Description of the chapters

The chapters of this book are designed to lead the reader in a logical progression

from learning the local properties of rings and curves (such as local principal

ideal domains and nonsingular points) to understanding the global properties of

rings and curves (through the study of class groups and of the Riemann-Roch

Theorem).

The first five chapters of the book are "amne" in nature. Basic notions in al-

gebra are introduced and are illustrated with examples coming alternately from

number theory and from algebraic geometry. The unifying theme of these first

five chapters is the concept of integral closure of a ring. Algebraic number the-

ory studies the integral closure OL of the ring of integers Z in a finite extension

L/Q, where L is obtained by adjoining to the field Q a root a of an irreducible

polynomial f(y) G Q[y\. In algebraic geometry, one associates to a nonsingular

plane curve given by an equation f(x,y) = 0 {/ € &[#,#], fc any field) its ring

of functions B := k[x,y]/(f) and its function field L (the field of fractions of

the domain B). The field L is obtained by adjoining to the field k(x) a root of

the polynomial / G k(x)[y}. If, in addition, f(x,y) is monic in y, then the ring

B is nothing but the integral closure of the ring k[x] in L. Both quadruples,

( Z , Q , O L , L ) and (k[x],k(x),B,L), are, from a commutative algebra perspec-

tive, instances of the same phenomenon. The reader gains insight into both

the number theoretic context and the algebraic-geometric context by being able,

very early on, to pass easily from one context to the other.

Since this book is not, after all, merely a course in commutative algebra,

but is intended to introduce the reader to arithmetic geometry, we also, in the

first five chapters, take a geometric perspective and discuss the fact that a ring

extension B/k[x] corresponds to a map of plane curves, namely, the projection to

the x-axis. This geometric perspective gives the reader added insight by allowing

the reader to draw "pictures" of ring extensions /maps of curves and of various

concepts attached to extensions, such as ramification.

The local properties of rings and curves having been studied in detail in the

first four chapters, Chapter V introduces one of the basic global invariants at-

tached to a Dedekind domain, its ideal class group. In addition, Chapter V is

l

http://dx.doi.org/10.1090/gsm/009/01