The fundamental problem studied in arithmetic algebraic geometry is
the solution of systems of algebraic equations. The notion of an
Algebraic Scheme is the essential geometric notion that incorporates
this question. We then introduce the notion of vector group schemes
and the *K*-group of such objects. With some additional constraints
these are the groups that seem to arise in many cryptographic
contexts.

While we cannot hope to introduce all the algebraic geometry and
commutative algebra that is necessary to study these *K*-groups here,
we give the fundamental definitions and some important examples. We
will also not give proofs as the subject is too vast to be covered
here. When we apply this theory to hyper-elliptic curves in the next
section we will be more precise.

- 8.1 Finite rings
- 8.2 Functors of points
- 8.3 Morphisms of schemes
- 8.4 Relativisation and categorical constructions
- 8.5 The category of vector space schemes
- 8.6 Vector Bundles and regular schemes
- 8.7 Action of correspondences
- 8.8 Cryptosystems