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 pullback homomorphisms by and fA: A(Y) A(X). As functors to abelian groups, K and A may also be covariant, with pushforward 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 RiemannRoch 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 RiemannRoch 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.
About William Fulton
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Author  :  William Fulton 
Publisher  :  SpringerVerlag New York Inc. 
Data Published  :  11 May 2011 
ISBN  :  1441930736 
EAN  :  9781441930736 
Format Book  :  PDF, Epub, DOCx, TXT 
Number of Pages  :  216 pages 
Age +  :  15 years 
Language  :  English 
Rating  : 
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