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## 8.7 Action of correspondences

If  0VWU0  is an exact sequence of vector space schemes over a scheme X and if YX is a morphism then the pull-back sequence of vector space schemes

0V×XYW×XYU×XY 0

is not in general exact. We say that YX 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 YX. Thus we have a homomorphism K0(X)K0(Y) for any morphism YX and a homomorphism G0(X)G0(Y) when YX is flat. An important property of tensor products is that the homomorphism K0(X)K0(Y) is a ring homomorphism and when XY is flat the homomorphism G0(X)G0(Y) is a homomorphism of K0(X) modules.

Now, let X be a closed subscheme of Z = n×Y. We want to construct a homomorphism G0(X)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)G0(Z).

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

Hk W = (Hk×Y) (n×W)

Let V be any vector space scheme on n×Y, the syzygy theorem asserts that there is a a sequence of positive integers k0, ..., kn and a sequence of vector space schemes Wn on Y which fit into an exact sequence

0VHk0 W0...Hkn Wn 0

Thus G0(n×Y is generated by G0(Y) as a module over K0(n). Moreover, to define the homomorphism G0(n×YG0(Y) it is enough to define the image of terms of the form Hk W (and check for consistency).

Consider the exact sequence which was introduced above

0(1×n - 1)n1×nH 0

By tensoring this with W and Hk - 1 we get an exact sequence on n×Y

0(Hk - 1|n - 1) WHk - 1 WHk W 0

This allows us to write the class of Hk W in G0(n×Y as

[Hk W] = [Hk - 1 W] - [Hk - 1|n - 1 W]

The second term on the right hand side can be thought of as an element of G0(n - 1×Y). By induction we can thus reduce the problem of defining the image of [Hk W] in G0(Y) to that of defining the image of [(1×m) 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(n×Y)G0(Y) and more generally for any closed subscheme X of n×Y we have G0(X)G0(Y).

Now let X be a projective scheme (i. e. a closed subscheme of n), and let Y be any scheme. Let Z 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)K0(Z); additionally, when ZX is flat we obtain a homomorphism G0(X)G0(Z). By using the sequence of closed inclusions Z X×Y n×Y we also have a homomorphism G0(Z)G0(Y). Thus we see that for any correspondence from a projective scheme X to a scheme Y we obtain a homomorphism K0(X)G0(Y) and when the correspondence is flat over X we get a homomorphism G0(X)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.

Next: 8.8 Cryptosystems Up: 8 Algebraic Schemes for Previous: 8.6 Vector Bundles and
Kapil Hari Paranjape 2002-10-20