Recall, that *L* was defined as the subscheme of
^{p + 1}×^{p} consisting of pairs of tuples
(*b*_{0},..., *b*_{p};*a*_{0} : ^{ ... } : *a*_{p}) such that
*a*_{i}*b*_{j} = *a*_{j}*b*_{i} for all *i*
and *j* between 0 and *p*. An open cover of
^{p} is given by the
open subschemes
*U*_{i} = *V*(0;*X*_{i}). We see easily that
*L*×_{p}*U*_{i} is given by the equations
*b*_{j} = (*a*_{j}/*a*_{i})*b*_{i} since *a*_{i} is a
unit. Thus the map from
_{a}×*U*_{i} to
*L*×_{p}*U*_{i}
given by

(*c*;*a*_{0} : ^{ ... } : *a*_{p}) (*a*_{0}/*a*_{i})*c*,...,(*a*_{p}/*a*_{0})*c*;*a*_{0} : ^{ ... } : *a*_{p}

gives an isomorphism. Thus
(*c*;*a*_{0} : ^{ ... } : *a*_{p}) (*a* : 0 : ^{ ... } : *a*_{p} : *c*^{ . }*a*_{i})

Note that this is an isomorphism outside the
The automorphisms of the vector space
^{n} are given as the closed
subscheme
GL_{n} of
^{n2 + 1} consisting of
((*X*_{ij})_{i, j = 1}^{n}, *T*) such that
det((*X*_{ij}))*T* = 1. For any scheme
*X*, any automorphism of the vector space scheme
^{n}×*X*
corresponds naturally to a morphism
*g* : *X*GL_{n}. Moreover, it is
clear that
GL_{n} is a group scheme.

Now let *E* be a vector bundle over a scheme *X*, {*U*_{i}} be an open
cover of *X* and be the isomorphism of vector space schemes
: *E*×_{X}*U*_{i}^{n}×*U*_{i}. For any *i* and *j* it is
clear that we get a morphism
: *U*_{i} *U*_{j}GL_{n} by
comparing the two isomorphisms of
*E*×_{X}(*U*_{i} *U*_{j}) with
^{n}×(*U*_{i} *U*_{j}). These morphisms satisfy
^{ . } = on
*U*_{i} *U*_{j} *U*_{k}.
Conversely, it is clear that we can use such a collection of morphisms
: *U*_{i} *U*_{j}*GL*_{n} to construct a vector bundle on *X* by
*patching* together the vector bundles
^{n}×*U*_{i}. More
generally, we can easily show that for any vector space scheme *V* on
*X*, the group scheme
GL_{n} operates on
*V*^{ n}. Thus we can
use the to patch together
*V*^{ n}×_{X}*U*_{i} to
obtain a vector space scheme. This vector space scheme is denoted
*E* *V* and is called the tensor product of *E* with *V*. It is
clear that
^{1} *V* = *V*. One can show that
*H*_{n} = *H*^{ n}
and
*H* *L* = ^{1}×^{p}.

As before we define the *K*-group of vector bundles of a scheme *S* as
the quotient *K*_{0}(*S*) of the free abelian group on isomorphism classes
of vector bundles by the subgroup generated by relations of the form
[*V*] + [*U*] - [*W*] where
0*V**W**U* 0 is an exact sequence of
vector bundles. Note that any vector bundle is a vector space scheme
and an exact sequence of vector bundles is also an exact sequence of
vector space schemes. Thus we have a natural homomorphism
*K*_{0}(*S*)*G*_{0}(*S*). When *S* is a *regular* scheme this is an isomorphism;
usually one gives a definition of regular schemes in terms of ring
theory and proves the equivalence, but we could equally well use this
as a definition. As a particular case we have the ``Jacobian
criterion'' which says that a scheme is regular if the Zariski tangent
vector space scheme is a vector bundle; note however that this is * not* in general necessary. For example the subscheme of
^{2}
defined by *XY* = *p* for some prime *p* is regular but its Zariski
tangent space is not a vector bundle.

In fact the tensor product construction makes *K*_{0}(*S*) into a ring and
*G*_{0}(*S*) a module over this ring.