An Algebraic Introduction to Complex Projective Geometry - Volume 1, Commutative algebra (Broché)

In this introduction to commutative algebra, the author has chosen a route that leads the beginning student through the essential ideas, without getting embroiled in technicalities. The route chosen leads the reader quickly to the fundamental concepts for understanding complex projective geometry, the only prerequisities being a basic knowledge of linear and multilinear algebra and some elementary group theory.
The book is in three parts. In the first, the general theory of Noetherian rings and modules is developed. A certain amount of homological algebra is included, and rings and modules of fractions are emphasized, as preparation for working with sheaves. In the second part, the central objects are polynomial rings in several variables with coefficients in the field of complex numbers. After Noether's normalization lemma and Hilbert's Nullstellensatz, affine complex schemes and thier morphisms are introduced ; Zariski's main theorem and Chevalley's semi-continuity theorem are then proved.
Finally, a detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.