By William Fulton
In a number of contexts of topology, algebraic geometry, and algebra (e.g. workforce representations), one meets the next scenario. One has contravariant functors ok and A from a undeniable type to the class of earrings, and a usual transformation p: K--+A of contravariant functors. The Chern personality being the important examination ple, we name the homomorphisms Px: K(X)--+ A(X) characters. Given f: X--+ Y, we denote the pull-back homomorphisms by way of and fA: A(Y)--+ A(X). As functors to abelian teams, okay and A can also be covariant, with push-forward homomorphisms and fA: A( X)--+ A(Y). frequently those maps don't go back and forth with the nature, yet there's a component r f E A(X) such that the next diagram is commutative: K(X) A(X) fK j J A ok( Y) ------p;-+ A( Y) The map within the most sensible line is p x improved by means of r f. while such commutativity holds, we are saying that Riemann-Roch holds for f. this kind of formula used to be first given by way of Grothendieck, extending the paintings of Hirzebruch to one of these relative, functorial environment. due to the fact then viii creation a number of different theorems of this Riemann-Roch sort have seemed. Un derlying each one of these there's a uncomplicated constitution having to do purely with ordinary algebra, self sufficient of the geometry. One function of this monograph is to explain this algebra independently of any context, in order that it will possibly serve axiomatically because the desire arises.
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