
The purpose of this book is to provide an introduction to Hopf algebras. This
book differs from other texts in that Hopf algebras are developed from notions of
topological spaces, sheaves, and representable functors. This approach has certain
pedagogical advantages, the foremost being that algebraic geometry and category
theory provide a smooth transition from modern algebra to Hopf algebras. For
example, the motivation for the definition of an exact sequence of Hopf algebras
is best established by first defining exactness for a sequence of representable group
functors.
Hopf algebras are attributed to the German mathematician Heinz Hopf
(1894–1971). The study of Hopf algebras spans many fields in mathematics,
including topology, algebraic geometry, algebraic number theory, Galois module
theory, cohomology of groups, and formal groups. In this work, we focus on
applications of Hopf algebras to algebraic number theory and Galois module
theory. By the end of the book, readers will be familiar with established results in
the field and should be poised to phrase research questions of their own.
An effort has been made to make this book as selfcontained as possible. That
said, readers should have an understanding of the material on groups, rings, and
fields normally covered in a basic course in modern algebra. Also, most of the
groups given here are Abelian. All of the rings are nonzero commutative rings
with unity, and we only consider commutative algebras. 