also a morphism. If Z X×Y is the graph of a morphism then the projection ZX is a morphism; its graph in Z×X X×Y×X is the intersection of the diagonal of the extreme terms (consisting of (x, y, x)) with Z×X. The map Z(A)X(A) is a bijection; let g : XZ be the inverse natural transformation. The graph of g in X×Z X×X×Y is the intersection of ×Y with X×Z. Thus g is also a morphism. In other words, there are morphisms ZX and XZ with composition either way being identity. Thus ZX is an isomorphism.

Now, let f : XY be a morphism and g : YZ be another morphism. Let W be the intersection of ×Z with X× in X×Y×Z. Under the above isomorphism X, we can identify W as a subscheme of X×Z. It clear that W(A) is the graph of the composite natural transformation gof. Thus, morphisms can be composed.

Let f : XY be a morphism and W Y be a subscheme. Then, we have a subscheme of given by its intersection with X×W. Since X is an isomorphism, we obtain a subscheme of X as well; this scheme is usually denoted f-1(W) and called the inverse image of W under f. In some cases it may happen that is contained in X×W so that f-1(W) = X. In this case we say that the morphism f factors through or lands inside W.

The theorem of Chevalley asserts that there is a smallest subscheme W of Y so that f factors through W; we can refer to W as the categorical image of f. Note that it may not be true that W(A) is the image of X(A) in Y(A) even for one non-zero finite ring A.

Given morphisms XW and XZ we easily check that the natural transformation XW×Z is a morphism. Given morphisms XS and YS, we obtain the compositions a : X×YXS and b : X×YYS. Thus we a morphism X×YS×S. The inverse image of the diagonal is denoted X×SY and is called the fibre product of X and Y over S. For any morphisms ZX×Y such that the resulting composites with a and b are equal, we see that the morphism actually lands in the subscheme X×SY.

One important example of a correspondence is the subscheme Z of p + q×q defined by the conditions XiYj = XjYi for 0 i, j q. Let U be the open subscheme of p + q given by U = V(0;X0, X1,..., Xq). For (a0 : ... : ap + q) in U(A), the tuple (a0,..., aq) generates the ring A, thus we see that we see that ((a0 : ... : ap + q),(a0 : ... : aq)) gives an element of p + q(Aq(A) which clearly lies in Z(A). Conversely, if ((a0 : ... : ap + q),(b0 : ... : bq)) lies in Z(A) and (a0,..., aq) generate the ring A, then the above equations show that there is a unit u in A so that bi = uai (apply the Chinese Remainder theorem for finite rings!). Thus, the projection Z(A)p + q(A) is a bijection over U(A) and gives a morphism Uq. This morphism is called the projection on p + q away from the linear subscheme (or subspace!) V(X0,..., Xq); more generally, we can refer to the above correspondence as the projection correspondence.

A natural generalisation of this is to consider a collection F0,...,Fq of homogeneous polynomials of the same degree in variables X0,...,Xp; we can then take the subscheme Z of p×q defined by the equations

Fi(X0,..., Xp)Yj = Fj(X0,..., Xp)Yi

for 0 i, j q. We can take U to be the open subscheme defined by U = V(0;F0,..., Fq). The correspondence Z restricts to a morphism Uq. The scheme Z is referred to as the blow-up of p along the closed subscheme Y = V(F1,..., Fq) and is sometimes denoted .

For any functor F on the category of finite rings we can introduce a new functor TF which associates to a finite ring A the set F(A[]) where A[] denotes the finite ring A[T]/(T2). The morphism A[]A that sends to induces a natural transformation of functors TFF. Now, if F = p is the projective space then Tp(A) consists of equivalence classes of p + 1-tuples

(a0 + b0,..., ap + bp) (ua0 + (a0b + ub0),..., uap + (apb + ub0)

where u is a unit in A and (a0,..., ap) generate the ring A (this is enough to ensure generation of A[] by the above p + 1-tuple). The elements sij = aiaj and tij = biaj - ajbi are invariants associated with the equivalents class upto simultaneous multiplication by a unit u in A. Thus, if we consider the equivalence classes (under multiplication by units in A) of pairs (S;T) where S is a symmetric matrix and T an anti-symmetric matrix; then the equations satisfied by S and T are
 sijskl - siksjl = 0 (1) tijskl + tjksil + tkisjl = 0 (2)

Moreover, the entries sij of S generate the ring A. Conversely, a pair of matrices (S, T) satisfying the two equations and the condition that the entries of S generate the ring can be seen to arise in from an element p(A[]).

Proof. Let us assume that A is a finite local ring (the other cases follow from the Chinese Remainder Theorem). In this case, at least one of the entries sij must be a unit (since a sum of nilpotent elements is nilpotent). The equation sijsij = siisjj shows that sii must also be a unit. Let us then define ak = sik/sii and bk = tki. The equation sjksii = sijsik implies that sjk = ajak as required. Moreover, the equation

tjksii = tjiski - tkisji

shows us that tjk = bjak - bkaj as required.

The collection of equivalence classes of pairs (S;T) under multiplication by units in A can be identified with p2 + 2p. Thus Tp is naturally isomorphic to the quasi-projective scheme

V(SijSkl - SikSjl, TijSkl + TjkSil + TkiSjl;Sij)

This quasi-projective scheme is the Zariski Tangent Scheme of p. More generally, for any quasi-projective scheme X given as a subscheme of p one can show that the functor TX is naturally isomorphic to a subscheme of Tp. In other words, TX is also a quasi-projective scheme; this scheme is called the Zariski Tangent scheme of X. Moreover, the natural transformation TXX (given by the natural map X(A[])X(A)) is a morphism of schemes. This gives an important example of a vector space scheme; a notion that we will introduce in the next section.

Next: 8.4 Relativisation and categorical Up: 8 Algebraic Schemes for Previous: 8.2 Functors of points
Kapil Hari Paranjape 2002-10-20