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,..., F_n;G₁,..., G_m) be a quasi-projective scheme in $\mathbb {P}$^p and K = V(D₁,..., D_k;E₁,..., E_l) be a quasi-projective scheme in $\mathbb {P}$^q. As before we can and do assume that the collections {D_i}, {E_i}, {F_i} and {G_i} have constant degrees. Let X_i's be the p + 1 variables for $\mathbb {P}$^p and Y_j be the q + 1 variables for $\mathbb {P}$^q. If d₁ is the degree of the D_t's then the bi-homogeneous polynomials of the form D_t ^. M where M is a monomial of degree d₁ in the variables X_i can be written as polynomials in the variables Z_ij = X_iY_j (by choosing some arbitrary pairing of X's with Y's for each term). Let {$\tilde{D_t}$} denote the resulting collection of polynomials in Z_ij as M varies over all possible monomials in the X's and F_t's vary. We have similar collections {$\tilde{E_t}$}, {$\tilde{F_t}$} and {$\tilde{G_t}$}. One then checks quite easily that L(A)×K(A) is the subset of $\mathbb {P}$^{pq + p + q}(A) defined by the conditions:

1. The equations Z_ijZ_kl - Z_ilZ_kj = 0 hold.
2. All the $\tilde{D_t}$'s and the $\tilde{F_t}$'s vanish.
3. The evaluation of the collection {$\tilde{E_i\cdot}$$\tilde{G_j}$} 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 $\Delta_{X}^{}$ $\subset$ X×X, the diagonal subscheme, which is defined by intersecting X×X with the diagonal subscheme of $\mathbb {P}$^q×$\mathbb {P}$^q when X is given a a subscheme of $\mathbb {P}$^q.

A correspondence from X to Y is a closed subscheme of X×Y. For any natural transformation f : X$\to$Y the graph $\Gamma_{f}^{}$ is the subfunctor of X×Y which gives for each finite ring A the graph of f (A) : X(A)$\to$Y(A). We say that f is a morphism if $\Gamma_{f}^{}$ is a closed subscheme of X×Y. In other words, a morphism is a natural transformation which is also a correspondence. Alternatively, if Z $\subset$ X×Y is a correspondence so that the projection Z(A)$\to$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 X$\to$X is a morphism with the diagonal as the associated correspondence. Moreover, each of the projections X×Y$\to$X and X×Y$\to$Y is a morphism. It is also clear that if W $\subset$ X is a subscheme then the intersection of W×Y with $\Gamma_{f}^{}$ gives the graph of the restriction of f : X$\to$Y to W; as a result this restriction is also a morphism. If Z $\subset$ X×Y is the graph of a morphism then the projection Z$\to$X is a morphism; its graph in Z×X $\subset$ 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)$\to$X(A) is a bijection; let g : X$\to$Z be the inverse natural transformation. The graph of g in X×Z $\subset$ X×X×Y is the intersection of $\Delta_{X}^{}$×Y with X×Z. Thus g is also a morphism. In other words, there are morphisms Z$\to$X and X$\to$Z with composition either way being identity. Thus Z$\to$X is an isomorphism.

