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
Previous Page Next Page