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 EF 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 example let n - 1 be considered as the closed subscheme of n defined by a single linear equation Xn = 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×nH given by the 1×1 matrix with entry Xn. One easily sees that this gives an exact sequence of vector space schemes
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 0UVW 0 is an exact sequence. Quillen has generalised this construction to define the groups Ki for any exact category. Grothendieck's K group then becomes K0. The K0 group of vector space schemes over S is denoted G0(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 G0(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 G0(n). Now the right hand side is independent of the linear equation chosen so that [M] becomes independent of the specific linear subspace M.