Now, let f : X$\to$Y be a morphism and g : Y$\to$Z be another morphism. Let W be the intersection of $\Gamma_{f}^{}$×Z with X×$\Gamma_{g}^{}$ in X×Y×Z. Under the above isomorphism X$\to$$\Gamma_{f}^{}$, 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 : X$\to$Y be a morphism and W $\subset$ Y be a subscheme. Then, we have a subscheme of $\Gamma_{f}^{}$ given by its intersection with X×W. Since $\Gamma_{f}^{}$$\to$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 $\Gamma_{f}^{}$ 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 X$\to$W and X$\to$Z we easily check that the natural transformation X$\to$W×Z is a morphism. Given morphisms X$\to$S and Y$\to$S, we obtain the compositions a : X×Y$\to$X$\to$S and b : X×Y$\to$Y$\to$S. Thus we a morphism X×Y$\to$S×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 Z$\to$X×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 $\mathbb {P}$^{p + q}×$\mathbb {P}$^q defined by the conditions X_iY_j = X_jY_i for 0 $\leq$ i, j $\leq$ q. Let U be the open subscheme of $\mathbb {P}$^{p + q} given by U = V(0;X₀, X₁,..., X_q). For (a₀ : ^... : a_{p + q}) in U(A), the tuple (a₀,..., a_q) generates the ring A, thus we see that we see that ((a₀ : ^... : a_{p + q}),(a₀ : ^... : a_q)) gives an element of $\mathbb {P}$^{p + q}(A)×$\mathbb {P}$^q(A) which clearly lies in Z(A). Conversely, if ((a₀ : ^... : a_{p + q}),(b₀ : ^... : b_q)) lies in Z(A) and (a₀,..., a_q) generate the ring A, then the above equations show that there is a unit u in A so that b_i = ua_i (apply the Chinese Remainder theorem for finite rings!). Thus, the projection Z(A)$\to$$\mathbb {P}$^{p + q}(A) is a bijection over U(A) and gives a morphism U$\to$$\mathbb {P}$^q. This morphism is called the projection on $\mathbb {P}$^{p + q} away from the linear subscheme (or subspace!) V(X₀,..., X_q); more generally, we can refer to the above correspondence as the projection correspondence.

A natural generalisation of this is to consider a collection F₀,...,F_q of homogeneous polynomials of the same degree in variables X₀,...,X_p; we can then take the subscheme Z of $\mathbb {P}$^p×$\mathbb {P}$^q defined by the equations

F_i(X₀,..., X_p)Y_j = F_j(X₀,..., X_p)Y_i

for 0 $\leq$ i, j $\leq$ q. We can take U to be the open subscheme defined by U = V(0;F₀,..., F_q). The correspondence Z restricts to a morphism U$\to$$\mathbb {P}$^q. The scheme Z is referred to as the blow-up of $\mathbb {P}$^p along the closed subscheme Y = V(F₁,..., F_q) and is sometimes denoted $\tilde{{\mathbb P}^p_Y}$.

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

(a₀ + b₀$$\epsilon$$,..., a_p + b_p$$\epsilon$$) $$\simeq$$ (ua₀ + (a₀b + ub₀)$$\epsilon$$,..., ua_p + (a_pb + ub₀)

where u is a unit in A and (a₀,..., a_p) generate the ring A (this is enough to ensure generation of A[$\epsilon$] by the above p + 1-tuple). The elements s_ij = a_ia_j and t_ij = b_ia_j - a_jb_i 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

s_ijs_kl - s_iks_jl = 0 (1)

t_ijs_kl + t_jks_il + t_kis_jl = 0 (2)

Moreover, the entries s_ij 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 $\mathbb {P}$^p(A[$\epsilon$]).

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 s_ij must be a unit (since a sum of nilpotent elements is nilpotent). The equation s_ijs_ij = s_iis_jj shows that s_ii must also be a unit. Let us then define a_k = s_ik/s_ii and b_k = t_ki. The equation s_jks_ii = s_ijs_ik implies that s_jk = a_ja_k as required. Moreover, the equation

t_jks_ii = t_jis_ki - t_kis_ji

shows us that t_jk = b_ja_k - b_ka_j as required. $\qedsymbol$

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

V(S_ijS_kl - S_ikS_jl, T_ijS_kl + T_jkS_il + T_kiS_jl;S_ij)

This quasi-projective scheme is the Zariski Tangent Scheme of $\mathbb {P}$^p. More generally, for any quasi-projective scheme X given as a subscheme of $\mathbb {P}$^p one can show that the functor T_X is naturally isomorphic to a subscheme of T_{$\mathbb {P}$^p}. In other words, T_X is also a quasi-projective scheme; this scheme is called the Zariski Tangent scheme of X. Moreover, the natural transformation T_X$\to$X (given by the natural map X(A[$\epsilon$])$\to$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.