Grothendieck¿s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.
The theory of schemes is the foundation for algebraic geometry proposed
and elaborated by Alexander Grothendieck and his coworkers. It has allowed
major progress in classical areas of algebraic geometry such as invariant
theory and the moduli of curves. It integrates algebraic number theory
with algebraic geometry, fulfilling the dreams of earlier generations of
number theorists. This integration has led to proofs of some of the major
conjectures in number theory (Deligne's proof of the Weil Conjectures,
Faltings proof of the Mordell Conjecture).This book is intended to bridge
the chasm between a first course in classical algebraic geometry and a
technical treatise on schemes. It focuses on examples, and strives to show
"what is going on" behind the definitions. There are many exercises to
test and extend the reader's understanding. The prerequisites are modest:
a little commutative algebra and an acquaintance with algebraic varieties,
roughly at the level of a one-semester course. The book aims to show schemes
in relation to other geometric ideas, such as the theory of manifolds.
Some familiarity with these ideas is helpful, though not required.
