By William Fulton
In a number of contexts of topology, algebraic geometry, and algebra (e.g. crew representations), one meets the subsequent scenario. One has contravariant functors ok and A from a definite classification to the class of earrings, and a common transformation p: K--+A of contravariant functors. The Chern personality being the relevant examination ple, we name the homomorphisms Px: K(X)--+ A(X) characters. Given f: X--+ Y, we denote the pull-back homomorphisms by means of and fA: A(Y)--+ A(X). As functors to abelian teams, ok and A can also be covariant, with push-forward homomorphisms and fA: A( X)--+ A(Y). frequently those maps don't travel with the nature, yet there's a component r f E A(X) such that the subsequent diagram is commutative: K(X) A(X) fK j J A okay( Y) ------p;-+ A( Y) The map within the best line is p x elevated via r f. whilst such commutativity holds, we are saying that Riemann-Roch holds for f. this kind of formula was once first given via Grothendieck, extending the paintings of Hirzebruch to this kind of relative, functorial atmosphere. due to the fact then viii advent a number of different theorems of this Riemann-Roch style have seemed. Un derlying every one of these there's a simple constitution having to do purely with hassle-free algebra, self sustaining of the geometry. One objective of this monograph is to explain this algebra independently of any context, in order that it might probably serve axiomatically because the desire arises.
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