In various contexts of topology algebraic geometry and algebra (e.g. group representations)
one meets the following situation. One has two contravariant functors K and A from a certain
category to the category of rings and a natural transformation p:K--+A of contravariant
functors. The Chern character being the central exam ple we call the homomorphisms Px: K(X)--+
A(X) characters. Given f: X--+ Y we denote the pull-back homomorphisms by and fA: A(Y)--+
A(X). As functors to abelian groups K and A may also be covariant with push-forward
homomorphisms and fA: A( X)--+ A(Y). Usually these maps do not commute with the character but
there is an element r f E A(X) such that the following diagram is commutative: K(X)~A(X) fK j
J~A K( Y) ------p -+ A( Y) The map in the top line is p x multiplied by r f. When such
commutativity holds we say that Riemann-Roch holds for f. This type of formulation was first
given by Grothendieck extending the work of Hirzebruch to such a relative functorial setting.
Since then viii INTRODUCTION several other theorems of this Riemann-Roch type have appeared. Un
derlying most of these there is a basic structure having to do only with elementary algebra
independent of the geometry. One purpose of this monograph is to describe this algebra
independently of any context so that it can serve axiomatically as the need arises.
