The inverse image of the zero section under such a homomorphism a
sub-vector space scheme of the domain of the homomorphism. This,
defines the *kernel* of a homomorphism of vector space scheme.
The image of a homomorphism *E**F* of vector space schemes over *S*
is also a sub-vector space scheme. In particular, we see that the
notion of *exact sequences* of vector space schemes can be defined
by saying the the image of one morphism is the kernel of the next.

In fact these objects form an *abelian* category. In order
to do this we need The Coherence theorem for vector space
schemes:

- For any vector space scheme
*V**S*there is an embedding*S*^{p}and an integer*m*and an injective homomorphism of vector space schemes*V**H*^{ m}; here by abuse of notation we use*H*to denote the restriction of the vector space scheme*H*^{p}defined earlier. - Given
*any*homomorphism*V**H*_{d}^{ m}, there is a homomorphism*H*_{d}^{ m}*H*_{d + e}^{ n}for some*e*and*n*so that the image of*V*is the kernel of the latter homomorphism.

For example let
^{n - 1} be considered as the closed subscheme of
^{n} defined by a single linear equation *X*_{n} = 0. The vector space
scheme
^{1}×^{n - 1} can be extended by zero to give a
vector space scheme on
^{n} which we denote by
(^{1}×^{n - 1})_{n}. We also have the morphism
^{1}×^{n}*H* given by the 1×1 matrix with entry
*X*_{n}. One easily sees that this gives an exact sequence of vector
space schemes

0(^{1}×^{n - 1})_{n}^{1}×^{n}*H* 0

More generally, this can be done with any linear polynomial in the
An irreducible (or atomic) object in an abelian category is defined as one which has no non-trivial sub-objects Ideally we would like to write every vector space scheme as a sum of irreducibles. However, it turns out that this is not possible. A compromise solution is to ``semi-simplify'' the operation as per a construction due to Grothendieck.

The Grothendieck *K*-group of a scheme *S* is the quotient of the free
group generated by isomorphism classes of vector space schemes over
*S* by the relations of the form
[*V*] = [*U*] + [*W*] when
0*U**V**W* 0 is an exact sequence. Quillen has generalised this
construction to define the groups *K*_{i} for any exact
category. Grothendieck's *K* group then becomes *K*_{0}. The *K*_{0} group
of vector space schemes over *S* is denoted *G*_{0}(*S*).

For any closed subscheme *T* of *S*, we have a vector space scheme on
*S* obtained by extending by zero the vector space scheme
^{1}×*T*; we use the symbol [*T*] to denote the corresponding
element of *G*_{0}(*S*). From the above exact sequence we see that for any
linear subscheme
*M* ^{n - 1} in
^{n} we have the equation
[*M*] = [^{n}] - [*H*] in
*G*_{0}(^{n}). Now the right hand side is * independent* of the linear equation chosen so that [*M*] becomes
independent of the specific linear subspace *M*.