8.7 Action of correspondences

If  0$\to$V$\to$W$\to$U$\to$0  is an exact sequence of vector space schemes over a scheme X and if Y$\to$X is a morphism then the pull-back sequence of vector space schemes

0$$\to$$V×_XY$$\to$$W×_XY$$\to$$U×_XY$$\to$$ 0

is not in general exact. We say that Y$\to$X is flat if this is so. However, if V is a vector bundle then the pull back sequence of vector space schemes is exact regardless of the nature of the morphism Y$\to$X. Thus we have a homomorphism K₀(X)$\to$K₀(Y) for any morphism Y$\to$X and a homomorphism G₀(X)$\to$G₀(Y) when Y$\to$X is flat. An important property of tensor products is that the homomorphism K₀(X)$\to$K₀(Y) is a ring homomorphism and when X$\to$Y is flat the homomorphism G₀(X)$\to$G₀(Y) is a homomorphism of K₀(X) modules.

Now, let X be a closed subscheme of Z = $\mathbb {P}$ⁿ×Y. We want to construct a homomorphism G₀(X)$\to$G₀(Y). This can be done in two steps (provided we prove that the construction is independent of the factorisation). The first step is to consider a vector space scheme on X as a vector space scheme on Z (of which it is a closed subscheme). We have already seen how to do this by ``extending by zero''; it is moreover clear that this preserves exact sequences. Thus we obtain a natural homomorphism G₀(X)$\to$G₀(Z).

Hilbert's syzygy theorem can be used to describe G₀($\mathbb {P}$ⁿ×Y) in terms of G₀(Y) as follows. For any integer n we have a line bundle Hⁿ on $\mathbb {P}$ⁿ as described above; let W be any vector space scheme on Y. We have a vector space scheme H_k $\boxtimes$ W on $\mathbb {P}$ⁿ×Y obtained as

H_k $$\boxtimes$$ W = (H_k×Y) $$\otimes$$ ($$\mathbb {P}$$ⁿ×W)

Let V be any vector space scheme on $\mathbb {P}$ⁿ×Y, the syzygy theorem asserts that there is a a sequence of positive integers k₀, ..., k_n and a sequence of vector space schemes W_n on Y which fit into an exact sequence

0$$\to$$V$$\to$$H_k0 $$\boxtimes$$ W₀$$\to$$...$$\to$$H_kn $$\boxtimes$$ W_n$$\to$$ 0

Thus G₀($\mathbb {P}$ⁿ×Y is generated by G₀(Y) as a module over K₀($\mathbb {P}$ⁿ). Moreover, to define the homomorphism G₀($\mathbb {P}$ⁿ×Y$\to$G₀(Y) it is enough to define the image of terms of the form H_k $\boxtimes$ W (and check for consistency).

Consider the exact sequence which was introduced above

0$$\to$$($$\mathbb {V}$$¹×$$\mathbb {P}$$^{n - 1})_{$\mathbb {P}$ⁿ}$$\to$$$$\mathbb {V}$$¹×$$\mathbb {P}$$ⁿ$$\to$$H$$\to$$ 0

By tensoring this with W and H_{k - 1} we get an exact sequence on $\mathbb {P}$ⁿ×Y

0$$\to$$(H_{k - 1}|_{$\mathbb {P}$^{n - 1}}) $$\boxtimes$$ W$$\to$$H_{k - 1} $$\boxtimes$$ W$$\to$$H_k $$\boxtimes$$ W$$\to$$ 0

This allows us to write the class of H_k $\boxtimes$ W in G₀($\mathbb {P}$ⁿ×Y as

[H_k $$\boxtimes$$ W] = [H_{k - 1} $$\boxtimes$$ W] - [H_{k - 1}|_{$\mathbb {P}$^{n - 1}} $$\boxtimes$$ W]

The second term on the right hand side can be thought of as an element of G₀($\mathbb {P}$^{n - 1}×Y). By induction we can thus reduce the problem of defining the image of [H_k $\boxtimes$ W] in G₀(Y) to that of defining the image of [($\mathbb {V}$¹×$\mathbb {P}$^m) $\boxtimes$ W]. The image of the latter is just [W]. The consistency of this definition can be checked by the theory of ``cohomology'' and higher direct images. Thus we have a homomorphism G₀($\mathbb {P}$ⁿ×Y)$\to$G₀(Y) and more generally for any closed subscheme X of $\mathbb {P}$ⁿ×Y we have G₀(X)$\to$G₀(Y).

Now let X be a projective scheme (i. e. a closed subscheme of $\mathbb {P}$ⁿ), and let Y be any scheme. Let Z $\subset$ X×Y be a correspondence from X to Y (i. e. Z is a closed subscheme of X×Y). We obtain a homomorphism K₀(X)$\to$K₀(Z); additionally, when Z$\to$X is flat we obtain a homomorphism G₀(X)$\to$G₀(Z). By using the sequence of closed inclusions Z $\subset$ X×Y $\subset$ $\mathbb {P}$ⁿ×Y we also have a homomorphism G₀(Z)$\to$G₀(Y). Thus we see that for any correspondence from a projective scheme X to a scheme Y we obtain a homomorphism K₀(X)$\to$G₀(Y) and when the correspondence is flat over X we get a homomorphism G₀(X)$\to$G₀(Y). In particular, correspondences from a regular scheme X to itself act as automorphisms of G₀(X) = K₀(X). This is a very useful tool in analysing the structure of K₀(X) for such schemes.