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

0*V*×_{X}*Y**W*×_{X}*Y**U*×_{X}*Y* 0

is
Now, let *X* be a closed subscheme of
*Z* = ^{n}×*Y*. We want to
construct a homomorphism
*G*_{0}(*X*)*G*_{0}(*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
*G*_{0}(*X*)*G*_{0}(*Z*).

Hilbert's syzygy theorem can be used to describe
*G*_{0}(^{n}×*Y*)
in terms of *G*_{0}(*Y*) as follows. For any integer *n* we have a line
bundle *H*^{n} on
^{n} as described above; let *W* be any vector
space scheme on *Y*. We have a vector space scheme
*H*_{k} *W*
on
^{n}×*Y* obtained as

0*V**H*_{k0} *W*_{0}...*H*_{kn} *W*_{n} 0

Thus
Consider the exact sequence which was introduced above

0(^{1}×^{n - 1})_{n}^{1}×^{n}*H* 0

By tensoring this with
0(*H*_{k - 1}|_{n - 1}) *W**H*_{k - 1} *W**H*_{k} *W* 0

This allows us to write the class of
[*H*_{k} *W*] = [*H*_{k - 1} *W*] - [*H*_{k - 1}|_{n - 1} *W*]

The second term on the right hand side can be thought of as an element
of
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
*K*_{0}(*X*)*K*_{0}(*Z*);
additionally, when *Z**X* is flat we obtain a homomorphism
*G*_{0}(*X*)*G*_{0}(*Z*). By using the sequence of closed inclusions
*Z* *X*×*Y* ^{n}×*Y* we also have a homomorphism
*G*_{0}(*Z*)*G*_{0}(*Y*). Thus we see that for any correspondence from a
projective scheme *X* to a scheme *Y* we obtain a homomorphism
*K*_{0}(*X*)*G*_{0}(*Y*) and when the correspondence is flat over *X* we get
a homomorphism
*G*_{0}(*X*)*G*_{0}(*Y*). In particular, correspondences from
a regular scheme *X* to itself act as automorphisms of
*G*_{0}(*X*) = *K*_{0}(*X*). This is a very useful tool in analysing the structure
of *K*_{0}(*X*) for such schemes.