8.5 The category of vector space schemes

One can easily ``relativise'' the notion of a homomorphism of modules to define the notion of a homomorphism of vector space schemes.

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$\to$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:

1. For any vector space scheme V$\to$S there is an embedding S $\subset$ $\mathbb {P}$^p and an integer m and an injective homomorphism of vector space schemes V$\to$H^{$\oplus$ m}; here by abuse of notation we use H to denote the restriction of the vector space scheme H$\to$$\mathbb {P}$^p defined earlier.
2. Given any homomorphism V$\to$H_d^{$\oplus$ m}, there is a homomorphism H_d^{$\oplus$ m}$\to$H_{d + e}^{$\oplus$ n} for some e and n so that the image of V is the kernel of the latter homomorphism.

Now a homomorphism H_d$\to$H_{d + e} can be identified with a homogeneous polynomial of degree e. Thus, the coherence theorem can be used to give a concrete definition of vector space schemes in terms of n×m matrices of homogeneous polynomials of degree e. Another application is the construction of cokernels. Given V $\subset$ W a sub-vector space scheme, we can write W as a sub-vector space scheme of H_d^{$\oplus$ n} and find a homomorphism H_d^{$\oplus$ n}$\to$H_{d + e}^{$\oplus$ m} so that V is the kernel. Then W/V is clearly identified with the image of W in H_{d + e}^{$\oplus$ m}.

For example let $\mathbb {P}$^{n - 1} be considered as the closed subscheme of $\mathbb {P}$ⁿ defined by a single linear equation X_n = 0. The vector space scheme $\mathbb {V}$¹×$\mathbb {P}$^{n - 1} can be extended by zero to give a vector space scheme on $\mathbb {P}$ⁿ which we denote by ($\mathbb {V}$¹×$\mathbb {P}$^{n - 1})_{$\mathbb {P}$ⁿ}. We also have the morphism $\mathbb {V}$¹×$\mathbb {P}$ⁿ$\to$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$$\to$$($$\mathbb {V}$$¹×$$\mathbb {P}$$^{n - 1})_{$\mathbb {P}$ⁿ}$$\to$$$$\mathbb {V}$$¹×$$\mathbb {P}$$ⁿ$$\to$$H$$\to$$ 0

More generally, this can be done with any linear polynomial in the X_i's that gives a surjective linear map $\mathbb {Z}$^{n + 1}$\to$$\mathbb {Z}$. The corresponding subscheme is isomorphic is again $\mathbb {P}$^{n - 1}.

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$\to$U$\to$V$\to$W$\to$ 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 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 $\mathbb {V}$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 $\cong$ $\mathbb {P}$^{n - 1} in $\mathbb {P}$ⁿ we have the equation [M] = [$\mathbb {P}$ⁿ] - [H] in G₀($\mathbb {P}$ⁿ). Now the right hand side is independent of the linear equation chosen so that [M] becomes independent of the specific linear subspace M.