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Additional info for Algebraic K-Theory III
Our treatment of this subject is very arithmetic. The geometric underpinnings will not be much in evidence. The whole subject can be dealt with under the aspect of curves over finite fields. We have chosen the arithmetic approach because our guiding theme 46 Michael Rosen in this book will be the exploration of the rich analogies that exist between algebraic number fields and global function fields. To begin with it is not necessary to restrict the constant field F to be finite. In fact, in this first part of the chapter we make no restrictions on F whatsoever.
For the rest of this section we assume that F = :IF is a finite field with q elements. ion field in one variable over a finite constant field is called a global function field. Our next goal is to define the zeta function of a global function field KIF and to investigate its properties. R. Schmidt, Schmidt , that a function field over a finite field always hM divisors of degree 1. \Ve will assume this, although it is possible to give a proof without introducing any new concepts. Using Schmidt's theorem) we have an exact sequence j (0) -+ ClK -+ GlK -+ Z -+ (0).
One of these estimates will involve invoking a deep result of A. Weil. The others are more elementary. We begin by writing down an identity which will be used repeatedly. Namely, d 00 (3) u du (log(1 ~ au)-l) = aku k . L k=l Here a is a complex number. The sum converges for all u such that lui < The proof of this identity is a simple exercise using the geometric series. For each character X modulo m define the numbers CN(X) by lal- 1 . We claim that The easy case is when X = Xo. Recall that L(s,Xo) = II (1 ~ IPI-S) (A(S) Plm Thus, C(u,Xo) = II (1- u Plm degP ) _1_ .
Algebraic K-Theory III by Bass