Algebraic Geometry and Commutative Algebra

ISBN-13: 9781447148289 / Angielski / Miękka / 2012 / 504 str.

Siegfried Bosch

Opis

Algebraic geometry is a fascinating branch of mathematics that combines methods from both, algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck's schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly seemed to be far away from geometry, like algebraic number theory. The new techniques paved the way to spectacular progress such as the proof of Fermat's Last Theorem by Wiles and Taylor.The scheme-theoretic approach to algebraic geometry is explained for non-experts. More advanced readers can use the book to broaden their view on the subject. A separate part deals with the necessary prerequisites from commutative algebra. On a whole, the book provides a very accessible and self-contained introduction to algebraic geometry, up to a quite advanced level.Every chapter of the book is preceded by a motivating introduction with an informal discussion of the contents. Typical examples and an abundance of exercises illustrate each section. This way the book is an excellent solution for learning by yourself or for complementing knowledge that is already present. It can equally be used as a convenient source for courses and seminars or as supplemental literature.

Recenzje

From the reviews:

"The book under review is a self-contained introduction to both fields and their relationship and is divided into two parts. ... The book is well written and there are exercises at the end of the chapter sections. ... this is certainly a welcome addition to the library of anyone interested in these fields." (Cícero Carvalho, Mathematical Reviews, September, 2013)

"Bosch (Wilhelms Univ., Westphalia, Germany) offers an introduction to scheme-theoretic algebraic geometry preceded by an extended treatment of the prerequisite commutative algebra. ... this is a useful, self-contained work for those interested in studying the modern approach to the subject. Summing Up: Recommended. Graduate students and researchers/faculty." (S. J. Colley, Choice, Vol. 50 (10), June, 2013)

"The book is carefully written, with good examples, detailed proofs, and plenty of exercises at the end of every one of its sections. Each chapter has an introduction where the author discusses informally and motivates the contents of the chapter. I liked the book and I believe that it can be used either as textbook for a two-semester introduction to algebraic geometry or for self-study by a motivated student." (Felipe Zaldivar, MAA Reviews, March, 2013)

"Book under review is to introduce the basic concepts and methods of modern algebraic geometry to novices in the field ... . each chapter comes with its own introduction, where the author motivates the respective contents by illustrating examples and spotlights the main aspects. ... a useful glossary of notations, a comprehensive index, and a list of hints for further reading facilitate working with this textbook considerably. ... the current book is an excellent primer on the elements of modern algebraic geometry and commutative algebra." (Werner Kleinert, Zentralblatt MATH, Vol. 1257, 2013)

Spis treści

Rings and Modules.- The Theory of Noetherian Rings.- Integral Extensions.- Extension of Coefficients and Descent.- Homological Methods: Ext and Tor.- Affine Schemes and Basic Constructions.- Techniques of Global Schemes.- Etale and Smooth Morphisms.- Projective Schemes and Proper Morphisms.

Blurb

Algebraic geometry is a fascinating branch of mathematics that combines methods from both algebra and geometry. It transcends the limited scope of pure algebra by means of geometric construction principles. Moreover, Grothendieck’s schemes invented in the late 1950s allowed the application of algebraic-geometric methods in fields that formerly seemed to be far away from geometry (algebraic number theory, for example). The new techniques paved the way to spectacular progress such as the proof of Fermat’s Last Theorem by Wiles and Taylor.

The scheme-theoretic approach to algebraic geometry is explained for non-experts whilst more advanced readers can use the book to broaden their view on the subject. A separate part studies the necessary prerequisites from commutative algebra. The book provides an accessible and self-contained introduction to algebraic geometry, up to an advanced level.

Every chapter of the book is preceded by a motivating introduction with an informal discussion of the contents. Typical examples and an abundance of exercises illustrate each section. Therefore the book is an excellent solution for learning by yourself or for complementing knowledge that is already present. It can equally be used as a convenient source for courses and seminars or as supplemental literature.