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Next: 8.4 Relativisation and categorical Up: 8 Algebraic Schemes for Previous: 8.2 Functors of points

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 $ \mathbb {P}$p and K = V(D1,..., Dk;E1,..., El) be a quasi-projective scheme in $ \mathbb {P}$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 $ \mathbb {P}$p and Yj be the q + 1 variables for $ \mathbb {P}$q. If d1 is the degree of the Dt's then the bi-homogeneous polynomials of the form Dt . M where M is a monomial of degree d1 in the variables Xi can be written as polynomials in the variables Zij = XiYj (by choosing some arbitrary pairing of X's with Y's for each term). Let {$ \tilde{D_t}$} denote the resulting collection of polynomials in Zij as M varies over all possible monomials in the X's and Ft's vary. We have similar collections {$ \tilde{E_t}$}, {$ \tilde{F_t}$} and {$ \tilde{G_t}$}. One then checks quite easily that L(AK(A) is the subset of $ \mathbb {P}$pq + p + q(A) defined by the conditions:

  1. The equations ZijZkl - ZilZkj = 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 XiYj = XjYi 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,..., 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 $ \mathbb {P}$p + q(A$ \mathbb {P}$q(A) which clearly lies in Z(A). Conversely, if ((a0 : ... : ap + q),(b0 : ... : bq)) lies in Z(A) and (a0,..., aq) generate the ring A, then the above equations show that there is a unit u in A so that bi = uai (apply the Chinese Remainder theorem for finite rings!). Thus, the projection Z(A)$ \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,..., 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 $ \mathbb {P}$p×$ \mathbb {P}$q defined by the equations

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

for 0 $ \leq$ i, j $ \leq$ q. We can take U to be the open subscheme defined by U = V(0;F0,..., Fq). 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,..., Fq) 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 TF 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 TF$ \to$F. Now, if F = $ \mathbb {P}$p is the projective space then T$\scriptstyle \mathbb {P}$p(A) consists of equivalence classes of p + 1-tuples

(a0 + b0$\displaystyle \epsilon$,..., ap + bp$\displaystyle \epsilon$) $\displaystyle \simeq$ (ua0 + (a0b + ub0)$\displaystyle \epsilon$,..., 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[$ \epsilon$] 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 $ \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 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. $ \qedsymbol$

The collection of equivalence classes of pairs (S;T) under multiplication by units in A can be identified with $ \mathbb {P}$p2 + 2p. Thus T$\scriptstyle \mathbb {P}$p 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 $ \mathbb {P}$p. More generally, for any quasi-projective scheme X given as a subscheme of $ \mathbb {P}$p one can show that the functor TX is naturally isomorphic to a subscheme of T$\scriptstyle \mathbb {P}$p. In other words, TX is also a quasi-projective scheme; this scheme is called the Zariski Tangent scheme of X. Moreover, the natural transformation TX$ \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.

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